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. Put together, this gives us a terminal condition of BG t+1 Gt + Gt+1 1 + rt = Tt + Tt+1 1 + rt (13.3) The government’s intertemporal budget constraint has exactly the same ﬂavor as the household’s intertemporal budget constraint. In words, it requires that the present discounted value of the stream of spending equals the present discount value of the stream of tax revenue. In other words, while the government’s budget need not balance (i.e. Gt = Tt or Gt+1 = Tt+1) in any particular period, it must balance in a present value sense. 13.2 Fiscal Policy in an Endowment Economy Let us ﬁrst incorporate ﬁscal policy into the endowment economy framework explored in Chapter 11. We will later move on to a production economy. There exists a representative household with a standard lifetime utility function. The household faces a sequence of budget constraints given by: Ct + St ≤ Yt − Tt Ct+1 + St+1 ≤ Yt+1 − Tt+1 + (1 + rt)St (13.4) (13.5) These are the same ﬂow budgets constraints we have already encountered, but include a tax payment to the government each period of Tt and Tt+1. These taxes are lump sum in the sense that they are additive in the budget constraint – the amount of tax that a household 296 pays is independent of its income or any other choices which it makes. We impose the terminal condition that St+1 = 0, and assume that the ﬂow budget constraints hold with equality in both periods. This gives rise to an intertemporal budget constraint for the household: Ct + Ct+1 1 + rt = Yt − Tt + Yt+1 − Tt+1 1 + rt (13.6) In words, (13.6) requires that the present discounted value of the stream of consumption equal the present discounted value of the stream of net income, where Yt − Tt denotes net income in period t (and similarly for period t + 1). The household’s lifetime utility optimization problem gives rise to the standard Euler equation: u′(Ct) = β(1 + rt)u′(Ct+1) (13.7) We can again use an
indiﬀerence curve - budget line setup to graphically think about what the consumption function will be. Before doing so, note that the household’s intertemporal budget constraint can be written: ] Ct + Ct+1 1 + rt − [Tt + Tt+1 1 + rt = Yt + Yt+1 1 + rt In other words, because the tax payments are additive (i.e. lump sum), we can split the income side of the intertemporal budget constraint into the present discounted value of the stream of income less the present discounted value of the stream of tax payments. But, since the household knows that the government’s intertemporal budget constraint must hold, the household knows that the present discounted value of tax payments must equal the present discounted value of government spending: (13.8) Ct + Ct+1 1 + rt (13.9) can be re-arranged to yield: = Yt + Yt+1 1 + rt − [Gt + Gt+1 1 + rt ] Ct + Ct+1 1 + rt = Yt − Gt + Yt+1 − Gt+1 1 + rt (13.9) (13.10) In other words, Tt and Tt+1 do not appear in the intertemporal budget constraint. From the household’s perspective, it is as if the government balances its budget each period, with Gt = Tt and Gt+1 = Tt+1. Figure 13.1 plots the budget line facing the household. It is simply a graphical depiction of (13.10). Points inside the budget line are feasible, points outside the budget line are infeasible. The slope of the budget line is −(1 + rt). 297 Figure 13.1: Budget Line The household’s objective is to choose a consumption bundle, (Ct, Ct+1), so as to locate on the highest possible indiﬀerence curve which does not violate the budget constraint. This involves locating at a point where the indiﬀerence curve is just tangent to the budget line (i.e. where the Euler equation holds). This is qualitative identical to what was seen in Chapter 9. From the household’s perspective, an increase in Gt is equivalent to a decrease in Yt (there are fewer resources available for the household to consume
) and similarly for Gt+1. Tt and Tt+1 are irrelevant from the household’s perspective. We can therefore intuit that the consumption function takes the form: Ct = C(Yt − Gt +, Yt+1 − Gt+1 + ), rt − (13.11) Now that we understand household optimality, let us turn to market-clearing. Market= St. In other words, household saving must equal government clearing requires that BG t borrowing (equivalently, household borrowing must equal government saving). From (13.1), we have that BG t = Gt − Tt. Inserting this for St into (13.4) yields: Yt = Ct + Gt (13.12) In other words, the aggregate market-clearing condition requires that total output equal the sum of private, Ct, and public, Gt, consumption. This is equivalent to imposing that aggregate saving is zero, where aggregate saving is St − BG t (i.e. household saving plus public saving, where −BG is public saving). t 298 𝐶𝐶𝑡𝑡+1 𝐶𝐶𝑡𝑡 (1+𝑟𝑟𝑡𝑡)(𝑌𝑌𝑡𝑡−𝐺𝐺𝑡𝑡)+𝑌𝑌𝑡𝑡+1−𝐺𝐺𝑡𝑡+1 𝑌𝑌𝑡𝑡−𝐺𝐺𝑡𝑡+𝑌𝑌𝑡𝑡+1−𝐺𝐺𝑡𝑡+11+𝑟𝑟𝑡𝑡 Feasible Infeasible Slope: −(1+𝑟𝑟𝑡𝑡) Equations (13.12) and (13.11) characterize the equilibrium of the economy. This is two equations in two endogenous variables, Ct and rt. Yt, Yt+1, Gt, and Gt+1 are all exogenous and hence taken as given. Note that Tt, Tt+1, and BG t (government debt issuance) do not appear in the equilibrium conditions.
This means that these variables are irrelevant for the determination of equilibrium prices and quantities. This does not mean that ﬁscal policy is irrelevant – Gt and Gt+1 are going to be relevant for equilibrium quantities and prices. But the level of taxes and debt are irrelevant. This discussion forms the basis of what is known as Ricardian Equivalence. Attributed to the famous early economist David Ricardo, this hypothesis was revived in its modern form by Robert Barro in a series of papers Barro (1974) and Barro (1979). The essential gist of Ricardian equivalence is that the method of government ﬁnance is irrelevant for understanding the eﬀects of changes in government expenditure. Put diﬀerently, a change in Gt will have the same eﬀect on the equilibrium of the economy whether it is ﬁnanced by an increase in taxes, by increasing debt, or some combination of the two. Corollaries are that the level of government debt is irrelevant for understanding the equilibrium behavior of the economy and that changes in taxes, not met by changes in either current or future government spending, will have no eﬀect on the equilibrium of the economy. The intuition for Ricardian Equivalence can be understood as follows. Suppose that the government increases Gt by issuing debt, with no change in taxes. This issuance of debt necessitates an increase in future taxes in an amount equal in present value to the current increase in spending. Since all the household cares about is the present discounted value of tax obligations, the household is indiﬀerent to whether the tax is paid in the present versus the future, so long as the present value of these payments are the same. In other words, from the household’s perspective, it is as if the government increases the tax in the present by an amount equal to the change in spending. Furthermore, suppose that the government cuts taxes in the present, Tt, with no announced change in current or future spending. For the government’s intertemporal budget constraint to hold, this will necessitate an increase in the future tax by an amount equal in present value to the decrease in current taxes. Since all the household cares about is the present discounted value of tax obligations, the cut in Tt is irrelevant for the household’s behavior. Finally, government debt is irrelevant. Suppose > 0. This is held by the household with St > 0.
that the government issues positive debt, BG t This stock of savings held by the household (i.e. its holdings of government debt) is not wealth for the household. Why not? The household will have to pay higher future taxes to pay oﬀ the debt – in essence, the household will pay itself principal plus interest on the outstanding debt in the future, through the government, in an amount equal in present value to the household’s current stock of savings. 299 Ricardian Equivalence is a stark proposition. It means that the level of government debt is irrelevant, that tax-ﬁnanced government spending changes have the same equilibrium eﬀects as deﬁcit-ﬁnanced changes in spending, and that the level of outstanding government debt is irrelevant. Does Ricardian Equivalence hold in the real world? Likely not. Ricardian Equivalence only holds in special cases. First, taxes must be lump sum (i.e. additive). If the amount of tax that households pay depends on actions they take, then Ricardian Equivalence will not hold. Second, Ricardian Equivalence requires that there be no liquidity constraints – i.e. households must be able to freely borrow and save at the same rate as the government. Third, Ricardian Equivalence requires that households are forward-looking and believe that the government’s intertemporal budget constraint must hold. Fourth, Ricardian Equivalence requires that the government and household have the same lifespan. If the government “outlives” households (as would be the case in what are called overlapping generations models, where each period one generation of households dies and another is born), then the timing of tax collection will matter to consumption and saving decisions of households. None of these conditions are likely to hold in the real world. Nevertheless, the insights from the Ricardian Equivalence are useful to keep in mind when thinking about real world ﬁscal policy. 13.2.1 Graphical Eﬀects of Changes in Gt and Gt+1 We can use the IS and Y s curves from Chapter 11 to analyze the equilibrium consequences of changes in current or future government spending. The IS curve shows the set of (rt, Yt) pairs consistent with total income equaling total expenditure, where total expenditure is Ct + Gt, when the household is behaving optimally. The presence of
government spending does not impact the derivation or qualitative shape of the IS curve. Since we are working in an endowment economy in which current production is exogenous, the Y s curve is simply a vertical line at some exogenous value of output, Y0,t. Total autonomous expenditure (i.e. desired expenditure independent of current income) is given by: E0 = C d(−Gt, Yt+1 − Gt+1, rt) + Gt (13.13) Changes in Gt or Gt+1 will inﬂuence autonomous expenditure (i.e. the intercept of the desired expenditure line), and will therefore impact the position of the IS curve. Consider ﬁrst an exogenous increase in Gt. This has two eﬀects on autonomous expenditure, as can be seen in (13.13). There is a direct eﬀect wherein an increase in Gt raises autonomous expenditure one-for-one. There is an indirect eﬀect wherein the increase in Gt depresses consumption. Which eﬀect dominates? It turns out that the direct eﬀect dominates, because the MPC is less than one. The partial derivative of autonomous expenditure with respect to 300 government spending is: ∂E0 ∂Gt = − ∂C d ∂Gt + 1 (13.14) Since we denote ∂Cd ∂Gt by MPC, which is less than one, this derivative works out to 1 − M P C > 0. Hence, autonomous expenditure increases when government spending increases, but by less than the increase in government spending. This shifts the vertical axis intercept of the expenditure line up, which in turn causes the IS curve to shift to the right – i.e. for a given real interest rate, r0,t, the level of income at which income equals expenditure is now larger. This is shown in Figure 13.2 below with the blue lines: Figure 13.2: Increase in Gt The rightward shift of the IS curve is shown in blue. There is no shift of the Y s curve since current output is exogenous. The rightward shift of the IS curve along a ﬁxed Y s curve means that the real interest rate must rise from r0,t to r1,t. The higher real interest rate 301 𝑌𝑌𝑡�
�𝑑𝑑 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝑌𝑌𝑡𝑡 𝑌𝑌0,𝑡𝑡 𝑌𝑌0,𝑡𝑡𝑑𝑑 𝑟𝑟0,𝑡𝑡 𝐼𝐼𝐼𝐼 𝑌𝑌𝑠𝑠 𝑟𝑟𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝐶𝐶𝑑𝑑�𝑌𝑌𝑡𝑡−𝐺𝐺1,𝑡𝑡,𝑌𝑌0,𝑡𝑡+1−𝐺𝐺0,𝑡𝑡+1,𝑟𝑟0,𝑡𝑡�+𝐺𝐺1,𝑡𝑡 𝑟𝑟1,𝑡𝑡 𝐼𝐼𝐼𝐼′ 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝐶𝐶𝑑𝑑�𝑌𝑌𝑡𝑡−𝐺𝐺0,𝑡𝑡,𝑌𝑌𝑡𝑡+1−𝐺𝐺0,𝑡𝑡+1,𝑟𝑟0,𝑡𝑡�+𝐺𝐺0,𝑡𝑡 =𝐶𝐶𝑑𝑑�𝑌𝑌𝑡𝑡−𝐺𝐺1,𝑡𝑡,𝑌𝑌𝑡𝑡+1−𝐺𝐺0,𝑡𝑡+1,𝑟�
�1,𝑡𝑡�+𝐺𝐺1,𝑡𝑡 reduces autonomous expenditure through an eﬀect on consumption, in such a way that the expenditure line shifts back down to where it started so as to be consistent with unchanged Yt. This eﬀect is shown in the ﬁgure with the green arrow. Since output is unchanged in equilibrium, it must be the case that consumption falls by the amount of the increase in government spending. In other words, consumption is completely “crowded out” by the increase in Gt. The complete crowding out of consumption is not a consequence of Ricardian Equivalence, but rather emerges because of the fact that total output is ﬁxed in this example. The intuition for this is the following. When Gt increases, the household feels poorer and acts as though its current tax obligations are higher. It would like to reduce its consumption some, but by less than the increase in Gt holding the interest rate ﬁxed (i.e. the MPC is less than 1). But in equilibrium, market-clearing dictates that consumption falls by the full amount of the increase in Gt (since Yt is ﬁxed). Hence, rt must rise to further discourage consumption, so that consumption falling by the full amount of the increase in Gt is consistent with the household’s consumption function. Next, consider an anticipated increase in future government spending, from G0,t+1 to G1,t+1. This only aﬀects current autonomous expenditure through an eﬀect on consumption. This eﬀect is negative. Hence, autonomous expenditure declines, so the expenditure line shifts down. This results in an inward shift of the IS curve. This is shown in blue in Figure 13.3. 302 Figure 13.3: Increase in Gt+1 The inward shift of the IS curve, coupled with no shift of the Y s curve, means that the real interest rate must fall in equilibrium, from r0,t to r1,t. The lower real interest rate boosts autonomous expenditure to the point where the expenditure line shifts back to where it began. In equilibrium, there is no change in Ct (since there is no change in Yt or current Gt). Eﬀectively, the anticipated increase in Gt+1 makes
the household want to reduces its current consumption and therefore increase its saving. In equilibrium, this is not possible. So the real interest rate must fall to discourage the household from increasing its saving. 13.2.2 Algebraic Example Suppose that the household has log utility over consumption. This means that the Euler equation is: 303 𝑌𝑌𝑡𝑡𝑑𝑑 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝐶𝐶𝑑𝑑�𝑌𝑌𝑡𝑡−𝐺𝐺0,𝑡𝑡,𝑌𝑌𝑡𝑡+1−𝐺𝐺0,𝑡𝑡+1,𝑟𝑟0,𝑡𝑡�+𝐺𝐺0,𝑡𝑡 =𝐶𝐶𝑑𝑑�𝑌𝑌𝑡𝑡−𝐺𝐺0,𝑡𝑡,𝑌𝑌𝑡𝑡+1−𝐺𝐺1,𝑡𝑡+1,𝑟𝑟1,𝑡𝑡�+𝐺𝐺0,𝑡𝑡 𝑌𝑌0,𝑡𝑡 𝑌𝑌0,𝑡𝑡𝑑𝑑 𝑟𝑟0,𝑡𝑡 𝐼𝐼𝐼𝐼 𝑌𝑌𝑠𝑠 𝑟𝑟𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝐶𝐶𝑑𝑑�𝑌𝑌𝑡𝑡−𝐺�
��0,𝑡𝑡,𝑌𝑌𝑡𝑡+1−𝐺𝐺1,𝑡𝑡+1,𝑟𝑟0,𝑡𝑡�+𝐺𝐺0,𝑡𝑡 𝐼𝐼𝐼𝐼′ 𝑟𝑟1,𝑡𝑡 Ct+1 Ct = β(1 + rt) (13.15) Solve the Euler equation for Ct+1, and plug back into the household’s intertemporal budget constraint, (13.10). Solving for Ct gives the consumption function for this utility speciﬁcation: Ct = 1 1 + β (Yt − Gt) + 1 1 + β Yt+1 − Gt+1 1 + rt Total desired expenditure is the sum of this plus current government spendingYt − Gt) + 1 1 + β Yt+1 − Gt+1 1 + rt + Gt (13.16) (13.17) Impose the equality between income and expenditure, and solve for Yt, which gives an expression for the IS curve: Now, solve for rt: Yt = Gt + 1 β Yt+1 − Gt+1 1 + rt 1 + rt = 1 β Yt+1 − Gt+1 Yt − Gt (13.18) (13.19) From (13.19), it is clear that an increase in Gt raises rt, while an increase in Gt+1 lowers rt. 13.3 Fiscal Policy in a Production Economy Now, we shall incorporate ﬁscal policy into the production economy outlined in Chapter 12. The government’s budget constraints are the same as outlined at the beginning of this chapter. There are now two types of private actors – the representative household and ﬁrm. We assume that only the household pays taxes, which are again assumed to be lump sum. It would not change the outcome of the model to instead assume that the ﬁrm paid taxes to the government so long as those taxes are also lump sum. The household faces the following sequence of budget constraints: Ct + St ≤ wtNt − Tt + Dt Ct+1 + St+1 ≤
wt+1Nt+1 − Tt+1 + Dt+1 + DI t+1 + (1 + rt)St (13.20) (13.21) This is the same as in Chapter 12, with the addition that the household pays taxes, Tt and 304 Tt+1, to the government in each period. wt denotes the real wage received by the household, while Dt is a dividend paid out from the household’s ownership of the ﬁrm. DI t+1 is again the dividend the household receives from its ownership in the ﬁnancial intermediary. Imposing the terminal condition that St+1 = 0 and assuming that each constraint holds with equality yields the intertemporal budget constraint for the household, which says that the present discounted value of net income for the household must equal the present discounted value of the stream of consumption: Ct + Ct+1 1 + rt = wtNt − Tt + Dt + wt+1Nt+1 − Tt+1 + Dt+1 + DI t+1 1 + rt (13.22) Because taxes paid in both periods t and t + 1 are additive, (13.22) can be written: Ct + Ct+1 1 + rt = wtNt + Dt + wt+1Nt+1 + Dt+1 + DI t+1 1 + rt − [Tt + Tt+1 1 + rt ] (13.23) Because the household will anticipate that the government’s intertemporal budget constraint will hold with equality, (13.3), we can re-write the household’s intertemporal budget constraint as: Ct + Ct+1 1 + rt = wtNt − Gt + Dt + wt+1Nt+1 − Gt+1 + Dt+1 + DI t+1 1 + rt (13.24) In other words, just like in the endowment economy, the household’s intertemporal budget constraint can be written as though the government balances its budget each period (with Tt = Gt and Tt+1 = Gt+1), whether the government does or does not in fact do this. The ﬁrst order conditions characterizing a solution to the household’s problem are an Euler equation for consumption and a static labor
supply ﬁrst order condition for both periods t and t + 1: uC(Ct, 1 − Nt) = β(1 + rt)uC(Ct+1, 1 − Nt+1) uL(Ct, 1 − Nt) = wtuC(Ct, 1 − Nt) uL(Ct+1, 1 − Nt+1) = wt+1uC(Ct+1, 1 − Nt+1) (13.25) (13.26) (13.27) These conditions are exactly the same as we encountered before. Neither government spending, nor government debt, nor taxes appear in these conditions. Mathematically, this is a consequence of the fact that the ﬁscal terms enter only additively into the household’s ﬂow budget constraints. From these conditions we can intuit that there exists a consumption function wherein the household cares about income net of government spending each period 305 and the interest rate, and a labor supply condition wherein the quantity of labor supplied depends on the real wage and an exogenous term which we have labeled θt: Ct = C d(Yt − Gt +, Yt+1 − Gt+1 + ), rt − (13.28) ), θt − Nt = N s(wt + (13.28) is qualitatively the same as in the endowment economy – the household behaves as though the government balances its budget each period, whether this is in fact the case or not. We make the same assumptions on labor supply – labor supply is increasing in the real wage and decreasing in θt, which we take to be an exogenous source of ﬂuctuations in labor supply such as a preference shock.1 (13.29) The ﬁrm side of the model is exactly the same as in Chapter 12. As such, the labor demand and investment demand curves are identical to what we had before: Nt = N d(wt, At, Kt) It = I d(rt, At+1, Kt) (13.30) (13.31) Market-clearing requires that household saving plus government saving (less government borrowing) equal investment: St − BG is the total amount saved t with the ﬁnancial intermediary, which must in turn equal investment from the representative = Gt − Tt, plugging this into the
household’s ﬁrst period budget constraint at ﬁrm. Since BG t equality yields: = It. In essence, St − BG t Ct + It + Gt − Tt = wtNt − Tt + Dt (13.32) In (13.32), the Tt terms on both sides of the equality cancel out. The dividend paid out by the ﬁrm equals Yt − wtNt. Plugging this into (13.32) yields the aggregate resource constraint: Yt = Ct + It + Gt (13.33) The full set of equilibrium conditions are given below: 1As noted in Chapter 12, in general labor supply should be impacted by anything relevant for consumption. Since higher Gt results in lower Ct for given values of Yt and rt, it would seem plausible that higher Gt would encourage the household to work more. This would be true in a preference speciﬁcation allowing for such wealth eﬀects, but as noted in the introduction to the book, we are implicitly focusing attention on preferences of the sort emphasized by Greenwood, Hercowitz, and Huﬀman (1988) where there are no wealth eﬀects and where (13.29) holds exactly. 306 Ct = C d(Yt − Gt, Yt+1 − Gt+1, rt) Nt = N s(wt, θt) Nt = N d(wt, At, Kt) It = I d(rt, At+1, Kt) Yt = AtF (Kt, Nt) Yt = Ct + It + Gt (13.34) (13.35) (13.36) (13.37) (13.38) (13.39) These are identical to the equilibrium conditions presented in Chapter 12, save for the fact that Gt and Gt+1 are arguments of the consumption function and that Gt appears in the aggregate resource constraint. These six equations feature six endogenous variables – Yt, Nt, Ct, It, wt, and rt – with the following exogenous variables: Gt, Gt+1, At, At+1, θt, and Kt. As in the endowment economy setup, government taxes, Tt and Tt+1, as well as government debt, BG t, do
not appear anywhere in these equilibrium conditions. Ricardian Equivalence still holds for the same intuitive reasons as in the endowment economy model. The level of government debt is again irrelevant. 13.4 Summary • The government ﬁnances its spending by collecting lump sum taxes and issuing debt. Although we could model useful government expenditure, we assume it is strictly wasteful. • Despite having no control over the time path of government expenditures, the household behaves as if the government balances its budget every period. That is, the household only cares about the present discounted value of its tax liability. Since the present discounted value of taxes equals the present discounted value of spending, the time path is irrelevant. • An increase in current government spending raises autonomous expenditure, but less than one for one. In an endowment economy equilibrium, consumption drops one for one with a rise in current government spending and the real interest rate increases. • Ricardian equivalence also holds in a production economy where output is endogenous. Key Terms 307 • Lump sum taxes • Ricardian equivalence theorem Questions for Review 1. Explain the extent you agree with this statement: Ricardian equivalence shows that government deﬁcits do not matter. 2. Explain the logical error in this statement: Government spending ﬁnanced by issuing bonds will not decrease desired consumption because bonds are simply debt obligations we owe to ourselves. 3. Politicians often talk about how tax cuts will stimulate consumption. Discuss why this claim is incomplete. 4. List the assumptions of the Ricardian Equivalence theorem. 5. Graphically analyze an increase in Gt in an endowment economy. Clearly explain the economic intuition. 6. Graphically analyze an increase in Gt+1 in an endowment economy. Clearly explain the economic intuition. Exercises 1. Suppose that we have an economy with many identical households. There is a government that exogenously consumes some output and pays for it with lump sum taxes. Lifetime utility for a household is: U = ln Ct + β ln Ct+1 The household faces two within period budget constraints given by: Ct + St = Yt − Tt Ct+1 = Yt+1 − Tt+1 + (1 + rt)St (a) Combine the two budget constraints into one intertemporal budget constraint. (b) Use this to ﬁnd the Euler equation. Is the Euler equation at all aﬀected by the presence of taxes,
Tt and Tt+1? 308 (c) Use the Euler equation and intertemporal budget constraint to derive an expression for the consumption function. The government faces two within period budget constraints: Gt + SG t = Tt Gt+1 = Tt+1 + (1 + rt)SG t (d) In equilibrium, what must be true about St and SG t? (e) Combine the two period budget constraints for the government into one intertemporal budget constraint. (f) Suppose that the representative household knows that the government’s intertemporal budget constraint must hold. Combine this information with the household’s consumption function you derived above. What happens to Tt and Tt+1? What is your intuition for this? (g) Equilibrium requires that Yt = Ct + Gt. Plug in your expression for the consumption function (assuming that the household knows the government’s intertemporal budget constraint must hold) to derive an expression for Yt. (h) Derive an expression for the “ﬁxed interest rate multiplier,” i.e. dYt dGt ∣drt=0 (i) Assuming that Yt is exogenous, what must happen to rt after an increase. in Gt? (j) Now, assume the same setup but suppose that the household does not anticipate that the government’s intertemporal budget constraint will hold – in other words, do not combine the government’s intertemporal budget constraint with the household’s consumption function as you did on part (f). Repeat part (h), deriving an expression for the “ﬁxed interest rate multiplier” while not assuming that the household anticipates the government’s budget constraint holding. Is it bigger or smaller than you found in (h)? (k) Since Yt is exogenous, what must happen to rt after an increase in Gt in this setup? Will the change in rt be bigger or smaller here than what you found in part (i)? (l) For the setup in which the household does not anticipate that the government’s intertemporal budget constraint must hold, what will be the “ﬁxed interest rate tax multiplier”, i.e. dYt? Is this diﬀerent dTt ∣drt=0 309 than what the tax multiplier would be
if the household were to anticipate that the government’s intertemporal budget constraint must bind? Is it smaller or larger than the ﬁxed interest rate multiplier for government spending (assuming that the household does not anticipate that the government’s intertemporal budget constraint will hold)? 2. Consider a representative agent with the utility function The budget constraint is U = ln Ct + θ 2 (1 − Nt)2 Ct = wtNt + Dt where wt is the wage and Dt is non-wage income (i.e. a dividend from ownership in the ﬁrm). The agent lives for only one period (period t), and hence its problem is static. (a) Derive an optimality condition characterizing optimal household behav- ior. (b) Solve for the optimal quantities of consumption and labor. (c) Suppose that the government implements a lump sum subsidy to all workers, Tt. It engages in no spending and has no budget constraint to worry about; hence it can choose Tt however it pleases. The household’s budget constraint is now: Ct = wtNt + Dt + Tt. How are the optimal quantities of Ct and Nt aﬀected by the introduction of the subsidy? Speciﬁcally, do people consume more or less leisure? What is the economic intuition for this? (d) Instead of a lump sum subsidy, suppose the government subsidizes work. With the subsidy, the workers receive an eﬀective wage rate of wt(1 + τt). The budget constraint is Ct = wt(1 + τt)Nt + Dt. How are the optimal quantities of C and N aﬀected by the introduction of the subsidy? Speciﬁcally, do people consume more or less leisure? What is the economic intuition for this? 310 (e) Suppose the government wants to help workers, but does not want to discourage work. Which of these subsidies will be more successful? 3. Consider a ﬁrm which operates for two periods. It produces output each period according to the following production function: Yt = AtK α t 0 < α < 1. The current capital stock is exogenously given. The ﬁrm can inﬂuence its future capital stock through investment. The two capital accumulation equations are: Kt+1 = It + (1
− δ)Kt. Kt+2 = It+1 + (1 − δ)Kt+1. The ﬁrm liquidates itself (i.e. sells oﬀ the remaining capital that has not depreciated during the period) at the end of the second period. The ﬁrm borrows to ﬁnance any investment in period t at rt. The ﬁrm’s objective is to maximize its value, given by: V = Dt + Dt+1 1 + rt where Πt denotes proﬁts, which are paid as dividends to its owners, and the ﬁrm takes the interest rate as given. (a) Write down the expressions for both current and future proﬁts, Dt and Dt+1. What is the terminal condition on Kt+2? (b) Write down the ﬁrm’s optimization problem. What are its choice vari- ables? (c) Algebraically solve for the ﬁrm’s optimal choice of investment, It. (d) Now suppose that there is a proportional tax rate (i.e. not a lump sum tax) on ﬁrm proﬁts, τt, which is the same in both periods (i.e. τt = τt+1). Re-do the above, solving for the optimal investment rule. What is the eﬀect of the tax rate on investment? (e) Instead, suppose that the tax rate is on revenue, not proﬁts. That is, after tax ﬁrm proﬁts in the ﬁrst period are now (1 − τt)Yt − It instead of (1 − τt)(Yt − It). In the second period output is again taxed, but the liquidated capital stock is not. In other words, after tax proﬁts in the second period are: (1 − τt+1)Yt+1 − It+1. Redo the problem. What is the 311 eﬀect of the tax rate on investment? How does your answer compare with your answer in Part d? 312 Chapter 14 Money Up until this point, we have completely ignored money. Isn’t economics all about money? In this chapter, we will deﬁne what economists
mean by money and will incorporate it into our micro-founded model of the macroeconomy. 14.1 What is Money? Deﬁning what money is (or is not) is not such an easy task. Generically, money is an asset that can be used in exchange for goods and services. For most things, we deﬁne them according to intrinsic characteristics of those things. For example, apples are round and red. This is not really so with money. Physical currency (i.e. the dollar bills in your wallet) can be used in exchange for goods and services, as can the electronic entries in your checking account through the process of writing checks or using a debit card. Both currency and checking accounts are money in the sense that they can be used in exchange, but intrinsically they are very diﬀerent things. When deﬁning what is (and is not) money, we think about the functions played by money. These functions are: 1. It serves a medium of exchange. This means that, rather than engaging in barter, one can trade money for goods and services. 2. It serves as a store of value. This means that money preserves at least some of its value across time, and is therefore a means by which a household can transfer resources across time. The store of value function of money means that money can serve a function like bonds – a way to shift resources across time. 3. It serves as a unit of account. This simpliﬁes economic decision-making, as we denominate the value of goods and services in terms of units of money. This makes it easy to compare value across diﬀerent types of goods. Suppose that an economy produces three goods – trucks, beer, and guns. Suppose a truck costs 10,000 units of money, a beer 1 unit of money, and a gun 10 units of money. We could equivalently say that a truck costs 10,000 cans of beer or 100 guns, but there are other ways to compare value. 313 For example, we might say that a beer costs 0.0001 trucks or 0.1 guns. By serving as the unit of account, money serves as the numeraire, or the thing which we price all other goods according to. Many diﬀerent kinds of assets can serve these purposes. Being an asset implies that something is a store of value. Whether or not an asset is a good medium of exchange is a di�
��erent story. We refer to an asset’s liquidity as measuring how easy it is to use it in exchange. Currency is the most liquid asset – it is the medium of exchange. A house is not as liquid. You could in principle sell the house to raise cash and use that cash to buy some other good or service, but doing so quickly and at a fair price may not be easy. Checking accounts, or what are often called demand deposits, are virtually equivalent to currency, because you are allow to exchange checking accounts for currency on demand. Because of this, in practice most measures of the money supply (see later sections of the book) count the value of checking accounts as money. In principle, almost anything could serve the aforementioned three functions, and hence almost anything could serve as money. In fact, in the past, many diﬀerent things have in fact served as money. For many years commodities served as money – things like cows, cigarettes, and precious metals (e.g. gold and silver). In more recent times, most economies have moved toward ﬁat money. Fiat money consists of pieces of paper (or electronic entries on a computer) which have no intrinsic value – they just have value because a government declares that they will serve as money, and they therefore have value to the extent to which they are accepted in exchange for goods and services. It is not hard to see why commodity-based money can become problematic. First, commodities have value independent from their role as money. Fluctuating commodity prices (say there is a drought which kills oﬀ cows, increasing the value of cows, or a new discovery of gold, which decreases the market value of gold) will generate ﬂuctuations in the price of all other goods, which can create confusion. This makes commodities which have value independent of their role in exchange problematic as a unit of account. Second, commodities may not store well, and hence may not be good stores of value. Third, commodities may not be easily divisible or transferable, and hence may not be very desirable as a medium of exchange. In the example listed above, it would not be easy to cut a cow up into 10,000 pieces in order to purchase a can of beer. Fiat money lacks these potential problems associated with commodity-based money. That being said, ﬁat money is prone to problems. Fiat money only has value because a government declares that it has value and people believe it. If people quit
believing that money has value (i.e. quit accepting it in exchange), then the money would cease to have value. This makes ﬁat money quite precarious. Second, ﬁat money is subject to manipulation by governments – if ﬁat money has no intrinsic value, a government could 314 simply create more of that ﬁat money for example to pay oﬀ debts, which would decrease the value of that ﬁat money and implicitly serve as a tax of the holders of that money. Most economists would agree that the medium of exchange role of money is the most important function played by money. In the real world, there are many potential stores of value – things like houses, stocks, bonds, etc. – so money is not really unique as a store of value. While it is convenient to adopt money as the numeraire, it would not be particularly problem to deﬁne some other good as the unit of account. Hence, in terms of a unit of account, nothing is all that unique about money. The crucial role that money serves is as a medium of exchange. Without money, we’d have to engage in barter, and this would be costly. For example, the professor teaching this course is providing educational services, and you (or your parents) are indirectly compensating that professor with your tuition money. Suppose there were no money, and we had to engage in barter instead. Suppose that your mother is a criminal defense attorney. To compensate the professor for educational services, she would like to trade criminal defense services in exchange for classroom instruction. But what if (hopefully) the professor is not currently in need of criminal defense? We refer to the potential mismatch between the resources a buyer of some good has available (in this case, criminal defense services) with the resources a seller has available (in this case, educational services) the double coincidence of wants problem. To successfully engage in barter, the buyer has to have something that the seller wants. With money, this is not so. The buyer can instead pay in money (e.g. money income from criminal services), and the seller can use that money to buy whatever he or she desires (e.g. a new house). The existence of money, by eliminating the double coincidence of wants problem, facilitates more trade (not trade in an international sense, but trade in the form of exchange of diﬀerent goods and services), which in turn leads to more specialization.
Increased specialization results in productivity gains that ultimately make everyone better oﬀ. It is no exaggeration to say that well-functioning medium of exchange is the most important thing to have developed in economic history, and it is diﬃcult to downplay the importance of money in a modern economy. We have not studied money to this point because, as long as it exists and functions well, it should not matter too much. Money really only becomes interesting if it does not work well or if there is some other friction with which it interacts. In this Chapter, we will study how to incorporate money into a micro-founded equilibrium model of the business cycle. We defer a discussion of how the quantity of money is measured, or how it interacts with the rest of the economy, until Chapter 21. 315 14.2 Modeling Money in our Production Economy It is not easy to incorporate money in a compelling way into the micro-founded equilibrium model of a production economy with which we have been working. Why is this? In our model, there is one representative household, one representative ﬁrm, and one kind of good (which one might think of as fruit). Because there is only one type of good and one type of household, exchange is pretty straightforward. Put a little diﬀerently, there is no double coincidence of wants problem for money to solve if there is only one kind of good in the economy. This means that the medium of exchange function of money, which in the real world is the most important role money plays, is not important in our model. With only one type of good in the economy, there is also not much compelling reason to use money as the numeraire – it is just as easy to price things in terms of units of goods (i.e. the real wage is ﬁve units of output per unit of time worked) as in money (i.e. the nominal wage is ten units of money per unit of time worked). One can use money as the unit of account in the model, but there is nothing special about it. What about money’s role as a store of value? One can introduce money into the model in this way, but there are competing stores of value – the household has access to bonds, and the ﬁrm can transfer resources across time through investment in new physical capital. We will introduce money into our model essentially as a store of value. Since things can be priced in terms of money, it also serves the unit
of account role. With only one kind of good, there is no important medium of exchange role. Eﬀectively, money is going to be an asset with which the household can transfer resources across time. In the revised version of the model, the household will be able to save through bonds (which pay interest) or money (which does not). If it helps to ﬁx ideas, one can think of saving through bonds as putting money “in the bank,” in exchange for the principal plus interest back in the future, whereas saving through money is stuﬃng cash under one’s mattress. If one puts a hundred dollars under one’s mattress, one will have a hundred dollars when one wakes up the next period. It is easy to see that it will be diﬃcult to get a household to actually want to hold money in this setup. Why? Because money is dominated as a store of value to the extent to which bonds pay positive interest. If one could put a hundred dollars in the bank and get back one hundred and ﬁve dollars next period (so a ﬁve percent interest rate), why would one choose to put a hundred dollars under the mattress, when this will yield one hundred dollars in the future? What is the beneﬁt of holding money? To introduce a beneﬁt of holding money, we will take a shortcut. In particular, we will assume that the household receives utility from holding money. To be speciﬁc, we will assume that the household receives utility from the quantity of real money balances which 316 the household holds, which is the number of goods a given stock of money could purchase. This shortcut can be motivated as a cheap way to model the beneﬁcial aspect of money as a medium of exchange. The basic idea is as follows. The more purchasing power the money one holds has, the lower will be utility costs associated with exchange. This results in higher overall utility. In the subsections below, we introduce money into our model and deﬁne a few important concepts. We conclude with a complete set of equilibrium decision rules, most of which look identical to what we previously encountered in Chapters 12 and 13. The new equations will be a money demand curve and an expression which relates the real interest rate to the nominal interest rate. 14.2.1 Household Let us begin with a discussion of how the introduction of money as a store of value
impacts the household’s budget constraint. First, some notation. Let Mt denote the quantity of money that the household chooses to hold. This quantity of money is taken between period t and t + 1, in an analogous way to savings, St (i.e. it is a stock ). Let Pt denote the price of goods measured in units of money (e.g. Pt would be two dollars per good). Let it be the nominal interest rate. If one puts one dollar in the bank in period t, one gets 1 + it dollars back in period t + 1. As we discussed in Chapter 1, real variables are measured in quantities of goods, whereas nominal variables are measured in units of money. Let Ct be the number of units of consumption (this is “real” in the sense that it is denominated in units of goods). Let St be the number of units of goods that one chooses to save via bonds (this is again real in the sense that it is denominated in units of goods). wt is the real wage (number of goods one gets in exchange for one unit of labor, Nt), Tt is the number of goods one has to pay to the government in the form of taxes, and Dt is the number of units of goods which the household receives in the form of a dividend from its ownership in the ﬁrm. All of these real quantities can be converted to nominal quantities by multiplying by Pt. So, for example, if Pt = 2 and Ct = 2, then the dollar value of consumption is 4. The period t ﬂow budget constraint for the household is given in (14.1): PtCt + PtSt + Mt ≤ PtwtNt − PtTt + PtDt (14.1) (14.1) says that the dollar value of consumption, PtCt, plus the dollar value of saving in bonds, PtSt, plus the dollar value of saving in money, Mt, cannot exceed the dollar value of net income. Net income is the dollar value of labor income, PtwtNt, less the dollar value of 317 tax obligations, PtTt, plus the dollar value of dividends received, PtDt. The period t + 1 budget constraint is given in (14.2): Pt+1Ct+1 ≤ Pt+1wt+1Nt+1 − Pt+1Tt+1 + (1 + it)PtSt + Pt+1Dt+
1 + Pt+1DI t+1 + Mt (14.2) (14.2) says that dollar value of period t + 1 consumption, Pt+1Ct+1, cannot exceed the dollar value of net income, Pt+1wt+1Nt+1 − Pt+1Tt+1, plus the dollar value of dividends received, Pt+1Dt+1 + Pt+1DI t+1, plus return on saving from bonds, which is (1 + it)PtSt (one puts PtSt dollars in the bank in period t, and gets back principal plus interest), plus the money one saved in period t, which is simply Mt. When looking at (14.1) and (14.2), it is important to note that PtSt (the dollar value of saving in bonds) and Mt (the dollar value of saving in money) enter the budget constraints in exactly the same way. The only diﬀerence is that bonds pay interest, it, whereas the eﬀective interest rate on money is zero. In writing the second period constraint, we have gone ahead and imposed the terminal conditions that the household not die with any positive or negative savings (i.e. St+1 = 0) and that the household not carry any money over into period t + 2 (i.e. Mt+1 = 0), since the household does not exist in period t + 2. Let’s re-write these budget constraints in real terms. Start by dividing (14.1) by Pt. Simplifying yields: Ct + St + Mt Pt ≤ wtNt − Tt + Dt (14.3) For the period t + 1 budget constraint, divide both sides of (14.2) by Pt+1. One gets: Ct+1 ≤ wt+1Nt+1 − Tt+1 + (1 + it) Pt Pt+1 St + Dt+1 + DI t+1 + Mt Pt+1 (14.4) Both the period t and t + 1 budget constraints are now expressed in real terms – the units of all entries are units of goods, not units of money. The period t constraint says that the household has real income from labor and distributed dividends, and pays taxes to a government. With this income, the household can consume, Ct, save in bonds, St, or save via money, Mt is referred to as real money balances (
or real balances for short). Mt Pt Pt equals the number of goods that the stock of money could purchase. For example, if Mt = 10 and Pt = 2, then the 10 units of money could purchases 10/2 = 5 units of goods. In period t + 1 the household has income from labor, income from its ownership of the ﬁrm, income from ownership of the ﬁnancial intermediary, interest income from its saving in bonds, and the real purchasing power of the money it brought between t and t + 1, equal to Mt. To be Pt+1. The term Mt Pt 318 concrete, the household brings Mt units of money into t + 1, which is the equivalent of Mt Pt+1 units of goods in period t + 1. In the period t + 1 constraint written in real terms, the term (1 + it) Pt Pt+1 multiplies the represents the (gross) real return on saving through bonds. As such we term St. (1 + it) Pt Pt+1 will deﬁne: 1 + rt = (1 + it) Pt Pt+1 (14.5) The expression in (14.5) is known as the Fisher relationship, after famous economist Irving Fisher. It relates the real interest rate, rt (which we have already encountered), to the nominal interest. The gross nominal interest rate is multiplied by Pt. Suppose that you Pt+1 put want to put one unit of goods into a saving bond in period t. This requires putting Pt units of money into the bond. This will generate (1 + it)Pt units of money in period t + 1. This will purchase (1 + it) Pt as the expected Pt+1 gross inﬂation rate between periods t and t + 1.1 This means that the Fisher relationship can equivalently be written: goods in period t + 1. Deﬁne 1 + πe = Pt+1 Pt t+1 1 + rt = 1 + it 1 + πe t+1 (14.6) Taking logs of (14.6) and using the approximation that the log of one plus a small number is the small number, the Fisher relationship can be approximated: Using the Fisher relationship, the period t + 1 budget constraint in real terms, (14.4) can rt = it − πe t+1 (14.7) equivalently be written: Ct+1 ≤
wt+1Nt+1 − Tt+1 + (1 + rt)St + Dt+1 + DI t+1 + 1 + rt 1 + it Mt Pt (14.8) Looking at (14.3) and (14.8), one sees that these are identical budget constraints to what term shows up in both the we encountered Chapter 13, with the only addition that the Mt Pt constraints. Let’s assume that (14.8) holds with equality. Solve for St from (14.8): 1Here and for most of the remainder of the book, we will take expected inﬂation to be exogenous. A thorny issue here concerns the equilibrium determination of Pt+1. As we will later see, Pt will be determined in equilibrium given Mt. But since the household would not want to hold any Mt+1 (i.e. money to take from t + 1 to t + 2), since the household ceases to exist after period t + 1, money will not have any value in period t + 1 and Pt+1 = 0. This is a generic problem with ﬁnite horizon models where money enters the utility function. We will ignore this, appealing to the fact that we are treating the two period model as an approximation to a multi-period model, and treat expected future inﬂation as exogenous. 319 − wt+1Nt+1 − Tt+1 + Dt+1 + DI t+1 1 + rt Now plug (14.9) into (14.3), assuming that it holds with equality. Doing so and simplifying St = Ct+1 1 + rt − 1 Mt Pt 1 + it (14.9) yields: Ct + Ct+1 1 + rt = wtNt − Tt + Dt + wt+1Nt+1 − Tt+1 + Dt+1 + DI t+1 1 + rt − it 1 + it Mt Pt (14.10) This is the intertemporal budget constraint for the household. It is identical to the real intertemporal budget constraint we encountered previously, with the addition of the term − it 1+it appearing on the right hand side. Mt Pt As noted earlier, we assume that the household receives utility from holding money, in particular the real purchasing power of money, Mt. A slight complication is that the way Pt in which we
have written the problem, money is held “across” periods (i.e. between t and t + 1), so it is not obvious whether the household should receive utility from holding money in period t or period t + 1. We will assume that this utility ﬂow accrues in period t and that the utility ﬂow from holdings of real money balances is additively separable from utility from consumption and leisure. Lifetime utility for the household is given by: U = u(Ct, 1 − Nt) + v (Mt Pt ) + βu(Ct+1, 1 − Nt+1) (14.11) Here, v(⋅) is a function which is increasing and concave which maps real money balances into utils. An example function is the natural log. The objective of the household will be to pick Ct, Ct+1, Nt, Nt+1, and now also Mt to maximize U, subject to the intertemporal budget constraint, (14.10). The household is a price taker and treats rt, wt, it, and Pt and Pt+1 (equivalently πe t+1) as given. Formally, the problem of the household is: max Ct,Ct+1,Nt,Nt+1,Mt U = u(Ct, 1 − Nt) + v (Mt Pt ) + βu(Ct+1, 1 − Nt+1) (14.12) s.t. Ct + Ct+1 1 + rt = wtNt − Tt + Dt + wt+1Nt+1 − Tt+1 + Dt+1 + DI t+1 1 + rt − it 1 + it Mt Pt (14.13) To ﬁnd the optimality conditions, solve for one of the choice variables in (14.13). We will solve for Ct+1. We get: Ct+1 = (1 + rt) [wtNt − Tt + Dt − Ct] + wt+1Nt+1 − Tt+1 + Dt+1 + DI t+1 − (1 + rt) it 1 + it Mt Pt (14.14) 320 Now plug this into the objective function, which transforms the problem into an uncon- strained one: max Ct,Nt,Nt
+1,Mt U = u(Ct, 1 − Nt) + v (Mt Pt ) +...... βu ((1 + rt) [wtNt − Tt + Dt − Ct] + wt+1Nt+1 − Tt+1 + Dt+1 + DI t+1 + (1 + rt) it 1 + it Mt Pt, 1 − Nt+1) (14.15) Take the derivatives of lifetime utility with respect to the choice variables. In doing so, we make use of the chain rule, but abbreviate the argument in the second period utility function as Ct+1 (the expression for which is given in (14.14)). ∂U ∂Ct ∂U ∂Nt ∂U ∂Nt+1 ∂U ∂Mt = uC(Ct, 1 − Nt) − βuC(Ct+1, 1 − Nt+1)(1 + rt) = 0 = −uL(Ct, 1 − Nt) + βuC(Ct+1, 1 − Nt+1)(1 + rt)wt = 0 = −uL(Ct+1, 1 − Nt+1) + uC(Ct+1, 1 − Nt+1)wt+1 = 0 1 = v′ ( Mt Pt Pt − βuC(Ct+1, 1 − Nt+1)(1 + rt) it 1 + it ) 1 Pt = 0 The ﬁrst three equations can be re-arranged to yield: uC(Ct, 1 − Nt) = β(1 + rt)uC(Ct+1, 1 − Nt+1) uL(Ct, 1 − Nt) = uC(Ct, 1 − Nt)wt uL(Ct+1, 1 − Nt+1) = uC(Ct+1, 1 − Nt+1)wt+1 (14.16) (14.17) (14.18) (14.19) (14.20) (14.21) (14.22) These are exactly the same ﬁrst order conditions we derived in Chapter 12 for the choices of consumption and labor. Each of these has the familiar “marginal beneﬁt = marginal cost” interpretation
. The new ﬁrst order conditions relates to the choice of how much money to hold across periods. We can re-write (14.19) as: v′ (Mt Pt ) = it 1 + it uC(Ct, 1 − Nt) (14.23) This condition also has the interpretation of “marginal beneﬁt = marginal cost,” though it takes a bit of work to see this. The left hand side is the marginal beneﬁt of holding an 321 additional unit of real money balances. This is the marginal utility of holding more money. What is the marginal cost of holding money? This is an opportunity cost. In the model, there are two savings vehicles – money and bonds, with the diﬀerence being that bonds pay interest, whereas money does not. If a household saves an additional unit of goods in money (i.e. chooses to hold an additional unit of real money balances), it is foregoing saving one unit in bonds. Saving one unit of goods in bonds would entail saving Pt units of money, which would yield (1 + it)Pt additional units of money in period t + 1. Saving one unit of goods in money entails saving Pt dollars, which yields Pt dollars in period t + 1. You can think about money and bonds as being identical, except bonds pay it whereas the interest rate on money is 0. The opportunity cost of saving in money is the diﬀerence between how much money you’d have in t + 1 from saving in bonds versus saving via money, goods in period t + 1, or itPt. This additional money in period t + 1 will purchase itPt Pt+1 which increases lifetime utility by the discounted marginal utility of future consumption, or βuC(Ct+1, 1 − Nt+1) itPt. Combining the Euler equation with the Fisher relationship, one can Pt+1 uC(Ct, 1 − Nt). Substituting this in yields the marginal write uC(Ct+1, 1 − Nt+1) = 1 1 1+it uC(Ct, 1 − Nt). Hence, (14.23) has the familiar marginal beneﬁt cost of holding money as = marginal cost interpretation. At an optimum, a household will hold money up until the point where the marginal beneﬁt of doing so equals the marginal cost. β it
1+it Pt+1 Pt Before proceeding, it is useful to conclude our analysis with a discussion of why it is important to assume that the household receives utility from holding money. Suppose that v′(⋅) = 0. This would mean that holding more (or fewer) real balances would not aﬀect a household’s lifetime utility. If this were the case, (14.23) could not hold, unless it = 0. If it > 0, then the marginal cost of holding money would always be positive, whereas the marginal beneﬁt of holding money would be zero. Put slightly diﬀerently, since bonds pay interest whereas money does not, if there is no marginal beneﬁt from money and the interest rate is positive, the household would choose to hold no money (i.e. we would be at a corner solution). If it = 0, then bonds and money would be perfect substitutes, and the household would be indiﬀerent between saving through bonds or saving through money. Hence, for the more general case in which the nominal interest rate is positive, to get the household to be willing to hold money we must have there be some beneﬁt of doing so, which we have built in to the model via real balances in the lifetime utility function. The ﬁrst two optimality conditions, (14.20) and (14.22), are identical to what we had before, and as such imply the same consumption and labor supply functions: Ct = C d(Yt − Tt, Yt+1 − Tt+1, rt) (14.24) 322 Nt = N s(wt, θt) (14.25) In (14.24) consumption demand is increasing in current and future net income and decreasing in the real interest rate (via the assumption that the substitution eﬀect of changes in the real interest rate dominates the income eﬀect). In (14.25), labor supply is increasing in the real wage and decreasing in θt, which we take to be an exogenous labor supply shifter. We can use (14.23) to think about how changes in diﬀerent variables impact the desired quantity of Mt a household would like to hold. First, we can see that the demand for Mt is proportional to Pt. If Pt goes up, this does not impact the amount
of Mt the household would Pt like to hold, and hence Mt is increasing in Pt. This is fairly intuitive – the more goods cost in terms of money, the more money a household would like to hold. Second, note that a higher it makes bigger. This means that the household needs to adjust its money holdings so as to make v′ ( Mt ) bigger. Since we assume v′′(⋅) < 0, this requires reducing Mt. This is again Pt fairly intuitive. The nominal interest rate represents the opportunity cost of holding money – the higher is it, the less money a household would like to hold. Finally, suppose that the household increases its consumption. This would make uC(⋅) decrease, which means that the ). Again, since household needs to adjust Mt in such a way as to generate a decrease in v′ ( Mt Pt we have assumed that v(⋅) is a concave function, this would entail increasing Mt. Hence, Mt is increasing in consumption. This again is fairly intuitive – the more stuﬀ a household is buying, the more money it is going to want to hold. Hence, we conclude that money demand ought to be increasing in Pt, decreasing in it, and increasing in consumption, Ct. it 1+it Based on this qualitative analysis of the FOC for money, we will write a money demand function as follows: Mt = PtM d(it −, Yt + ) (14.26) In writing (14.26), we have written money demand as a function of Yt rather than Ct. Since Ct depends on Yt, this is not such a bad simpliﬁcation. We are, however, abstracting from things other than Yt which would impact Ct, and hence money demand. We make this abstraction for simplicity and to facilitate comparison with money demand speciﬁcations used in empirical work, which typically are speciﬁed as depending on Yt, rather than Ct. Money demand is decreasing in the nominal interest rate and increasing in income. It is proportional (and hence increasing) in Pt. Money demand depends on the nominal interest rate (whereas consumption demand depends on the real interest rate). We can, however, specify money demand in terms of the real interest rate using the approximate version of the Fisher relationship, where it = rt + πe t+1. 323 To the extent to which expected inﬂation is close to
constant, the real and nominal interest rates will move together: Example − Mt = PtM d(rt + πe t+1 ), Yt + (14.27) Suppose that we have an endowment economy in which labor is ﬁxed. Suppose that lifetime utility is given by: U = ln Ct + ψt ln (Mt Pt ) Then the ﬁrst order optimality conditions work out to: 1 Ct ψt = β(1 + rt) 1 Ct+1 1 Ct it 1 + it Pt Mt = (14.30) can be re-arranged to yield: Mt = Ptψt 1 + it it Ct (14.28) (14.29) (14.30) (14.31) In (14.31), desired Mt is increasing in Pt, decreasing in it, and increasing in Ct. 14.2.2 Firm In our model, the ﬁrm does not use money as a means to transfer resources across time. In other words, the ﬁrm does not hold money. As such, its problem is identical to what we previously encountered. The problem can be written in nominal or real terms. Using the Fisher relationship, the ﬁrst order conditions for the ﬁrm’s problem are exactly the same as we had previously encountered. These are repeated below for convenience. wt = AtFN (Kt, Nt) wt+1 = At+1FN (Kt+1, Nt+1) 1 + rt = At+1FK(Kt+1, Nt+1) + (1 − δ) (14.32) (14.33) (14.34) 324 These ﬁrst order conditions implicitly deﬁne labor and investment demand functions of the sort (where underscores denote the sign of the partial derivatives): Nt = N d(wt, At − +, At+1 + It = I d(rt − ), Kt + ), Kt − (14.35) (14.36) 14.2.3 Government The third actor in our model economy is the government. As in Chapter 13, this government chooses an exogenous amount of spending each period, Gt and Gt+1. It uses lump sum taxes levied on the household, Tt and Tt+1, to ﬁnance this
expenditure. In addition to its ﬁscal responsibility, we assume that the government can set the money supply. One can think about this as “printing” money, though in reality most money these days is simply electronic. The amount of money supplied by the government is assumed to be exogenous, M s t = Mt. There is no cost of the government of “printing” money. Hence, “printing” money is essentially a form of revenue for a government. This form of revenue is referred to as seignorage. The government’s period t and t + 1 budget constraints, expressed in nominal terms, are: PtGt ≤ PtTt + PtBt + Mt Pt+1Gt+1 + (1 + it)PtBt + Mt ≤ Pt+1Tt+1 (14.37) (14.38) In (14.37), Bt is the amount of real debt issued by the government; multiplication by Pt puts it in nominal terms. As one can see, the inclusion of Mt on the right hand side means that Mt is a source of nominal revenue for the government. We have imposed that the government not issue any debt in period t + 1 (i.e. Bt+1 = 0) and also that it not issue any money in period t + 1 (i.e. Mt+1 = 0). In the second period, the government can purchase goods (expenditure of Pt+1Gt+1) but must pay oﬀ its debt. It brings PtBt dollars of debt into period t + 1, and pays back the principal plus nominal interest, so (1 + it)PtBt is its nominal interest expense in period t + 1. In addition, we can think about the government “buying back” the money it printed in period t, so Mt is an expense for the government in period t. One can think about the government creating money and selling it in period t, and then buying it back (or “retiring it”) in period t + 1. It raises nominal revenue Pt+1Tt+1. Each of these constraints can be written in real terms by dividing each budget constraint by the price level in that period: 325 Gt ≤ Tt + Bt + Mt Pt Gt+1 + (1 + it) Pt Pt+1 Bt + Mt Pt+1
≤ Tt+1 (14.39) (14.40) (14.39) is the same real budget constraint we encountered for the government in Chapter = (1+rt), appears on the right hand side as real seignorage revenue. Since (1+it) Pt Pt+1 13, but Mt Pt (14.40) is the same as we encountered previously, but Mt Pt+1 appears on the left hand side. We can combine these constraints into an intertemporal budget constraint for the govern- ment. Solve for Bt in the period t + 1 constraint: Bt = Tt+1 − Gt+1 1 + rt − 1 1 + rt Mt Pt+1 Plugging this in to (14.39), we get: Gt + Gt+1 1 + rt = Tt + Tt+1 1 + rt + Mt Pt − 1 1 + rt Mt Pt+1 Since 1 1+rt = 1 1+it Pt+1 Pt, the term involving money can be re-written, yielding: (14.41) (14.42) Mt Pt + it 1 + it Gt + Gt+1 1 + rt = Tt + Tt+1 1 + rt (14.43) is the government’s intertemporal budget constraint, and is analogous to the household’s intertemporal budget constraint, (14.10). The real presented discounted value of government spending (consumption for the household) must equal the real present discounted value of revenue (income for the household), plus a term related to real balances for each. This term is the same for both the government and household,, though it enters with a positive sign in the government’s IBC and a negative sign in the household’s IBC. (14.43) it 1+it Mt Pt 14.2.4 Equilibrium The equilibrium is a set of prices and allocations for which all agents are behaving optimally and all markets simultaneously clear. Household optimization requires that the consumption and labor supply functions, (14.24) and (14.25) hold. Firm optimization requires that the labor demand and investment demand functions, (14.35) and (14.36), both hold. There is an additional optimality condition related to the household’s demand for money, given by (14.26). The Fisher relationship, written in its approximate form, (14
.7), relates the nominal and real interest rates with the rate of expected inﬂation, which we take as 326 exogenous to the model. Market-clearing requires that all budget constraints hold with equality. In real terms, the household’s period t budget constraint is: Ct + St + Mt Pt ≤ wtNt − Tt + Dt Market-clearing in the market for bonds requires that: Real ﬁrm proﬁt is: Combining these yields: St − Bt = It Dt = Yt − wtNt Ct + It + Bt + Mt Pt = Yt − Tt From the government’s period t budget constraint, we have: Combining (14.48) with (14.47), we have: Tt = Gt − Bt − Mt Pt Yt = Ct + It + Gt (14.44) (14.45) (14.46) (14.47) (14.48) (14.49) This is a standard aggregate resource constraint. Note that money does not appear. Recall the household’s intertemporal budget constraint, repeated here for convenience: Ct + Ct+1 1 + rt = wtNt − Tt + Dt + wt+1Nt+1 − Tt+1 + Dt+1 + DI t+1 1 + rt − it 1 + it Mt Pt (14.50) Because taxes are lump sum, this can equivalently be written: Ct + Ct+1 1 + rt = wtNt + Dt + wt+1Nt+1 + Dt+1 + DI t+1 1 + rt − (Tt + Tt+1 1 + rt ) − it 1 + it Mt Pt (14.51) From the government’s intertemporal budget constraint, (14.43), we have: Tt + Tt+1 1 + rt = Gt + Gt+1 1 + rt − it 1 + it Mt Pt (14.52) 327 Combining (14.52) with (14.51), we get: Ct + Ct+1 1 + rt = wtNt + Dt + wt+1Nt+1 + Dt+1 + DI t+1 1 + rt − (Gt + Gt+1 1
+ rt ) (14.53) There are two things worth noting. First, the real balance term, Mt Pt, drops out. Second, taxes disappear, leaving only the present discounted value of government expenditures on the right hand side. These terms can be re-arranged to yield: Ct + Ct+1 1 + rt = wtNt − Gt + Dt + wt+1Nt+1 − Gt+1 + Dt+1 + DI t+1 1 + rt (14.54) Just as we saw in Chapter 13, both government and household budget constraints holding, along with taxes being lump sum (i.e. additive), means that, from the household’s perspective, it is as though Tt = Gt. In other words, Ricardian Equivalence continues to hold – the household behaves as though the government balances its budget each period, whether the government does so or not. This means that the consumption function, (14.24), can instead be written: Ct = C d(Yt − Gt, Yt+1 − Gt+1, rt) The full set of equilibrium conditions can be written: Ct = C d(Yt − Gt, Yt+1 − Gt, rt) Nt = N s(wt, θt) Nt = N d(wt, At, Kt) It = I d(rt, At+1, Kt) Yt = AtF (Kt, Nt) Yt = Ct + It + Gt Mt = PtM d(it, Yt) rt = it − πe t+1 (14.55) (14.56) (14.57) (14.58) (14.59) (14.60) (14.61) (14.62) (14.63) The ﬁrst six of these expressions are identical to what we encountered in Chapter 13. There are two new equations – the money demand speciﬁcation (14.62), and the Fisher relationship relating the real and nominal interest rates to one another, (14.63). There are eight endogenous variables – Yt, Ct, It, Nt, wt, rt, Pt, and it. The ﬁrst six of these are the same as we had before, with the two new endogenous nominal variables, Pt
and it. The 328 exogenous variables are At, At+1, Gt, Gt+1, Kt, Mt, and πe t+1. The ﬁrst six of these are the same as we previously encountered, with the addition of the two new exogenous nominal variables, Mt and πe t+1. 14.3 Summary • Money is a store of value, unit of account, and medium exchange. The medium of exchange is the primary reason money is valuable as it allows people to avoid the double coincidence of wants problem. That is, one can exchange money for a good or service rather than bartering. • The medium of exchange motive is diﬃcult to model since we only have one good and a representative agent. As a shortcut, we assume the representative agent receives utility from holding real money balances. • The Fisher relationship says that the real interest rate is approximately equal to the nominal interest rate minus expected inﬂation. • A higher nominal interest rate increases the opportunity cost of holding money. Hence, money demand is decreasing in the nominal interest rate. Conversely, as income goes up, the household wants to make more exchanges which means the demand for money increases. • The government sells money to the household in period t and buys it back in t + 1. The rest of government and the entire ﬁrm optimization problem are exactly the same as in previous chapters. Key Terms • Store of value • Unit of account • Medium of exchange • Double coincidence of wants problem • Commodity-based money • Fiat money • Fisher relationship Questions for Review 329 1. Explain why bartering is ineﬃcient. 2. Explain some of the problems associated with commodity-based money. 3. Can the real interest rate be negative? Why? 4. Can the nominal interest rate be negative? Why? 5. In our model, households can save through bonds or money. If households do not receive utility from holding real money balances, how much will they save in money? 6. Write down the demand function for real money balances. How is it aﬀected by income and the nominal interest rate? 7. Derive the government’s intertemporal budget constraint. How is it diﬀerent than the intertemporal constraint in Chapter 13. Exercises 1. In our basic model with money, the money demand curve is implicitly deﬁned by: ) = φ′ (Mt Pt
(a) Suppose that the functional forms are as follows: φ ( Mt Pt and u(Ct) = ln Ct. The parameter θ is a positive constant. Write the money demand curve using these functional forms. ) = θ ln Mt Pt it 1 + it u′(Ct) The “quantity equation” is a celebrated identity in economics that says that the money supply times a term called “velocity” must equal nominal GDP: MtVt = PtYt Velocity, Vt is deﬁned as the number of times the average unit of money is used. Here’s the basic idea. Suppose that nominal GDP is 100 dollars, and that the money supply is ten dollars. If money must be used for all transactions, then it must be the case that velocity equals 10: Vt = PtYt 10. The quantity equation is an identity because it is deﬁned Mt to hold. We do not measure Vt in the data, but can back it out of the data given measurement on nominal GDP and the money supply. = 100 330 (b) Take your money demand expression you derived in part (a). Assume that Ct = Yt. Use this expression to derive the quantity equation. In terms of the model, what must Vt equal? (c) What is the relationship between the nominal interest rate, it, and your model-implied expression for velocity, Vt (i.e. take the derivative of Vt with respect to it and determine whether it is positive, zero, or negative). Given the way velocity is deﬁned conceptually (the number of times the average unit of money is used), explain why the sign of the derivative of Vt with respect to it does or does not make sense. 2. [Excel Problem] Assuming log utility, the basic consumption Euler equa- tion can be written: Ct+1 Ct = β(1 + rt) If we take logs of this, and use the approximation that the natural log of one plus a small number is approximately the small number, then we can write this as: rt = gC t+1 − ln β In other words, the real interest rate ought to equal the expected growth rate of consumption minus the log of the discount factor. (a) For the period 1947 through 2015, download annual data on the GDP price deﬂator (here), annual data on real consumption
growth (here), and data on the the 3-Month Treasury Bill rate (here, this series is available at a higher frequency than annual, so to get it in annual terms, click “edit graph” and modify frequency to annual using the average method). The approximate real interest rate is rt = it − πe t+1. Measure it by the 3-Month T-Bill rate and assume expected inﬂation equals realized inﬂation one period ahead (i.e. the interest rate observation in 1947 will be the 3-Month T-Bill in 1947, while you will use realized inﬂation in 1948 for expected inﬂation in 1947). Compute a series for the real interest rate. Plot this series. What is the average real interest rate? How often has it been negative? Has it been negative or positive recently? (b) What is the correlation between the real interest rate series you create 331 and expected consumption growth (i.e. compute the correlation between consumption growth in 1948:2015 and the real interest rate between 1947:2014)? Is the sign of this correlation qualitatively in line with the predictions of the Euler equation? Is this correlation strong? 3. [Excel Problem] In this problem, we will investigate the velocity of money in the data. (a) Download quarterly data on the money supply and nominal GDP. Do this for the period 1960-2015. Deﬁne the money supply as M2. You can get this from the St. Louis Fed Fred website. Simply go to the website, type “M2” into the search box, and it’ll be the ﬁrst hit. You’ll want to click on “Monthly, seasonally adjusted.” Then it’ll take you to a page and you can click “Download data” in the upper left part of the screen. There will be a box on that page labeled “Frequency.” You will want to click down to go to “quarterly” using “average” as the “aggregation method” (this is the default). To get the GDP data just type “GDP” into the search box. “Gross Domestic Product” will be the ﬁrst hit. You’ll want to make sure that you’re downloaded “Gross Domestic Product�
� not “Real Gross Domestic Product.” After you have downloaded these series, deﬁne log velocity as log nominal GDP minus the log money supply. Produce a plot of log velocity over time. (b) The so-called “Monetarists” were a group of economists who advocated using the quantity equation to think about aggregate economy policy. A central tenet of monetarism was the belief that velocity was roughly constant, and that we could therefore think about changes in the money supply as mapping one-to-one into nominal GDP. Does velocity look roughly constant in your time series graph? Are there any sub-periods where velocity looks roughly constant? What has been happening to velocity recently? (c) Download data on the three month treasury bill rate as a measure of the nominal interest. To get this, go to FRED and type “treasury bill” into the search box. The ﬁrst hit will be the “Three Month Treasury Bill, Secondary Market Rate.” Click on the “monthly” series, and then on the next page click “Download Data.” You will again need to change 332 the frequency to quarterly in the relevant box as you did above. Interest rates are quoted as percentages at an annualized frequency. To make the concept consistent with what is in the model, you will need to divide the interest rate series by 400 (dividing by 4 puts it into quarterly units, as opposed to annualized, and dividing by 100 gets it out of percentage units, so you are dividing by 4×100 = 400). Now, use your model implied money demand function from part (b) to derive a model-implied time series for velocity. Use the M2 series and the nominal GDP series, along with your downloaded measure of the interest rate, to create a velocity series. Assume that the parameter θ = 0.005. Produce a plot of the model-implied log velocity series. Does it look kind of like the velocity series you backed out in the data? What is the correlation between the model-implied log velocity series and the actual log velocity series you created in part (d)? Is the model roughly consistent with the data? 333 Chapter 15 Equilibrium Eﬃciency The conditions of the equilibrium model of production which we have been developing through Part III, expressed as supply and demand decision rules, are repeated below for convenience: Ct = C
d(Yt − Gt, Yt+1 − Gt+1, rt) Nt = N s(wt, θt) Nt = N d(wt, At, Kt) It = I d(rt, At+1, Kt) Yt = AtF (Kt, Nt) Yt = Ct + It + Gt Mt = PtM d(rt + πe t+1, Yt) rt = it − πe t+1 (15.1) (15.2) (15.3) (15.4) (15.5) (15.6) (15.7) (15.8) These decision rules come from ﬁrst order optimality conditions from the household and ﬁrm problems. These ﬁrst order conditions implicitly deﬁne the above decision rules. The ﬁrst order optimality conditions for the household are given below: uC(Ct, 1 − Nt) = β(1 + rt)uC(Ct+1, 1 − Nt+1) uL(Ct, 1 − Nt) = uC(Ct, 1 − Nt)wt v′ (Mt Pt uC(Ct, 1 − Nt) it 1 + it ) = (15.9) (15.10) (15.11) Equation (15.9) is the consumption Euler equation. It says that, when behaving optimally, the household ought to equate the current marginal utility of consumption, uC(Ct, 1 − Nt), to the discounted marginal utility of next period’s consumption, βuC(Ct+1, 1 − Nt+1), times the gross real interest rate. This ﬁrst order condition, when combined with the household’s budget 334 constraint, implicitly deﬁnes the consumption function, (15.1), which says that consumption is an increasing function of current and future perceived net income and a decreasing function of the real interest rate. (15.10) is the ﬁrst order conditions for optimal labor supply, which would look the same (only with t+1 subscripts) in the future. This says to equate the marginal rate of substitution between leisure and consumption (the ratio of uL/uC) to the relative price of leisure in terms of
consumption, which is the real wage. This condition implicitly deﬁnes labor supply. Labor supply is increasing in the real wage (under the assumption that the substitution eﬀect dominates) and decreasing in an exogenous variable θt, which can be interpreted as a parameter of the utility function governing how much utility the household gets from leisure. (15.11) is the ﬁrst order condition for money holdings, and implicitly deﬁnes the money demand function, (15.7). The ﬁrst order optimality conditions coming out of the ﬁrm’s proﬁt maximization problem are: wt = AtFN (Kt, Nt) wt+1 = At+1FN (Kt+1, Nt+1) rt + δ = At+1FK(Kt+1, Nt+1) (15.12) (15.13) (15.14) Expressions (15.12)-(15.13) are the ﬁrm’s optimality conditions for the choice of labor. These say to hire labor up until the point at which the real wage equals the marginal product of labor. These expressions are identical for period t and t + 1. These implicitly deﬁne the labor demand function, (15.3). Labor demand is decreasing in the real wage, increasing in current productivity, and increasing in the current capital stock. Expression (15.14) is the ﬁrst order optimality condition for the choice of next period’s capital stock. This implicitly deﬁnes the investment demand function, (15.4). Investment demand is decreasing in the real interest rate, increasing in future productivity, and decreasing in Kt. In the equilibrium, the household and ﬁrm take rt, wt, it, and Pt as given – i.e. they behave as price-takers, and their decision rules are deﬁned as functions of these prices. In equilibrium, these price adjust so that markets clear when agents are behaving according to their decision rules. 15.1 The Social Planner’s Problem In a market economy, prices adjust to equilibrate markets. Does this price adjustment bring about socially desirable outcomes? We explore this question in this section. Let us suppose that there exists a hypothetical social planner. This social planner is 335 benevolent and chooses
allocations to maximize the lifetime utility of the representative household, subject to the constraints that aggregate expenditure not exceed aggregate production each period. The social planner’s problem is also constrained by the capital accumulation equation. The question we want to examine is the following. Would this benevolent social planner choose diﬀerent allocations than the ones which emerge as the equilibrium outcome of a market economy? The objective of the social planner is to maximize the lifetime utility of the representative household. Lifetime utility is: U = u(Ct, 1 − Nt) + v (Mt Pt The planner faces a sequence of two resource constraints. Noting that Kt+1 = It + (1 − δ)Kt, and that It+1 = −(1 − δ)Kt+1 to impose zero left over capital, these resource constraints can be written: ) + βu(Ct+1, 1 − Nt+1) (15.15) Ct + Kt+1 − (1 − δ)Kt + Gt ≤ AtF (Kt, Nt) Ct+1 − (1 − δ)Kt+1 + Gt+1 ≤ At+1F (Kt+1, Nt+1) (15.16) (15.17) The social planner’s problem consists of choosing quantities to maximize (15.15) subject to (15.16)-(15.17). Note that there are no prices in the social planner’s problem (other than the presence of Pt, to which we shall return more below). The hypothetical planner directly chooses quantities, unlike the market economy which relies on prices to equilibrate markets. We will consider a couple of diﬀerent versions of the social planner’s problem. In the ﬁrst, we will treat Mt, Gt, and Gt+1 as given and not things which the planner can control. In this scenario, the planner gets to choose Ct, Ct+1, Nt, Nt+1, and Kt+1 (which in turn determines It), taking the money supply, the sequence of government spending, and other exogenous variables as given. Then we will do a version wherein the planner can also choose the money supply. This will permit a discussion of the optimal supply of money (at least in a model with no other frictions). Finally, we will discuss ways in which to modify the problem so that
government spending is beneﬁcial. 15.1.1 The Basic Planner’s Problem The social planner’s problem, taking Mt, Gt, and Gt+1 as given, is: max Ct,Ct+1,Nt,Nt+1,Kt+1 U = u(Ct, 1 − Nt) + v ( Mt Pt ) + βu(Ct+1, 1 − Nt+1) (15.18) s.t. 336 Ct + Kt+1 − (1 − δ)Kt + Gt ≤ AtF (Kt, Nt) Ct+1 − (1 − δ)Kt+1 + Gt+1 ≤ At+1F (Kt+1, Nt+1) (15.19) (15.20) This is a constrained optimization problem with two constraints. Assume that each constraint holds with equality, and solve for Ct and Ct+1 in terms of other variables in each constraint: Ct = AtF (Kt, Nt) − Kt+1 + (1 − δ)Kt − Gt Ct+1 = At+1F (Kt+1, Nt+1) + (1 − δ)Kt+1 − Gt+1 (15.21) (15.22) Now plug (15.21) and (15.22) into (15.18), turning the problem into an unconstrained one: max Nt,Nt+1,Kt+1 U = u (AtF (Kt, Nt) − Kt+1 + (1 − δ)Kt − Gt, 1 − Nt) + v ( Mt Pt ) +... ⋅ ⋅ ⋅ + βu (At+1F (Kt+1, Nt+1) + (1 − δ)Kt+1 − Gt+1, 1 − Nt+1) (15.23) Take the derivatives with respect to the remaining choice variables. When doing so, we will abbreviate the partial derivatives with respect to Ct and Ct+1 using just Ct or Ct+1, but one must use the chain rule when taking these derivatives, in the process making use of (15.21)-(15.22). ∂U ∂Nt = uC(
Ct, 1 − Nt)AtFN (Kt, Nt) − uL(Ct, 1 − Nt) ∂U ∂Nt+1 = βuC(Ct+1, 1 − Nt+1)At+1FN (Kt+1, Nt+1) − βuL(Ct+1, 1 − Nt+1) = −uC(Ct, 1 − Nt) + βuC(Ct+1, 1 − Nt+1) (At+1FK(Kt+1, Nt+1) + (1 − δ)) ∂U ∂Kt+1 Setting these derivatives equal to zero and simplifying yields: uL(Ct, 1 − Nt) = uC(Ct, 1 − Nt)AtFN (Kt, Nt) uL(Ct+1, 1 − Nt+1) = uC(Ct+1, 1 − Nt+1)At+1FN (Kt+1, Nt+1) 1 = βuC(Ct+1, 1 − Nt+1) uC(Ct, 1 − Nt) [At+1FK(Kt+1, Nt+1) + (1 − δ)] (15.24) (15.25) (15.26) (15.27) (15.28) (15.29) Expressions (15.27)-(15.29) implicitly characterize the optimal allocations of the planner’s problem. How do these compare to what obtains as the outcome of the decentralized 337 equilibrium? To see this, combine the ﬁrst order conditions for the household, (15.9)-(15.10), with the ﬁrst order conditions for the ﬁrm, (15.12)-(15.14). Do this in such a way as to eliminate rt, wt, and wt+1 (i.e. solve for wt from the ﬁrm’s ﬁrst order condition, and then plug that in to the household’s ﬁrst order condition wherever wt shows up). Doing so yields: uL(Ct, 1 − Nt) = uC(Ct, 1 − Nt)AtFN (Kt, Nt) uL(Ct
+1, 1 − Nt+1) = uC(Ct+1, 1 − Nt+1)At+1FN (Kt+1, Nt+1) uC(Ct, 1 − Nt) = βuC(Ct+1, 1 − Nt+1) [At+1FK(Kt+1, Nt+1) + (1 − δ)] (15.30) (15.31) (15.32) This emerges because wt = AtFN (Kt, Nt) and wt+1 = At+1FN (Kt+1, Nt+1) from the ﬁrm’s from the household’s problem. One ought to notice that problem, and 1 1+rt (15.27)-(15.29) are identical to (15.30)-(15.32). = βuC (Ct+1,1−Nt+1) uC (Ct,1−Nt) The fact that the optimality conditions of the benevolent planner’s problem are identical to those of the decrentalized competitive equilibrium once prices are eliminated means that a benevolent social planner can do no better than the private economy left to its own device. In a sense this is a formalization of Adam Smith’s laissez fair idea – a private economy left to its own devices achieves a Pareto eﬃcient allocation, by which is meant that it would not be possible to improve upon the equilibrium allocations, taking as given the scarcity embodied in the resource constraints and the exogenous variables. In modern economics this result is formalized in the First Welfare Theorem. The First Welfare theorem holds that, under some conditions, a competitive decentralized equilibrium is eﬃcient (in the sense of coinciding with the solution to a benevolent planner’s problem). These conditions include price-taking behavior (i.e. no monopoly), no distortionary taxation (the taxes in our model are lump sum in the sense of being independent of any actions taken by agents, whereas a distortionary tax is a tax whose value is a function of actions taken by an agent, such as a labor income tax), and perfect ﬁnancial markets. These conditions are satisﬁed in the model with which we have been working. The result that the First Welfare Theorem holds in our model has a very important implication.
It means that there is no role for economic policy to improve upon the decentralized equilibrium. Activists policies can only make things worse. There is no justiﬁcation for government policies (either monetary or ﬁscal). This was (and is) a controversial idea. We will discuss in more depth these implications and some critiques of these conclusions in Chapter 22. 338 15.1.2 Planner Gets to Choose Mt Now, let us consider a version of the hypothetical social planner’s problem in which the planner gets to choose Mt, in addition to Ct, Ct+1, Nt, Nt+1, and Kt+1. We continue to consider Gt and Gt+1 as being exogenously ﬁxed. The revised version of the problem can be written: max Ct,Ct+1,Nt,Nt+1,Kt+1,Mt U = u(Ct, 1 − Nt) + v ( Mt Pt ) + βu(Ct+1, 1 − Nt+1) (15.33) s.t. Ct + Kt+1 − (1 − δ)Kt + Gt ≤ AtF (Kt, Nt) Ct+1 − (1 − δ)Kt+1 + Gt+1 ≤ At+1F (Kt+1, Nt+1) (15.34) (15.35) This is the same as we had before, except now Mt is a choice variable. We can proceed in characterizing the optimum in the same way. Solve for Ct and Ct+1 in the constraints and transform the problem into an unconstrained one: max Nt,Nt+1,Kt+1,Mt U = u (AtF (Kt, Nt) − Kt+1 + (1 − δ)Kt − Gt, 1 − Nt) + v ( Mt Pt ) +... ⋅ ⋅ ⋅ + βu (At+1F (Kt+1, Nt+1) + (1 − δ)Kt+1 − Gt+1, 1 − Nt+1) (15.36) Because Mt enters utility in an additive way from Ct and Nt, and because it does not appear in the constraints, the ﬁrst order conditions with respect to
Nt, Nt+1, and Kt+1 are the same as above, (15.27)-(15.29). The remaining ﬁrst order condition is with respect to money. It is: ∂U ∂Mt Setting this derivative equal to zero implies, for a ﬁnite price level, that: = v′ ( Mt Pt ) 1 Pt v′ (Mt Pt ) = 0 (15.37) (15.38) In other words, the social planner would like to set the marginal utility of real balances equal to zero. While this condition may look a little odd, it is just a marginal beneﬁt equals marginal cost condition. v′ ( Mt ) is the marginal beneﬁt of holding money. What is the Pt marginal cost? From the planner’s perspective, there is no marginal cost of money – money is 339 literally costless to “print.” This diﬀers from the household’s perspective, where the marginal cost of money is foregone interest on bonds. To get the marginal utility of real balances to go to zero, the quantity of real balances must go to inﬁnity. Again, in a sense this is quite intuitive. If it is costless to create real balances but they provide some beneﬁt, why not create an inﬁnite amount of real balances? When we compare (15.38) with (15.11), we see that in general the planner’s solution and the equilibrium outcome will not coincide. So while the equilibrium allocations of consumption, labor, and investment will be eﬃcient, in general there will be an ineﬃcient amount of money in the economy. There is one special circumstance in which these solutions will coincide, however. This is when it = 0 – i.e. the nominal interest rate is zero. If it = 0, then (15.11) holding requires that v′ ( Mt Pt ) = 0, which then coincides with the planner’s solution. What real world implication does this have? It suggests that if a government (or more speciﬁcally a central bank) wants to maximize household welfare, it should conduct monetary policy to be consistent with a nominal interest rate of zero. This kind of policy is called the “Friedman Rule” after Nobel Prize winning economist Milton Friedman. Friedman
’s essential argument was that a positive nominal interest rate means that the private cost of holding money exceeds the public cost of creating additional money. At an optimum, the private cost should be brought in line with the public cost, which necessitates an interest rate of zero. Since the approximate Fisher relationship is given by rt = it − πe t+1, if it is negative and rt is positive, it must be that expected inﬂation is negative. While we have taken expected inﬂation as given, over long periods of time one might expect the expected rate of inﬂation to equal average realized inﬂation. If the real interest rate is positive on average over long periods of time, implementation of the Friedman rule therefore requires deﬂation (continuous decreases in the price level). One might ask an obvious question: if the Friedman rule is optimal, then why don’t we observe central banks implementing it? For most of the last 60 or 70 years, nominal interest rates in the US and other developed economies have been positive, as have inﬂation rates. Only recently have nominal interest rates gone down toward zero, and this has been considered a problem to be avoided by central bankers and other policymakers. As we will discuss later in Part V, in particular Chapter 29, in the short run the central bank may want to adjust the money supply (and hence the nominal interest rate) to stabilize the short run economy. Doing so requires the ﬂexibility to lower interest rates. For reasons we will discuss further in Chapter 29, nominal interest rates cannot go negative (or at least cannot go very negative). Implementation of the Friedman rule would give a central bank no “wiggle room” to temporarily cut interest rates in the short run. For this reason, most central banks have decided that the Friedman rule is too strong a prescription for a modern economy. 340 However, one can nevertheless observe that central banks do evidently ﬁnd it desirable to not veer too far from the Friedman rule. Most central banks prefer low inﬂation rates and low nominal interest rates; countries with very high inﬂation rates tend to have poor economic performance. That most central banks prefer low inﬂation and nominal interest rates suggest that there is some real-world logic in the Friedman rule, although in its strict form it is too strong. 15.1.3 Planner Gets to Choose Gt and G
t+1 Let us now take our analysis of a hypothetical social planner’s problem a step further. Whereas we have heretofore taken Gt and Gt+1 as given, let us now think about how a planner would optimally choose government expenditure. As written, the planner’s problem of choosing Gt and Gt+1 ends up being trivial – the planner would seek out a corner solution in which Gt = 0 and Gt+1 = 0. Why? As we have written down the model, there is no beneﬁt from government spending. Government spending in the model is completely wasteful in the sense that higher Gt reduces Ct and It, without any beneﬁt. As such, the planner would want to have Gt = 0. This is obviously not a good description of reality. While one can argue about the optimal size of government spending, it is surely the case that there are at least some beneﬁts to government expenditure. From a micro perspective, government expenditure is useful to the extent to which it resolves “public good” problems. Public goods are goods which are both non-excludable and non-rivalrous. Non-excludability means that it is diﬃcult or impossible to exclude people from using a good once it has been produced. Non-rivalrous means that use of a good by one person does not prohibit another from using the good. A classic example of a public good is military defense. If there is a town with 100 people in it protected by an army, it is diﬃcult to use the army to defend the 50 people in the town who are paying for the army while not defending the 50 people who are not paying. Rather, if the army provides defense, it provides defense to all the people in the town, whether they pay for it or not. Likewise, 50 of the people in the town enjoying the defense provided by the army does not exclude the other 50 people in the town from also enjoying that defense. Other examples of public goods are things like roads, bridges, and parks. While it may be possible to exclude people from using these (one can charge a toll for a road or an entry fee for a park), and while these goods may not be strictly non-rivalrous (a ton of people on the road may make it diﬃcult for someone else to drive onto the road), in practice goods like these have characteristics similar to a strict public
good like military defense. Public goods will be under-provided if left to private market forces. Because of the non-excludability, private 341 ﬁrms will not ﬁnd it optimal to produce public goods – why produce something if you can’t make people pay for it? Governments can step in and provide public goods and therefore increase private welfare. As with our discussion of money, it is not an easy task to model in a compelling yet tractable way the public good provision problem in a macroeconomic model. As with money, it is common to take short cuts. In particular, it is common to assume that the household receives utility from government spending. In particular, let h(⋅) be a function mapping the quantity of government expenditure into the utility of a household, where it is assumed that h′(⋅) > 0 and h′′(⋅) < 0. Let household lifetime utility be given by: U = u(Ct, 1 − Nt) + v ( Mt Pt ) + h(Gt) + βu(Ct+1, 1 − Nt+1) + βh(Gt+1) (15.39) In (15.39), in each period the household receives a utility ﬂow from government spending. The future utility ﬂow is discounted by β. The exact form of the function h(⋅) is not particularly important (other than that it is increasing and concave), though it is important that utility from government spending is additively separable with respect to utility from other things. Additive separability means that the solution to the household’s optimization problem (or the planner’s optimization problem) with respect to non-government spending variables is the same whether the household gets utility from government spending or not. What this means is that ignoring utility from government spending, as we have done to this point, does not aﬀect the optimality conditions for other variables. The budget constraints faced by the planner are unaﬀected by the inclusion of utility from government spending in the speciﬁcation of lifetime utility. We can write the modiﬁed unconstrained version of the optimization problem (after substituting out the constraints) as: max Nt,Nt+1,Kt+1,Mt U = u (AtF (Kt, Nt) − Kt+
1 + (1 − δ)Kt − Gt, 1 − Nt) + v ( Mt Pt ⋅ ⋅ ⋅ + βu (At+1F (Kt+1, Nt+1) + (1 − δ)Kt+1 − Gt+1, 1 − Nt+1) + βh(Gt+1) (15.40) ) + h(Gt) +... The optimality conditions for the planner with respect to Nt, Nt+1, Kt+1, and Mt are the same as above. The new optimality conditions with respect to the choices of Gt and Gt+1 are: ∂U ∂Gt = −uC(Ct, 1 − Nt) + h′(Gt) ∂U ∂Gt+1 = −βuC(Ct+1, 1 − Nt+1) + βh′(Gt+1) (15.41) (15.42) 342 Setting these equal to zero and simplifying yields: h′(Gt) = uC(Ct, 1 − Nt) h′(Gt+1) = uC(Ct+1, 1 − Nt+1) (15.43) (15.44) Expressions (15.43)-(15.44) say that, an optimum, government spending should be chosen so as to equate the marginal utility of government spending with the marginal utility of private consumption. We can think about these optimality conditions as also representing marginal beneﬁt equals marginal cost conditions. h′(Gt) is the marginal beneﬁt of extra government expenditure. What is the marginal cost? From the resource constraint, holding everything else ﬁxed, an increase in Gt requires a reduction in Ct. Hence, the marginal utility of consumption is the marginal cost of extra government spending. At an optimum, the marginal beneﬁt ought to equal the marginal cost. The optimality condition looks the same in both period t and t + 1. Let us relate these optimality conditions back to the simpler case where the benevolent planner takes Gt and Gt+1 as given. In that case, the planner would choose the same allocations that emerge as the outcome of a decentralized equilibrium. As such, the equilibrium outcome is eﬃcient and
there is no role for changing Gt or Gt+1 in response to changing exogenous variables. That is not going to be the case when Gt and Gt+1 can be optimally chosen by the planner. In particular, (15.43)-(15.44) indicate that government spending ought to move in the same direction as consumption. In periods when consumption is high, the marginal utility of consumption is low. Provided h′′(⋅) < 0, this means that government spending ought to be adjusted in such a way as to make the marginal utility of government spending low, which requires increasing Gt. The opposite would occur in periods in which Ct is low. Put another way, procylical ﬁscal policy is optimal (by procyclical we mean moving government expenditure in the same direction as consumption) when the beneﬁts of government expenditure are modeled in this way. Note that this runs counter to conventional wisdom, which holds that government spending ought to be high in periods where the economy is doing poorly (i.e. periods in which consumption is low). 15.2 Summary • In the production economy, prices adjust to simultaneously equilibrate all markets. As it turns out, this equilibrium is eﬃcient in the sense that the allocations correspond to the allocations a social planner would decide if his or her goal was to maximize the 343 representative household’s present discounted value of lifetime utility. • The result that the competitive equilibrium is eﬃcient implies that a social planer cannot do a superior job relative to the market economy in allocating resources. We say that the competitive equilibrium is Pareto eﬃcient which means the allocations cannot be reallocated by a social planner to improve welfare. • This is an example of the First Welfare Theorem which says under conditions of perfect competition, no distortionary taxation, and perfect ﬁnancial markets, all competitive equilibrium are eﬃcient. An implication is that activist policy, whether monetary or ﬁscal, can only reduce welfare. • The marginal beneﬁt to holding real money balances is positive and the marginal cost to printing money is zero. Therefore, the social planner would like expand real money balances as much as possible. This condition can be implemented by setting the nominal interest rate equal to 0. Provided the long-run real interest rate is positive, this implies the long-run rate of inﬂation should be
negative. This is called the Friedman rule. • Until this chapter, we have assumed government spending is neither productive nor provides people utility. These assumptions imply the optimal level of government spending is zero. On the other hand, if people receive utility from government spending then the social planner would equate the marginal utility of consumption and the marginal utility of government spending. Somewhat paradoxically, this implies that government spending should be procyclical. That is, the social planner would raise government spending in booms and decrease spending during recessions. Key Terms • Pareto eﬃcient allocation • First Welfare Theorem • Friedman rule • Public goods Questions for Review 1. What does it mean for allocations to be Pareto eﬃcient? What are the policy implications? 2. Explain the economic logic of the Friedman rule. What does the Friedman rule imply about the time path of prices? 344 3. What assumptions imbedded in the Neoclassical model are essential for the Pareto optimality of the allocations? 4. If the representative household receives utility from government expenditures, should a benevolent government increase or decrease expenditures during recessions? Exercises 1. In the text we have assumed the representative agent does not derive utility from government expenditures. Instead, consider the one period problem where the representative agent derives utility from consumption and government spending U = u(C) + v(G) Both u(C) and v(G) are increasing and concave. The household is exogenously endowed with Y. Since this is a one period model, the government balances its budget in every period. Once the government chooses a level of expenditure, the representative agent consumes whatever remains of the endowment. Hence, we can think about this problem as one where the government chooses the level of government spending and consumption to maximize the represntative agent’s utility function. Formally, the problem is u(C) + v(G) max C,G s.t. C + G = Y. (a) Write this as an unconstrained problem where the government chooses G. (b) Derive the ﬁrst order condition. (c) Suppose u(C) + v(G) = ln C + ln G. Solve for the optimal levels of G and C. (d) If the economy is in a recession (i.e. low Y ), should a benevolent government increase or decrease government expenditures? What is the economic intuition for this?
2. One of the assumptions that goes into the Neoclassical model is that there are no externalities. Here we discard that assumption. Suppose that the process for turning output into productive capital entails damage to the environment of Dt = φ(It) where φ(It) > 0 provided It > 0 and I ′ > 0, I ′′ > 0. 345 We assume this cost provides disutility to the consumer so that the present discounted value of utility is U = u(Ct) − Dt + βu(Ct+1) where we have assumed labor is not a factor of production. Note that the household only receives disutility in the ﬁrst period since investment is negative in the second period (we also assume a parameter restriction such that It > 0 is optimal in period t). The production function is Yt = AtF (Kt). The capital accumulation equations are Kt+1 = It + (1 − δ)Kt Kt+2 = It+1 + (1 − δ)Kt+1 The terminal condition continues to be Kt+2 = 0. The market clearing condition is Yt = Ct + It. (a) Formulate this as a social planner’s problem in which the only choice variable is Kt+1. (b) Derive the ﬁrst order condition on Kt+1. (c) If ﬁrms do not account for the environmental damage will there be too much or too little investment? Prove this by ﬁnding the ﬁrst order condition of the ﬁrm’s proﬁt maximization problem. (d) How might an activist government restore the Pareto optimal allocation? Be as speciﬁc as possible. 346 Chapter 16 Monopolistic Competition In the previous chapter we showed that, with some assumptions, the competitive equilibrium is eﬃcient. One of the assumptions was that all actors in the economy are price takers. In reality, of course, this is violated. In this chapter we consider the consequences of ﬁrms having market power. 16.1 The Microeconomics of Monopoly You likely studied monopolies in a microeconomics class. But as a reminder, a monopoly is when one ﬁrm supplies the entire market. Let’s say a ﬁrm is a monopoly that sells a
product Q, at price, P, and for a cost C(Q). We assume that C(0) = 0, C ′(Q) > 0, and C ′′(Q) ≥ 0. The ﬁrst assumption says that a ﬁrm only incurs a cost if it produces a positive amount of output. C ′(Q) has the interpretation of marginal cost which we assume is always positive and nondecreasing. The critical diﬀerences between a monopoly and a perfectly competitive ﬁrm is that the monopoly knows that producing more a product reduces its price whereas the perfect competitor understands that its production has no eﬀect on price. To be concrete about this, let’s start with the proﬁt maximization of the perfect competitor which takes P as a parameter. The problem is The ﬁrst order condition is Π = P Q − C(Q) max Q ∂Π ∂Q = P − C ′(Q) = 0 (16.1) This has the interpretation that the ﬁrm produces up to a point where price equals marginal cost. Now, let’s contrast this with the proﬁt maximization problem for the monopolist. Assume that the ﬁrm faces an inverse demand curve of P (Q) where P ′(Q) < 0. This is an “inverse” demand curve because we write price as a function of quantity rather than quantity as a function of price. The assumption that P ′(Q) < 0 simply says that the demand curve is 347 downward sloping. The proﬁt maximization problem for the monopolist is P (Q)Q − C(Q) max Q The ﬁrst order condition is ∂Π ∂Q = P ′(Q)Q + P (Q) − C ′(Q) = 0 (16.2) The term P ′(Q)Q + P (Q) is marginal revenue, i.e. the revenue that the monopolist earns for producing an additional unit of output. The “direct eﬀect” of producing an additional unit is P, but the monopolist understands that producing another unit reduces the price that can be charged on all units of output. Thus, the “indirect eﬀect” is
P ′(Q)Q. Since P ′(Q)Q < 0, the monopolist produces less output that the perfect competitor. How much less? Recall that the elasticity of output with respect price is the percent change in quantity, for a percent change in price. Mathematically, the elasticty, is Q′(P ) P Q. It follows that, the inverse elasticity is P ′(Q) P. We assume that < −1 so that a one Q percent increase in price leads to a more than one percent decrease in quantity. Rewrite (16.2) as = 1 P ′(Q)Q + P (Q) = C ′(Q) ⇔ + 1] = C ′(Q) ⇔ P (Q) [ 1 P (Q) C ′(Q) P (Q) − C ′(Q) C ′(Q) ⇔ = = 1 + −1 1 + The term on the right-hand side is the “price markup”, i.e. the percent price is above marginal cost. The closer is to negative one, the bigger is the markup. When → −∞, the markup goes to zero. Intuitively, if a small increase in price leads to a huge drop in quantity demanded, the ﬁrm will price at marginal cost. This limiting case corresponds exactly to the case of perfect competition since the perfectly competitive ﬁrm, in eﬀect, faces a horizontal demand curve. Outside of the limit case, the monopolist will underproduce output and sell at a price above marginal cost. 348 Figure 16.1: Perfect Competition Versus Monopoly Figure 16.1 depicts the solution graphically when marginal cost is equal to a constant. Since selling additional units decreases the price of existing units, the marginal revenue line lies below the demand curve (which can also be interpreted as average revenue). The monopoly sets marginal revenue equal to marginal cost and produces at level Qmon which is below Qcomp, the perfectly competitive equilibrium quantity. The monopolist captures Qmon(Pmon − Pcomp) as proﬁts. The triangle with base Qcomp − Qmon is deadweight loss. In this area, consumers are willing to pay above the marginal cost of production to buy the product, but the monopolist ﬁnds it optimal not to produce since doing so would require them to reduce the price on
every other unit. This lost surplus is the eﬃciency cost of the monopoly.1 We have now covered the perfectly competitive case and monopoly. Does this bring us any closer to the real world? In reality, both of these stark cases are rare. Think about the market for cars. As a consumer you can go to any number of ﬁrms (e.g. Ford, Honda, Subaru) which compete on price which means the car market is not dominated by a monopoly. At the same time, each of these ﬁrms produce slightly diﬀerent products to appeal to all sorts of customers. This product diﬀerentiation gives each ﬁrm some market power which means 1We are ignoring all forms of price discrimination. 349 𝑃 𝑄 𝑀𝐶 𝐷 𝑀𝑅 𝑃’()* 𝑄’()* 𝑄)(+ 𝑃)(+ DWL the market can not really be characterized as one of perfect competition either. This leads to an “in between” case which is called monopolistic competition. We discuss monopolistic competition in the next section and incorporate it into a general equilibrium model. 16.2 A General Equilibrium Model of Monopolistic Competition The monopolistic competition model assumes that there are many ﬁrms in a market each of which has a diﬀerentiated product and that these products are substitutes (think of the market for cars). A critical assumption is that each ﬁrm only considers how setting its price aﬀects its own sales rather than the sales of its competitors. Continuing with the car example, we assume that Subaru understands that a lower price means they will sell more cars, but does not take into account how this pricing decision will aﬀect the pricing decisions of Ford and Honda. Models of oligopoly which incorporate strategic decision making by ﬁrms are signiﬁcantly more diﬃcult to work with in the context of macroeconomic models but are very popular (and insightful) in the ﬁeld of industrial organization. We make a few simplifying assumptions relative to the perfectly competitive model we discussed in Chapter 12. First, we assume that there is only one period and that labor is the only factor of production. This allows us ignore capital accumulation and consumption smoothing
. We also assume that there is no money and no government spending. Adding any of these ingredients does not signiﬁcantly change the results. 16.2.1 Households The household problem, if anything, is simpler than the one in Chapter 12. Households get utility from a consumption good, Ct and leisure, 1 − Nt. The utility function is u(Ct, 1 − Nt) and, as before, assume that utility is increasing (at a diminishing rate) with respect to each argument. The household’s income includes wage income, wtNt, and proﬁts from owning ﬁrms, Dt. Formally, their utility maximization problem is max Ct,Nt U = u(Ct, 1 − Nt) s.t. Ct = wtNt + Dt Just as in Chapter 12, the household supplies labor up to the point where the marginal rate of substitution of leisure for consumption equals the wage rate. uL(Ct, 1 − Nt) = wtuC(Ct, 1 − Nt) (16.3) 350 This gives rise to the labor supply curve, Nt = N s(wt, θt) where, as before, θt is an exogenous labor supply shifter. 16.2.2 Production Part I: The Product Bundler The idea of monopolistic competition is that there are many ﬁrms selling diﬀerentiated products. This requires us to step away from the one-ﬁrm, one-good model. Instead, we assume that there is a continuum of ﬁrms with total output given by −1 Mt −1 j,t dj) y Yt = (∫ 0 This integral plus the language of a “continuum of ﬁrms” may look intimidating. It helps to think of an integral as a sum. In other words, think of it as, Yt = Mt ∑ j=0 yj,t = y1,t + y2,t +... + yNt,t (16.4) In the sum, the distance between the j’s is an integer. That is, we go from j = 1 to j = 2, and so on. With an integral, we assume that there is that the distance between the j’s goes to 0. Figure 16.3 shows the intuition
on the number line. The discrete interval goes from j = 1 to j = 2, etc. while the continuum is the entire line. The integral simply “adds up” over the entire continuum. 351 Figure 16.2: Continuum Versus an Interval The parameter governs how substitutable these products are. The bigger is the more substitutable are the products. We assume that > 1. For now, we ﬁx Mt = 1 which sets the number of ﬁrms exogenously. After analyzing the model with a ﬁxed number of ﬁrms, we endogenize entry. It turns out that the model is easier to analyze if we assume the existence of a “product bundler” which aggregates the output of all the j’s according to (16.4) and sells them as a ﬁnal consumption good. An equally valid, and perhaps more intuitive approach, is to assume that the households directly purchase each of the j products. Both approaches give the same solution but working with the “product bundler” is a bit easier to work with mathematically. Each yj,t is sold at a yet to be determined price, pj,t. The problem of the bundler is to choose yj,t to maximize revenue minus costs. Formally, the problem is max yj,t (∫ 0 1 −1 j,t dj) y −1 − ∫ 1 0 pj,tyj,tdj The ﬁrst order condition is −1 j,t y (∫ 0 1 −1 j,t dj) y 1 −1 − pj,t = 0. 352 𝑗=1 𝑗=2 𝑗=3 This can be simpliﬁed to or 1 −1 j,t Y t y − pj,t = 0 yj,t = Ytp− j,t. (16.5) (16.5) is the demand curve for product j. Demand for yj,t is decreasing in its price and increasing in aggregate output. Since aggregate output is equal to income, we can say that yj,t is increasing in income. We assume that the product bundler is a perfect competitor and therefore makes no proﬁt. This means, 1 Yt = ∫ 0 pj,tyj,tdj Substit
uting in the demand curve for ﬁrm j gives Yt = ∫ 1 0 Ytp1− j,t dj. Pulling the Yt out of the inegral and cancelling gives 1 = ∫ 1 0 p1− j,t dj (16.6) 16.2.3 Production Part II: Intermediate Good Firms Each yj,t is produced by a ﬁrm hiring labor using the production function yj,t = Atnj,t. While it is feasible for ﬁrms to hire diﬀerent amounts of labor, they all produce with a common labor enhancing productivity term, At. Each unit of labor costs wt. Since labor is the only factor of production, the total cost, T C of producing yj,t units of output is T Cj,t = wtnj,t = wt yj,t At. The marginal cost is the cost of producing one additional unit of output. This can be found by taking the derivative of the total cost function with respect to yj,t. M Cj,t = ∂T Cj,t ∂yj,t = wt At Consequently, the marginal cost is the same across all ﬁrms, i.e. M Cj,t = M Ct and is equal to a constant. Proﬁts of the intermediate goods ﬁrm are given by dj,t = pj,tyj,t − mctyj,t 353 Each intermediate goods ﬁrm chooses pj,t and yj,t to maximize proﬁts. Remember though, this is not an unconstrained problem. Each ﬁrm only sells the quantity demanded at a given price. In other words, the ﬁrm chooses price and quantity subject to being on their demand curve. Formally, max pj,t,yj,t dj,t = pj,tyj,t − mctyj,t s.t. yj,t = Ytp− j,t Substituting in the constraint gives max pj,t Dj,t = Ytp1− j,t − mctYtp− j,t. The ﬁrst order condition is (1 − )Ytp− j,t + Ytp−−1 j,t = 0. Rearranging and cancelling Yt gives p
j,t = − 1 mct. (16.7) There are two insights here. One is that each ﬁrm chooses the same price since −1 mct is independent of j. Second, price is a markup over marginal cost. The bigger is, the smaller is the markup. As → ∞, price converges to marginal cost. This is intuitive since a very large means these products are very substitutable. If the products are close to perfect substitutes, ﬁrms must charge lower markups to stay competitive. Finally, given that all ﬁrms charge the same markup, we can return to the demand curve for intermediate good j to derive output. yj,t = p− j,tYt = − 1 mctYt Hence, all ﬁrms produce the same amount of output. Since output is the same across ﬁrms, they must also hire the same amount of labor so nj,t = nt. 354 16.2.4 Aggregation and Equilibrium The endogenous variables are Yt, Ct, Nt, mct, wt, and pt. In equilibrium, labor supply must equal labor demand. Labor supply is simply Nt. Labor demand is 1 ∫ 0 1 nj,tdj = ∫ 0 = nt ∫ ntdj 1 dj 0 = nt = Nt Thus, the amount of labor hired by each ﬁrm equals the total amount of labor supplied in the economy. This may sound puzzling, but it’s a consequence of ﬁxing the measure of ﬁrms at 1, i.e. Mt = 1. The integral is essentially a weighted average and each ﬁrm is “small” and thus receives a small weight. Aggregate output is given by Yt = (∫ 1 1 0 −1 −1 j,t dj) y (Atnt) −1 dj) −1 = (∫ 0 = AtNt The dividends issued by ﬁrm j to the household are dj,t = pj,tyj,t − mctyj,t = yj,t(pj,t − mct) mct = yj,t = Yt (mct − 1 − mct − 1 ) which is also independent of j. This means that ∫ 1 derive an expression for marginal cost. 0 dj,td
j = dt ∫ 1 0 dj = Dt. Return to (16.6) to 1 1 = ∫ p1− j,t dj 1 0 = p1− t ∫ = ( mct − 1 0 dj 1− ) 355 Solving for mct gives mct = − 1 (16.8) Going to the ﬂow budget constraint of the household, we have 1 Ct = ∫ 0 wtnj,t + ∫ 1 0 dj,tdj ) − mct − 1 = wtNt + Dt = mctAtNt + Yt ( mct − 1 = mctAtNt (1 + 1 ) − 1 − 1 = mctAtNt = AtNt Thus, total output equals total expenditure. The equilibrium can be summarized by the six equations, mct Nt = N s(wt, θt) pt = − 1 mct = − 1 wt = Atmct Yt = AtNt Yt = Ct (16.9) (16.10) (16.11) (16.12) (16.13) (16.14) The ﬁrst equation is the labor supply curve. The second is the optimal pricing rule by the intermediate goods ﬁrm. The third is marginal cost and the fourth is the equilibrium wage rate. The ﬁfth describes total output and the sixth says output equals expenditure. 16.2.5 Example Suppose U = Ct + θt ln(1 − Nt). The ﬁrst order condition is wt − θt 1 − Nt = 0 356 On the ﬁrm side, mct = −1 amount of labor by substituting the wage into the ﬁrst order condition above.. This means pt = 1 and wt = −1. We can solve for the equilibrium Nt = 1 − θt wt = 1 − θt ( − 1)At Thus, the equilibrium quantity of labor is increasing in TFP, At, and is decreasing in the disutility of labor, θt, and the markup, −1. The ﬁrst two comparative static results are the same as in earlier chapters. Why is the equilibrium quantity of labor decreasing in the markup? When ﬁrms charge higher markups, demand for each of the j goods is lower. This causes
ﬁrms to hire less labor and produce less output. The expressions for output and consumption are given by Ct = Yt = AtNt = At − θt − 1 16.2.6 Eﬃciency In the previous example we saw that higher markups cause ﬁrms to demand less labor and, in equilibrium, produce less output. In Chapter 15 we showed that the competitive equilibrium was Pareto optimal. Although we have an equilibrium in the model with monopolistic competition it is not a competitive equilibrium in the sense of Chapter 15. The reason is that intermediate good ﬁrms are not price takers. Each one sells a diﬀerentiated product which gives them some market power to set their own price. Hence, the equilibrium concept in this chapter needs to be adapted accordingly. More formally, an equilibrium with monopolistic competition is a set of prices and allocations such that: i) the household optimizes taking prices as given. ii) intermediate goods ﬁrms optimize choosing how much labor to hire and a price of their product subject to their demand curve, iii) the product bundler aggregates the intermediate good into a ﬁnal good taking prices as given, and iv) markets clear. Despite the diﬀerent equilibrium concept, we can ask the same question from Chapter 15; namely, is the equilibrium eﬃcient? To do this, we focus on the situation where the planner hires the same amount of labor to produce each product.2 The planner’s problem is to choose 2This can be shown to be optimal. 357 Ct and Nt to maximize utility subject to the resource constraint. In math, u(Ct, 1 − Nt) max Ct,Nt s.t. Ct = Yt Yt = (∫ 1 0 (Atnj,t) −1 ) −1 = AtNt Substituting the constraint into the objective function, we have The ﬁrst order condition is u(AtNt, 1 − Nt) max Nt Atuc(Ct, 1 − Nt) − uL(Ct, 1 − Nt) = 0. Rearranging gives uL(Ct, 1 − Nt) uc(Ct, 1 − Nt) = At The left hand side is the marginal rate of substitution of leisure for consumption. The right hand side is the marginal product of labor. The planner hires labor
up to a point where these are equal. How does this compare to the decentralized equilibrium? Recall, the household supplies labor up to the point where the marginal rate of substitution equals the wage rate. uL(Ct, 1 − Nt) uc(Ct, 1 − Nt) = wt. At < At. In other words, the equilibrium wage rate is less that Above, we showed that wt = −1 the marginal product of labor. The decentralized equilibrium produces a lower level of output than what is optimal. As gets larger the markup gets smaller and the wage approaches the marginal product of labor. How do we make sense of the results? It actually helps to think back to the case of the monopolist. The monopolist produces less than is socially optimal because it knows that with each added unit of production it lowers the sales price of its output. This leads the monopolist to produce less output than what would be provided in an equilibrium with competitive ﬁrms who take the price as given. The same logic is at work with monopolistic competition. Each ﬁrm ﬁnds it optimal to restrict output to maximize proﬁts. The distortion is the smallest when is very large. In that case, the intermediate goods are very close substitutes and 358 cannot aﬀord to charge big markups. In an exercise at the end of the chapter you will prove that the optimal government policy is to subsidize the cost of labor. The intuition is that since each ﬁrm underproduces, the optimal policy pays ﬁrms to produce more. 16.3 Markups and Labor’s Share of Income One of Kaldor’s Facts stated in Chapter 4 is the relative constancy of labor’s share of income. Kaldor was making this observation in the 1950s. While this was an accurate characterization for the years Kaldor was describing, it is deﬁnitively not an accurate description of the US over the last several decades (and especially since 2000) and the empirical work by Karbarbounis and Neiman (2014) shows that this decline in the labor share is a global phenomenon. Figure 16.3: Labor Share in US In our perfectly competitive model with a Cobb Douglas production function, Yt = t N 1−α t, from Chapter 5, labor’s share on income is AtK α wtNt Yt = 1 − α 359 19601
97019801990200020100.580.590.600.610.620.630.640.65RatioShare of Labour Compensation in GDP at Current National Prices for United StatesShaded areas indicate U.S. recessions.*Source: University of Groningenfred.stlouisfed.org In contrast, in the model with monopolistic competition that we just outlined, labor’s share. Implicitly, our production function was Cobb Douglas with α = 0. If we of income is −1 generalized this by allowing 0 < α < 1, it turns out that labor’s share in the economy with market power is wtNt Yt = (1 − α) − 1. Thus, labor’s share declines as the markup increases. Measuring to what extent markups may have increased since 2000 is an active area of economic research with Basu (2019) summarizing some of the recent ﬁndings. De Loecker et al. (2020) use data from publicly traded ﬁrms in the US to argue that markups have indeed increased. In a view consistent with this, Autor et al. (2020) show that market shares have increasingly ﬂowed to “superstar ﬁrms” which results in a higher aggregate markup. Indirect evidence from the rising importance of markups comes from looking at the share of income going to capital versus proﬁts. In the model with markups and Cobb Douglas production, capital’s share of income is given by rtKt Yt = α − 1. After subtracting oﬀ the shares going to labor and capital, the share of income going to proﬁts is 1. Consequently, an increase in markups should be accompanied by an increase in the share of income going to proﬁt. Moreover, the shares of income going to capital and labor should decline in the same proportions. That is both the capital and labor share should both decline but their ratio should stay constant. The evidence on these propositions is mixed. Barkai (2020) shows that, in the US case, capital’s share of income declined and the share going to proﬁts increased. The aforementioned paper by Karbarbounis and Neiman argue that the markup increased but the increase explains relatively little of the decline in the labor share. Researchers confront a couple diﬀerent challenges
in thinking about markups and shares of income. First, when the elasticity of substitution between capital and labor is not one, i.e. the production function is not Cobb Douglas, the shares going to capital and labor depend on the capital to labor ratio. Indeed, Karbarbounis and Neiman estimate that capital and labor are substitutes so that the decline in the price of computers and investment more generally have caused ﬁrms to substitute away from labor and the labor share to decline. However, for technical reasons, the elasticity of substitution is diﬃcult to estimate empirically. Other researchers, for instance, have argued that the elasticity of substitution between capital and 360 labor is below one.3 The second challenge might be more mundane but it is just as critical and that is how we classify some income as “capital” and other income as “proﬁt”. This is diﬃcult because ﬁrms often own rather than lease their capital so economic proﬁts (what we want to measure) diﬀer from accounting proﬁts. Moreover, some capital is intangible such as research and development or simply good teamwork, i.e. “cultural capital.” Barkai (2020) deals with some of these challenges and argues that the “pure” (i.e. economic) proﬁt share has increased since the 1980s. In summary, our model predicts that an increase in markups should be associated with a decrease in the labor share. Thus, an increase in markups could explain the empirical decline in the labor share in the US. Recent research shows that markups may have increased but their quantitative eﬀects on the labor share is still open for debate. 16.4 Endogenous Entry and the Gains to Variety Now we generalize the model in the previous section to allow for ﬁrm entry. This means Mt is endogenous and not equal to 1. Nothing changes with the household, product bundler or the optimization problem of the monopolistic competitors. Recall, the price of good j is yj,t = Ytp1− j,t The price is equal to a constant markup over marginal cost. Now things get interesting. Go to (16.6). pj,t = − 1 mct Substitute the solution for pj,t. Mt p1− j,t dj − mct) Mt
dj ∫ 0 3Oberﬁeld and Raval (2019) estimate an elasticity of substitution in the manufacturing sector of about 0.7. 361 Integrating over the constant gives 1 = ( − 1 1− mct) Mt This implies mct = − 1 In other words, as more ﬁrms come enter the market to compete for labor, marginal costs go up. The equilibrium wage is 1 M −1 t wt = − 1 1 AtM −1 t Since each intermediate good ﬁrm charges the same price, they produce the same level of output and hire the same amount of labor. In equilibrium it must be the case that labor demand equals labor supply. Mt dj Mt ∫ 0 nj,tdj = nt ∫ 0 = Mtnt = Nt Total output can be calculated using (16.4). Mt Mt Yt = (∫ 0 = (∫ 0 −1 −1 j,t dj) y (Atnt) −1 dj) −1 dj) − 1 Mt = Atnt (∫ 0 = AtntM −1 t 1 = AtNtM −1 t Since > 1, output is increasing in the number of intermediate good ﬁrms, Mt. What’s the intuition for this result? There are a couple ways of thinking about it. Each intermediate good ﬁrm specializes in producing their product. A greater number of ﬁrms allows for a greater degree of specialization. With more specialization, output goes up for despite the physical factors of production (just labor here) not changing. An alternative way to think about it is to just forgo the notion of a product bundler all together and write aggregate 362 consumption as a function of each individual consumption unit, cj,t. Ct = (∫ 0 −1 Mt −1 j,t c ) (16.15) Now think of each intermediate goods ﬁrm as a “variety”. With these preferences, utility is increasing in the number of varieties. Why does this make sense? Think about two economies with equal sized labor forces and the same At that just make food. In one economy, the labor force is entirely devoted to making pizza. In the other the labor force is split into making pizza, hamburgers, and pad Thai. With the preferences of (16.15), people would get higher utility in the second economy than the ﬁ
rst. Generalizing this, we can say that utility is increasing in the number of varieties. This result is quite intuitive but powerful and is the building block in many models of international trade and economic growth. Critical to quantifying the “gains from variety” is the parameter. The closer is to 1 the bigger are the gains from variety. The reason is that as gets bigger the individual products are better substitutes for each other. Returning to the food example, people are better oﬀ in an economy with pizza and falafel than in an economy with pizza and calzones. In the context of international trade, increasing variety is one of the ways people are made better oﬀ through trade. The US manufactures Fords and Germany manufactures Mercedes. People in both countries gain from variety. 16.4.1 Free Entry Condition What remains is to pin down the number of ﬁrms in equilibrium. Recall that ﬂow proﬁts are dj,t = pj,tyj,t − mctyj,t j,tYt(pj,t − mct) = p− = ( − 1 − mct) AtM 1 −1 t Nt mct − 1 = M − −1 t AtM 1 −1 t Nt ( Mt 1 −1 ) = M 2− −1 t AtNt 1 1− This means that proﬁts are bigger than 0 for all Mt > 0. That is, intermediate good ﬁrms would always ﬁnd it worthwhile to enter. As a more realistic alternative, suppose ﬁrms have to pay a ﬁxed cost, f to enter the 363 market. Flow proﬁts in this case are dj,t = M 2− −1 t AtNt 1 1− − f Firms will ﬁnd it optimal to enter whenever dj,t > 0. Thus, in an equilibrium with free entry ﬁrms continue to enter until dt is driven to 0. 2− −1 M t AtNt 1 1− − ft = 0 Solving for Mt yields −1 2− Mt = ( ft AtNt Provided, > 2, the number of ﬁrms is decreasing in ft and and increasing in At and Nt. If the ﬁxed cost to starting a business increases, fewer potential ﬁrms ﬁ
nd it optimal to enter. As goes up, markups go down so, again, entry is less proﬁtable. As the eﬀective market size increases (AtNt) more ﬁrms enter the market. (16.16) 1 2− ) These comparative static results are reversed if < 2 and there is some interesting economics as to why. When a ﬁrm enters the market, they drive up marginal cost which pushes down ﬂow proﬁts but they also increase output since output depends positively on the number of ﬁrms. If the second eﬀect always suﬃciently dominates the ﬁrst, ﬂow proﬁts are a positive function of Mt and we will again have a case where an inﬁnite number of ﬁrms enter. The two cases are illustrated in Figure 16.4. When < 2 there is an Mt where ﬂow proﬁts equal zero but that is not an equilibrium since more ﬁrms would ﬁnd it optimal to enter the market. 364 Figure 16.4: Proﬁts a Function of Mt 16.4.2 Eﬃciency Consider the problem of the social planner who chooses Nt and Mt to maximize utility subject to the resource constraint, Ct + ftMt = AtM 1 −1 t Nt. Note that the resource constraint now includes the ﬁxed cost, ft of starting each business. Substituting consumption out of the utility function, the maximization problem is U = u(AtM t Nt − ftMt, 1 − Nt) 1 −1 max Nt,Mt The ﬁrst order conditions are ∂U ∂Mt = 1 − 1 2− AtNtM −1 t − ft = 0 ∂U ∂Nt = AtM t uC(Ct, 1 − Nt) − uL(Ct, 1 − Nt) = 0 1 −1 365 𝑑" 𝑀" −𝑓" 𝜀<2 𝜀>2 𝑀"∗ The second FOC says that the planner hires labor up to a point where the marginal product equals the marginal rate of substitution of leisure for consumption. This was exactly the same optimization condition
we had when Mt = 1. Since the equilibrium wage is lower than the marginal product, we can infer, for a given Mt, that the quantity of labor supplied in equilibrium is ineﬃciently low. How does the equilibrium number of varieties compare to the optimal number. To ﬁnd out, rearrange the ﬁrst FOC, M opt t = ( ft AtNt −1 ) 2− ( − 1) −1 2− (16.17) Call the equilibrium number of ﬁrms, M eq t. Comparing (17.40) to (16.17) we see that the socially optimal M opt t is bigger than the equilibrium M eq t when ( − 1) −1 2− > 1 2− This is satisﬁed for some values of but not others. The larger is the more likely it is to be satisﬁed. Here is the intuition for the results. In the last section we showed that for a given number of varieties, ﬁrms under produce compared to the socially optimal level. Firms ﬁnd it optimal to charge a markup and that lowers demand relative to what it would be had the ﬁrm been pricing at marginal cost. The same mechanism is at work here. What determines the optimal number of varieties in relation to the equilibrium number? When a ﬁrm decides to enter in the decentralized equilibrium it raises marginal costs for all the other ﬁrms but also raises their productivity by adding variety. The social planner internalizes all of this when deciding whether to add another variety. One might be tempted to blame the ineﬃcient equilibrium on the market power of the ﬁrms. This is a mistake. Because of the free entry condition ﬁrms earn zero proﬁts in equilibrium, which is what the perfectly competitive ﬁrm earned in Chapter 12 and in our earlier models of economic growth. Instead, the ineﬃciency is a result of increasing returns to scale in production. It costs ft + mct to produce one unit of any of the varieties. Firms only have to pay the ﬁxed cost once so the cost to producing two units is ft + 2mct and so on. The key feature here is that average cost is always above marginal cost. If a ﬁrm priced at marginal cost, they would earn negative pro�
��ts. The only way for ﬁrms to break even is to charge a markup, which is consistent with what we found in equilibrium. In summary, the ineﬃciencies associated with monopolistic competition (once one allows for free entry) are more subtle than simply ascribing the markups to undesirable market power. 366 16.5 Firm Dynamics: Theory and Evidence The number of ﬁrms in the free entry equilibrium is given by Mt = ( ft AtNt ) −1 2− 1 2− In logs, this can be expressed as: ln Mt = − 1 − 2 (ln At + ln Nt − ln ft) + 1 2 − ln Subtract this from ln Mt−1. ln Mt − ln Mt−1 = − 1 − 2 (ln At + ln Nt − ln ft) − − 1 − 2 (ln At−1 + ln Nt−1 − ln ft−1) Recall, that the growth rate of a variable x, gx t is approximated by ln xt − ln xt−1. This means, gm t = − 1 − 2 (ga t + gn t − gf t ). The growth rate in the number of ﬁrms is a positive function of population and TFP growth and decreases with growth in the ﬁxed cost of starting a ﬁrm. There is a well documented decline in business dynamism which is characterized by declining startup and exit rates of ﬁrms, a decline in the net entry rate (startup rate minus exit rate), and an increase in the average ﬁrm age.4 While our model is not rich enough to discuss things like ﬁrm age and ﬁrms do not exit in our model, we can think of gm t as roughly corresponding to the empirical startup rate. Despite our model’s limitations, there is some correlational evidence that the decline in the startup rate has coincided with a decline in the growth rate of the working age population. Figure 16.5 shows the data between 1979-2016. The correlation between the two series is about 0.5. 4Decker et al. (2016) summarizes the main evidence. 367 Figure 16.5: Startup and working age population growth rates While the correlational evidence is suggestive, correlation does not imply causation. In a recent paper, Karahan
et al. (2019) provide evidence that the correlation is indeed causal. In particular, they show that states that experienced low population growth rates in the 70s have lower startup rates today. In their quantitative model, declining population growth accounts for more than half of the decrease in the startup rate. 16.6 Summary • A monopolist ﬁnds it optimal to set price above marginal cost and produce an ineﬃciently low level of output. • The model of monopolistic competition features a continuum of ﬁrms each producing a unique variety. Firms ﬁnd it optimal to charge a markup and output is ineﬃciently low relative to the social planner’s solution. • In the data, labor’s share of income has been declining. Our model predicts that labor’s share is decreasing in markups. Although economists have yet to reach consensus, there is some evidence that markups by US ﬁrms have increased over the last couple decades. 368 1975198019851990199520002005201020152020Year910111213141516Start Up Rate1975198019851990199520002005201020152020Year00.511.522.5Growth Rate of Working Age Population • Holding the factors of production ﬁxed, economic welfare is increasing in the number of products. This can either be thought of as ﬁrms using a wider variety of intermediate inputs or households consuming a broader variety of products. • In the free entry equilibrium, ﬁrms earn no proﬁts. Despite this, the equilibrium produces too few varieties of products and does not supply as much labor compared to their Pareto optimal levels. • The startup rate of new ﬁrms has decreased over the last few decades. The growth rate of ﬁrms in our model is positively related to the growth rates of population and TFP. This is consistent with some recent empirical work. Key Terms • Markup • Monopolistic Competition • Product variety Questions for Review • Explain why monopolist produces less output than the socially eﬃcient level. • How does monopolistic competition diﬀer from oligopoly? • Describe what has happened to the labor share over the last 20 years. What are some potential explanations for the decline? • Why does expanding product variety raise welfare? • True or false: A ﬁrm charging a price
over marginal cost is earning economic proﬁt. 369 Chapter 17 Search, Matching, and Unemployment An individual is classiﬁed as unemployed if she is not working but actively searching for work. In the microeconomically-founded model we have considered in this section of the book, there is no unemployment. In equilibrium, the real wage adjusts so that the quantity of labor supplied equals the quantity of labor demanded. There is no such thing as an individual who would like to work at the prevailing market wage but cannot ﬁnd work. Indeed, the most common macroeconomic models (both the neoclassical and New Keynesian models, as described and studied in Parts IV and V) used among academics and policymakers have this feature that there is no unemployment as it is deﬁned in the data. While these models can speak to labor market variables like wages and total hours worked, they are silent on the issue of unemployment. This is a potentially important omission, for at least two reasons. First, the ﬁnancial and business press, as well as non-academic policymakers, are quite focused on unemployment statistics. Second, as documented in Chapter 1, the majority of variation in total hours worked comes from individuals transitioning into and out of work as opposed to already employed workers varying the intensity of their work. Part of the reason that macroeconomic models often abstract from unemployment is that it is not trivial to incorporate unemployment, as it is deﬁned into the data, into relatively simple modeling frameworks. In particular, thinking about unemployment requires moving away from the representative agent assumption and allowing for heterogeneity among both workers and ﬁrms. In this chapter, we build a model in which there are a continuum of workers and ﬁrms.1 In equilibrium, some workers are unmatched with ﬁrms and some ﬁrms are unmatched with workers – i.e. there exists unemployment. We can use the model to think about issues related to the level of unemployment and why it changes over time. Before introducing the model, we start with some stylized facts about the labor market. 1“Continuum” is a fancy word that roughly means “a lot.” Formally, assuming a continuum of agents allows for there to be many agents, but there are so many agents that each individual agent behaves as a price-taker. 370 17.1 Stylized Facts We have already deﬁned the unemployment rate and
showed how it varies over time (see, e.g., Figure 1.6 in Chapter 1). Here we focus on the determinants of the unemployment rate – job creation, job destruction, and separations. Before 2000, there was very little high frequency data available on the demand side of the labor market. The Job Openings and Labor Turnover Survey (JOLTS), conducted by the Bureau of Labor Statistics changed that. JOLTS is a monthly establishment level survey that collects data on job postings, new hires, and separations at the private sector. A separation occurs when an employer and a worker end their relationship. The other data set we use comes from the Current Population Survey (CPS) which is a monthly survey that keeps track of labor market outcomes across individuals. The monthly CPS started in 1979. The data displayed below can be downloaded from the Federal Reserve Bank of St. Louis FRED. 1. There are an enormous number of jobs are created and destroyed each month. Figure 17.1: Total Hires and separations in the US 2000-2016 Take a careful look at Figure 17.1. Note that the vertical axis is in thousands. That means over ﬁve million jobs were created and ﬁve million jobs were destroyed in the ﬁrst month of 2001. Even during the depths of the Great Recession, more than three million jobs were created each month, which may be counterintuitive given how much discussion there was about the underperformance of the job market. However, as our next item shows, the fact that many jobs were created during the Great Recession does not mean the labor market was healthy. 2. Net job creation is procyclical When the number of hires exceed the number of separations, employment expands. Another way of saying this is that there is positive net job creation. The reverse is 371 myf.red/g/5bFe3,2003,6004,0004,4004,8005,2005,6006,000Jan 2001Jan 2002Jan 2003Jan 2004Jan 2005Jan 2006Jan 2007Jan 2008Jan 2009Jan 2010Jan 2011Jan 2012Jan 2013Jan 2014Jan 2015Jan 2016fred.stlouisfed.orgHires: Total PrivateTotal Separations: Total PrivateLevel in Thousands true if separations exceed new hires. Figure 17.2 plots net job creation, which is the diﬀerence between total hires and total separations in Figure 17.1. We observe that net job creation is clearly countercycl
ical. For example, at the height of the Great Recession there were close to one million net jobs destroyed. This more closely jives with our basic intuition about the labor market. However, as we have just seen, these net creation numbers mask the large amount of jobs created and destroyed each month. Figure 17.2: Net job creation in the US 2000-2016. 3. During recessions, the number of quits fall and the number of layoﬀs rise. Separations can be divided into two categories: quits and layoﬀs/discharges. A separation is classiﬁed as a quit when an employee departs voluntarily. Layoﬀs and discharges occur when the employee leaves involuntarily. In other words, when the employer decides to end the relationship, the separation is called a layoﬀ or discharge; when the employee ends the relationship, the separation is called a quit. Sometimes separations do not clearly fall into either of these categories. For instance, if an employee is forced to resign, would that show up as a quit or a layoﬀ? All of these ambiguous cases are classiﬁed as “other separations.” Retirements also go into the “other separations” category. 372 myf.red/g/5ccp-1,000-800-600-400-2000200400Jan 2001Jan 2002Jan 2003Jan 2004Jan 2005Jan 2006Jan 2007Jan 2008Jan 2009Jan 2010Jan 2011Jan 2012Jan 2013Jan 2014Jan 2015Jan 2016fred.stlouisfed.orgHires: Total Private-Total Separations: Total PrivateLevel in Thous.-Level in Thous. Figure 17.3: Separations in the US 2000-2016. Figure 17.3 shows that the number of quits usually exceeds the number of layoﬀs and discharges. In other words, most of the time the majority of job separations are voluntary. An exception is the latter half of the Great Recession, when discharges exceeded quits by a sizeable amount. We observe that quits fall during recessions and layoﬀs rise. One reason for this is that workers are less likely to quit their job and search for greener pastures when the job market is poor. At the same time, during recessions ﬁrms tend to demand less labor. One way in which this lower demand
for labor is manifested is through the termination of existing employment relationships. 4. Not all people looking for work ﬁnd a job This may be obvious (especially for those of you who have looked for a summer employment), but not everyone looking for a job is immediately successful. Mechanically, the job ﬁnding rate equals the number of working-age people transitioning from unemployment to employment divided by the number of working-age unemployed people. During expansions the job ﬁnding rate rises and during recessions it falls. Figure 17.4 plots the job ﬁnding rate over time in the US. In addition to being procyclical, another interesting feature of 17.4 is that the job ﬁnding rate has generally been declining over time. This is important for several reasons. First, the longer a person remains unemployed the more their skill depreciates. If a carpenter is unemployed for one month, he is unlikely to forget how to install wood ﬂoors. However, if the same carpenter is unemployed for one year, he is likely to be more rusty when he starts installing ﬂoors again. Second, less output can be produced if it takes longer for ﬁrms and workers to meet each other. Finally, if the unemployed person is receiving unemployment beneﬁts, a lower job ﬁnding rate implies a longer unemployment duration. Therefore, there are higher ﬁscal costs when the job ﬁnding rate is low. The job creation rate reached a nadir in the Great Recession and has subsequently recovered although not to all the 373 myf.red/g/5cd91,4001,6001,8002,0002,2002,4002,6002,8003,0003,2003,400Jan 2001Jan 2002Jan 2003Jan 2004Jan 2005Jan 2006Jan 2007Jan 2008Jan 2009Jan 2010Jan 2011Jan 2012Jan 2013Jan 2014Jan 2015Jan 2016fred.stlouisfed.orgQuits: Total PrivateLayoffs and Discharges: Total PrivateLevel in Thousands way to its high in 2000. Figure 17.4: Job ﬁnding rate in the US 2000-2016. 5. There is a negative relationship between the vacancy and unemployment rates Recall that the unemployment rate is the number of unemployed individuals divided by the labor force. The vacancy rate is the number of job postings divided by the sum of the number of job
postings and employment. If employment is low and the number of job postings are high, the vacancy rate is also high. Figure 17.5 shows that there is a negative relationship between the unemployment rate and the vacancy rate in the data. When the vacancy rate is high and the unemployment rate is low, the prospects for individuals looking for work are good. When the reverse is true, prospects are bad. The negative relationship between vacancies and unemployment is called the Beveridge curve after the British economist William Beveridge. Figure 17.5: Job ﬁnding rate in the US 2000-2016. A ﬁnal note about the Beveridge curve is that it gives us some insight into the eﬃciency of the labor market. Points to the northeast are in some sense less eﬃcient than points 374 myf.red/g/5cjg0.150.200.250.300.350.40200020022004200620082010201220142016fred.stlouisfed.orgLabor Force Flows Unemployed to Employed: 16 Years and Over/Unemployment LevelThous. of Persons/Thous. of Personsmyf.red/g/5dOpPercentPercent1.52.02.53.03.54.04.544.555.566.577.588.599.510fred.stlouisfed.orgJob Openings: Total Private (left), Civilian Unemployment Rate (bottom), Dec 2000 Apr 2016Rate closer to the origin. Why is this? At the origin no ﬁrms are posting jobs and no one is unemployed. Consequently, everyone in the economy who wants to work and every ﬁrm that wants to employ a worker are in fact matched together and are producing output. Moving farther away from the origin means that there are more unmatched workers and ﬁrms so less output is being produced. One caveat is that just because workers and ﬁrms are matched does not imply that those matches are productive. For example, workers could be stuck in a low productivity job because they do not have the resources to migrate to more productive localities. 17.2 One-Sided Search: The McCall Model Should you accept a job oﬀer today or keep searching? When is it optimal to quit? Why do some unemployed workers search more intensely than others? These are the questions that the McCall model of unemployment is designed to answer.2.
The unit of analysis is an unemployed worker faced with a job oﬀer. The worker must decide to accept the wage oﬀer and work at that wage forever or keep looking for work. Before launching into the model, we take a slight detour to review random variables and distribution functions. 17.2.1 Probability: A Review A random variable is a variable whose value depends on outcomes of a random phenomenon.3 For instance, a roll of a die is a random variable. The outcome, i.e. whether a one, two, three, etc. is rolled is random. We do not know the outcome before we roll the die. Random variables are called “discrete” if the number of outcomes is ﬁnite, as in the die case. Random variables are “continuous” if the number of outcomes is inﬁnite. Where might this be the case? Think about height in a population. Even between narrowly speciﬁed heights, say 72 inches and 73 inches, there exists an inﬁnite number of potential heights, e.g. 72.1, 72.111, 72.1111 and so on. We focus on continuous variables in this section. A density function for a random variable, X, is denoted by f (x). The density function can intuitively be understood as the likelihood that X takes a value around x. Since x is continuous, the probability that it equals any one number in particular is 0. Why? The rough idea is that if X can take on an inﬁnite number of values, the probability that it equals one of those values is exactly 0. We call all values that X can take the “support”. If the lowest value that X can take is x and the highest value is x then the support is all numbers in the range [x, x]. It must be the case that the probability that X takes some value in the 2See McCall (1974) 3Wikipedia cite. 375 support is equal to 1. In math, this says. x ∫ x f (x)dx = 1. The probability that X is less than or equal to some number c is P r(X ≤ c) = ∫ c x f (x)dx (17.1) Recall, the mean is the average value of X. Since f (x) tells us the relative likelihood, the mean can be understood
as a weighted average where the weighting function is f (x). Consequently the mean, E[X] is deﬁned as E[X] = ∫ x x xf (x)dx (17.2) One ﬁnal concept that we will occasionally have use for is the cumulative distribution function, or distribution function for short. A distribution function for a random variable, X, is the probability that X is less than or equal to some number x and is denoted by F (x). Fortunately, we already solved for this in (17.1). Doing a change of variables, P r(X ≤ x) = ∫ x x f (s)ds = F (x) − F (x) = F (x) where the last equality follows from the fact that the probability that X exactly equals x is 0. Since F is the antiderivative of f it follows that F ′(x) = f (x). It is always true that F (x) = 1 and F ′(x) ≥ 0. It is sometimes easier to work with the distribution function istead of the density function. For instance, the mean of x can be written as E[X] = ∫ x x xdF (x). (17.3) This follows because dF (x) = F ′(x)dx = f (x)dx. 17.2.2 Example The uniform distribution has a density function 1 x−x. That is, all values of x are equally likely. The density function is shown in 17.6. 376 Figure 17.6: Uniform Probability Distribution First, verify the integral over the entire support is one. x 1 x − x ∫ x ∣x x=x dx = Now, let’s calculate the mean. x x x − x ∫ x dx = ∣x x=x x2 2(x − x) = x2 − x2 2(x − x) (x + x)(x − x) 2(x − x) = = (x + x) 2. In other words, the mean is just the midpoint of the lower and upper bounds. The intuition 377 𝑥 𝑓(𝑥) 1𝑥−𝑥 𝑥 𝑥 is that because each point has equal density, the mean is just a simple average between the lowest
value and the highest value. Finally, the distribution function is x 1 x − x ∫ x ∣x x=x dc = x) 17.2.3 The Model The model lasts for two periods. An unemployed worker comes into the ﬁrst period with a wage oﬀer, w. The decision is to accept the oﬀer and work at w in t and t + 1 or to reject the oﬀer, accept an unemployment beneﬁt of b and draw another wage in t + 1 from the distribution function F (w). The individual’s decision is to compare lifetime utility working under the wage w with the expected utility of accepting the unemployment beneﬁt in t and taking a draw from F (w) next period. To keep things simple, we assume linear utility and that the only source of income is from wages or unemployment beneﬁts. Mathematically, the problem is U = max{w + βw, b + β ∫ w w max{w, b}dF (w)} (17.4) (1 + β)w is the lifetime utility of working at wage w. b + β ∫ w w max{w, b}dF (w) is the utility of accepting unemployment beneﬁts in t plus the expected utility of drawing a wage at t + 1. Note that the worker would reject any oﬀer less than b, hence the max operator in the integral. Deﬁne the wage at which the individual would be indiﬀerent between accept and reject as the “reservation wage”, wr. Analytically, the reservation wage solves (1 + β)wr = b + β ∫ w w max{w, b}dF (w) (17.5) The reservation wage gives us a ﬂoor on expected utility as a function of w. Graphically, lifetime utility looks like Figure 17.7. 378 Figure 17.7: Expected Utility of Reservation Wage For any w < wr, the individual rejects the wage oﬀer which explains why utility does not decline for w < wr. Of course, if the individual receives a wage oﬀer such that w > wr then the decision is to accept and utility is increasing in w. For now, let’s assume w > b. Then, solving for the reservation
wage, w w wdF (w)] wr = [ + βµ 1 + β where µ = ∫ w w wdF (w) is the average wage. Performing some comparative statics, we see ∂wr ∂β = µ − b (1 + β)2 > 0 ∂wr ∂µ = β 1 + β ∂wr ∂ 379 (17.6) (17.7) (17.8) 𝑤 𝑈 𝑤𝑟(1+𝛽) 𝑤𝑟 An increase in β means the agent is more patient and hence willing to reject a wider range of oﬀers in t in the hopes of better oﬀer in t + 1. It is also more worthwhile to wait the higher are the mean wage and unemployment beneﬁts. This points to one reason why unemployment beneﬁts may increase unemployment – namely they make individual’s more choosy over jobs. Without a general equilibrium model, we cannot say whether this is “good” or “bad”. Higher unemployment beneﬁts is costly from a ﬁscal perspective, but they also allow people to hold out for good jobs. This also ignores ﬁrms’ incentives to post vacancies (something we return to later in the chapter). 17.2.4 Example: Uniform Distribution The distribution function for the uniform distribution is f (w) = w − w w − w If b < w, the reservation wage is characterized by wr = = 1 1 + β 1 1 + β [ dw] [b + β (w + w) 2 ] As expected, the reservation wage is increasing in b, β, and µ = (w+w) Finally, let’s allow for the possibility that b > w. This is arguably the more realistic case since there are plenty of jobs that pay below the ﬂow beneﬁt of unemployment (especially once one allows for home production and leisure). The reservation wage is characterized by:. 2 wr(1 + β) = b + β ∫ w w max{w, b}dF (w) = b + β ∫ b w bdF (w) + β ∫ w b wdF (w) = b + βbF (b) + β ∫ w b
wdF (w) where the second line follows because b is a constant that can be pulled out of the integral. 380 Substituting in the distribution function, we have (1 + β)wr = b + β [b b + ∫ = b + β [b 1 w − w + w2 − b2 2(w − w [ w2 + b2 − 2wb 2(w − w) ] w w] ] = b + β [ (w − b)2 + 2b(w − w) 2(w − w) ] = b + β [b + (w − b)2 2(w − w) ] where the fourth line follows from adding and subtracting 2bw. Solving for the reservation wage yields wr = b + β 1 + β (w − b)2 2(w − w) One can verify that this is increasing in β and b. Suppose that the distribution for w gets riskier. Formally, suppose we add a constant, c to w and subtract a constant c from w. The mean of the wage distribution stays the same, but how is the reservation wage aﬀected? ww + c − b)2 2(w + c − w − c) (w + c − b)2 2(w − w) (w − b)2 2(w − w) = wr Thus, the reservation wage increases. This is an example of a “mean preserving spread”. A mean preserving spread of a distribution is formed by spreading out some of the probability mass while leaving the mean the same. The intuition here is that the individual can reject any wage oﬀer less than b in t + 1 so it does not matter how low w gets conditional on it being less than b. On the other hand, the individual beneﬁts from the “upside” of the risk since it potentially means a higher wage. The intuition is shown graphically in Figure 17.8. Both distributions have a common mean, µ, but the distribution with the wider variance has a higher reservation wage. 381 Figure 17.8: Eﬀects of Mean Preserving Spread The beauty of the McCall model is that we can add a number of features to it including: job separations, endogenous probability of receiving job oﬀers, and the possibility of receiving multiple oﬀers per period. This makes
it very attractive as a microeconomic model of unemployment. However, we have taken the distribution of wages and job vacancies as given. To be useful as a macroeconomic model, these objects must be endogenized. 17.3 The Bathtub Model of Unemployment Before launching into an equilibrium model of unemployment with optimizing agents, we can put a little more structure on how we think about the relationship between the unemployment rate, separation rate, and the job ﬁnding rate. Imagine a bathtub that is partially full with water. The faucet is turned on putting more water in the tub, but the drain is also unplugged allowing water to escape. If more water is coming through the faucet than leaving, the tub deepens. If more water is going down the drain, the tub gets more shallow. Job creation is like water coming from the faucet and increases total employment. Separations are like water going down the drain as they subtract from the water in the tub. The water in the tub is akin to the level of employment. It only changes to the extent jobs 382 𝑥 𝑓(𝑥) 1𝑥−𝑥 𝑥 𝑥 𝑥−𝑐 𝑥+𝑐 𝑥𝑟 𝑥𝑟′ 𝜇 are created or destroyed.4 This is conceptually similar to the process of capital accumulation studied in Part II. In other words, we ought to think about there being a stock of employed workers. Separations and matches represents ﬂows. Compared to long run models of capital accumulation, employment plays the role of the capital stock, new job matches the role of investment, and separations the role of depreciation. The unemployed are those who have been involuntarily separated from a job match or have been searching and unable to ﬁnd a match. Let us now be more formal in describing the evolution of the stock of unemployed workers. Let ut denote the number of unemployed workers in period t. This is predetermined and cannot change within period t. Denote the job separation rate by s and the job ﬁnding rate by f. Think of these as time invariant parameters. Unemployment evolves over time according to the following law of motion: ut+1 − ut = −f ut + s(1 − ut). (17.9) (17.9) says that the change in
unemployment between t and t + 1, i.e. ut+1 − ut, depends negatively on matches, f ut, and positively on separations, where s(1 − ut) denotes the number of separated workers in period t. If matches exceed separations, the unemployment rate declines. If separations exceed job ﬁnding, the unemployment rate increases. Similarly to the growth models considered in Part II, there exists a steady state in which the unemployment rate is constant, which means ut+1 = ut = u∗. Imposing this condition yields 0 = −f u∗ + s(1 − u∗) ⇔ u∗ = s s + f (17.10) Equation 17.10 shows that the steady state unemployment rate is higher when the separation rate is higher or the ﬁnding rate is lower. Similar to models of long run growth, if ut < u∗ then the economy will transition to a higher unemployment rate and vice-versa. Let’s consider some numbers based on real-world data. The job ﬁnding rate for the ﬁrst few months of 2016 hovered around 25 percent. The layoﬀs and discharges rate was around 1.2 percent and the quit rate was around 2 percent. Since many people who quit are transferring to another job rather than entering unemployment, it would not be correct to count them as separations in this model. Suppose one quarter of quits transition to unemployment. Then, the separation rate is 0.25(2) + 1.2 = 1.7 percent. If the separation and job ﬁnding rate remain the same going forward in time, the long run unemployment rate would be: 4It is not quite clear whether quits should be analogized to water leaving the tub since presumably many of these workers are moving to other jobs. 383 u∗ = 0.017 0.017 + 0.25 = 0.064 Hence, the long run unemployment rate is around 6.4 percent. At the beginning of 2016 the unemployment rate was 4.7 percent. If our numbers for the separation and job ﬁnding rates are correct and constant, we would have expected the unemployment rate to rise from that time. This is in fact not what has happened, where the unemployment rate has continued to decline into 2018. 17.3.1 Transition Dynamics: A Quantitative Experiment As above, suppose that our initial
value of unemployment is 4.7 percent and the steadystate value is 6.4. Assume that the separation and job ﬁnding rates are constant. Similarly to growth models, we can compute the transition dynamics sequentially. u1 = u0 − f u0 + s(1 − u0) ⇔ u1 = 0.047 − 0.25(0.047) + 0.017(1 − 0.047) = 0.0515 If the current unemployment rate is 4.7 percent, our simple descriptive model would predict it should rise to 5.15 percent a month later. Going two months into the future, we would have: u2 = u1 − f u1 + s(1 − u1) ⇔ u2 = 0.0515 − 0.25(0.0515) + 0.017(1 − 0.0515) = 0.0547. Note that the change between u2 and u1 is smaller than the diﬀerence between u1 and u0 and with each change the unemployment rate comes closer and closer to the steady-state unemployment rate. Formally, one can show that this diﬀerence equation converges to a unique steady state. However, we will leave the details to the more mathematically inclined reader. The transition dynamics can be plotted in Excel or some other software as we do below. The top panel of Figure 17.9 shows that the unemployment rate converges monotonically to its steady state. Within about a year, the unemployment rate almost completely converges. Now consider the following counterfactual experiment. Suppose that after ten months, the separation rate drops to 0.01 and is expected to forever remain that this lower level. Then, the unemployment rate in month 11 is computed by 384 u11 = u10 − 0.25u10 + 0.01u10 We can then iterated forward in time using this new separation rate. The results are displayed in the bottom panel of Figure 17.9. In the ﬁrst ten periods, the unemployment rate is exactly the same because the parameters are identical. In period 11, the counterfactual separation rate drops to 0.01. This results in the unemployment rate falling and continuing to fall as it transitions to a new steady state. The new transition dynamics are outlined with a dotted line while the solid line traces out the transition dynamics when the separation rate stays the same. The solid line simply converges to the
old steady state of 6.4 percent, but the dashed line converges to 3.9 percent reﬂecting the lower separation rate. 385 Figure 17.9: The top panel shows the transition dynamics starting at u0 = 0.047. The bottom panel shows the transition dynamics associated with moving to a lower separation rate. The bathtub model gives us a way to account for movements between employment and unemployment and dynamic behavior over time. However, it is atheoretical – there is no optimization that results in endogenous values of the job ﬁnding and separation rates. In the next section, we go over a theory of unemployment and how the job ﬁnding rate is endogenously determined in equilibrium. For simplicity, we will continue to assume a constant separation rate. 386 051015202530Time0.0450.050.0550.060.065Unemployment Rate051015202530Time0.0350.040.0450.050.0550.060.065Unemployment RateHigh sLow s 17.4 Two Sided Matching: The Diamond-Mortensen-Pissarides Model Hiring employees is not costless. A business must recruit qualiﬁed applicants, interview them, and ultimately decide whom to hire. Creating a job consumes resources. Also, even when the business posts a vacancy, it does not always ﬁll the job. Sometimes they just do not meet the right person or the right person accepts a better job oﬀer. Similarly, sometimes a prospective employee just does not ﬁnd the right place to work. The models we have considered to this point ignore these features of reality. Two sided matching models were developed by Peter Diamond, Dale Mortensen, and Christopher Pissarides to address these attributes that characterize labor markets.5 These models make the assumption that vacancy creation is costly and that there are frictions impeding how prospective employees and employers meet each other. These frictions are embodied in something called the matching function. 17.4.1 The Matching Function Recall that the production function takes capital and labor as inputs to create some output. Higher productivity increases the eﬃciency with which inputs are transformed into output. Much in the same way, we assume the existence of a matching function that takes the number of unemployed individuals and vacancy postings as inputs and creates new hires as an output. Formally, the matching function we assume is: H
t = ψtM (ut, Vt) (17.11) Ht, ut, and Vt are the number of news hires, the existing stock of unemployed workers, and vacancies posted by ﬁrms, respectively. ψt measures the eﬃciency of the matching function. The higher is ψt the more matches are created for a given number vacancies and unemployment. We assume the following properties of the matching function. 1. The matching function is bounded below by 0. 0 ≤ ψtM (ut, Vt) 2. If there are no vacancies or no unemployment, there are no hires. 5See Diamond (1982), Pissarides (1985), and Pissarides and Mortenson (1994). ψtM (0, Vt) = ψtM (ut, 0) = 0 387 3. The number of matches cannot exceed the minimum of the number of vacancies and unemployment. Ht ≤ min [ut, Vt] 4. The matching function is increasing at a decreasing rate in vacancies and unemployment. ψtMu(ut, Vt) > 0, ψtMv(ut, Vt) > 0 ψtMuu(ut, Vt) < 0, ψtMvv(ut, Vt) < 0 5. The matching function has constant returns to scale in unemployment and vacancies. Hence, for λ > 0 ψtM (λut, λVt) = λψtM (ut, Vt) The ﬁrst point says that the number of matches cannot be negative. Point two says that if there are no unemployed people or no vacancies, there are no matches. Point three says that the number of hires cannot exceed the number of unemployed or the number of vacancies. Note that if there were no frictions in the labor market, the number of matches would equal the lesser of the number of vacancies and unemployed. Given matching frictions, this will generally not be true. The fourth point says that the matching function is increasing in both inputs at a decreasing rate. The ﬁnal point says that the matching function is constant returns to scale. If you double the number of vacancies and double the number of unemployed you exactly double the number of matches. The matching function quite obviously looks similar to the production function we have used to map labor and capital inputs into output. Similar to a production function, the
matching function can be a bit of a black box. Reality is complex. We do not know how exactly ﬁrms combine inputs into making output just like we do not know what precise frictions cause unemployment. However, much like the production function, the matching function is a useful abstraction as a reduced form way to model the frictions giving rise to unemployment. Given this abstraction we can consider the eﬀects of various exogenous changes to the labor market. As above, we can deﬁne the job ﬁnding rate as the ratio of new hires to the initial stock of unemployed, or ft = Ht ut. Formally: ft = ψtM (ut, Vt) ut = ψtM (1, Vt ut ). (17.12) The second equality in (17.12) follows from the constant returns to scale assumption. Note that the job ﬁnding rate is increasing in the ratio of vacancies to unemployment. This 388 makes sense: the more jobs postings there are relative to people looking for work, the easier it is to ﬁnd a job. We can also deﬁne a new term which we shall call the job ﬁlling rate. We will denote this with qt. It is equal to the fraction of news hires relative to total vacancies. Formally: qt = ψtM (ut, Vt) Vt = ψtM ( ut Vt, 1). (17.13) The job ﬁlling rate is decreasing in the vacancy to unemployment ratio. The more vacancies there are relative to unemployment, the more diﬃcult it will be to ﬁll any one vacancy. The ratio of vacancies to unemployment, or Vt, is often referred to as “labor market ut tightness.” When the labor market is “tight” it is relatively easy for workers to ﬁnd a job (i.e. ft is high). Conversely, when the labor market is “slack” it is easy for ﬁrms to ﬁll a job, but diﬃcult for workers to ﬁnd a job (i.e qt is high but ft is low). 17.4.2 Household and Firm Behavior While the more general version of this model is cast in an inﬁnite horizon context, we consider
a one-period version of the model. A one-period version of the model generalizes to the multi-period version if the separation rate is s = 1. In other words, at the beginning of each period all households are unemployed. During a period, some households ﬁnd jobs and others do not. At the end of the period, those households who did ﬁnd jobs separate. Then, the next period begins exactly as the ﬁrst period began, with all households unemployed. With a separation rate of 1, nothing a household does in t aﬀects its choice set in t + 1, so the model can eﬀectively be thought of as static. Much of the logic we develop in the simple one-period model nevertheless generalizes to a multi-period framework with a separation rate less than one. There exists a continuum of households. Technically, these households are indexed by j along the unit interval, [0, 1]. The continuum assumption along the unit interval technically means that (i) there are an inﬁnite number of households, so that each household is “small” relative to the total population and consequently takes prices as given; and (ii) the total “number” of households is nevertheless normalized to one. This is convenient because if the total number of households is normalized to one, then there is no distinction between the level of unemployment and the unemployment rate. Because of the implicit assumption of a separation of one each period, all households begin a period as unemployed. They are therefore ex-ante identical and we need not keep track of the household index j. This also means that ut = 1, so that Vt ut = Vt. For simplicity assume that a household’s utility function is simply linear in its consumption 389 (i.e. we are not modeling disutility from work or utility from leisure), which in a static context simply equals total income. If the household searches for a job, it ﬁnds a job with probability ft and this job pays it wt units of income. The household does not ﬁnd a job with probability 1 − ft. If it doesn’t ﬁnd a job, a household receives b ≥ 0 in unemployment beneﬁts. Therefore, a household’s utility is: U = ftwt + (1 − ft)b = b + ft(wt − b) (17
.14) There similarly exist a continuum of ﬁrms with total size normalized to one. Firms can be indexed by k. Each ﬁrm decides at the beginning of the period whether or not to enter the labor market. Upon entering the market, they post a vacancy at a cost of κ. With probability qt, they ﬁll the vacancy and produce zt units of output. zt is exogenous.6 In the case of a match, the ﬁrm earns a proﬁt of zt − wt − κ. With probability 1 − qt, the ﬁrm fails to match with a worker and produces nothing, thereby earning proﬁt of −κ. Because none of the payoﬀs depend on ﬁrm-speciﬁc characteristics, the expected proﬁt for a ﬁrm choosing to post a vacancy is therefore: Π = −κ + qt(zt − wt) + (1 − qt)0. (17.15) In other words, ﬁrms that enter the market make an investment of κ for the potential to earn zt − wt with probability qt. 17.4.3 Equilibrium The unemployment rate (equivalent to the level of unemployed) at the beginning of a period is ut = 1. Let u′ t denote the unemployment level/rate at the end of a period. We use a ′ superscript rather than a t + 1 because all of the action in this model occurs within a period. The endogenous variables of interest are Vt, u′ t, and wt. To determine the values of these variables in equilibrium, we must take a stand on how ﬁrms enter the labor market (i.e. decide whether or not to post a vacancy). We shall assume what is called a free-entry condition. This free-entry condition requires that, in equilibrium, the proﬁt of any ﬁrm posting a vacancy is Π = 0. If this were not the case, all ﬁrms would want to post a vacancy (if Π > 0) or no ﬁrms would want to post a vacancy (if Π < 0). Π = 0 means that ﬁrms are indiﬀerent between trying to hire in a period or not
. Making use of this free-entry condition, we can solve (17.15) for the job ﬁlling rate, qt: 6Technically, we can think about a ﬁrm’s production as being linear in labor input, i.e. yt(k) = ztnt(k). If a ﬁrm gets a match, it has nt(k) = 1 and produces zt. If the ﬁrm does not ﬁnd a match, it produces 0. 390 qt = κ zt − wt Remember that qt is decreasing in the number of vacancies. If wages go up, the right hand side of equation 17.40 goes up. Firms post fewer vacancies, increasing qt. The logic is reversed if zt increases. However, since wages are an equilibrium object, we have not solved the model yet. (17.16). How should wages be determined? In your principles of economics classes (and indeed throughout much of this book) the wage is equal to the marginal product of labor. This relies on some notion of perfect competition in the labor market. In a model like this, however, with random search and matching, the assumption of perfect competition is untenable. A worker randomly matches with a ﬁrm. If they cannot agree over a wage, the match dissolves. The ﬁrm earns zero proﬁt and the worker earns an outside option of b. No one gets an opportunity to “rematch.” Consequently, we need a rule for how the workers and ﬁrms split the surplus of successful matches. We follow the most common approach for dividing the surplus, which is called Nash Bargaining.7 In the Nash Bargaining problem, a matched worker and ﬁrm choose the wt to maximize the joint product of their individual payoﬀs to producing minus their “outside options.” An outside option is whatever the best alternative to not engaging in the match is. Since the ﬁrm produces nothing if it does not match with a worker, its outside option is zero.8 Hence, a ﬁrm’s payoﬀ from a successful match is zt − wt. This is the same across all ﬁrms. If a worker does not match with a ﬁrm, it receives unemployment beneﬁts of b. If
it does match, it receives wt. So the worker’s payoﬀ to the match minus its outside option is wt − b. Like for the ﬁrm, this payoﬀ is identical for all households and we needn’t keep track of j or k indexes. Formally, the Nash bargaining problem can be written: max wt (zt − wt)χ(wt − b)1−χ. (17.17) χ is a parameter between zero and one. It represents the relative bargaining weights of households and ﬁrms. The closer χ is to one, the greater the bargaining weight of the ﬁrm. Conversely, as χ approaches zero, the household has all the bargaining power. The ﬁrst order optimality condition can be found by taking the derivative with respect to wt and equating it to zero. Doing so, we obtain: −χ(zt − wt)χ−1(wt − b)1−χ + (1 − χ)(zt − wt)χ(wt − b)−χ = 0 7See Nash (1950). 8It is true that a ﬁrm pays a ﬁxed cost to post a vacancy, but this cost is sunk once the decision to post a vacancy has been made and is therefore not relevant for a ﬁrm’s outside option. 391 Simplifying, one obtains: χ(wt − b) = (1 − χ)(zt − wt) ⇔ wt = (1 − χ)zt + χb. (17.18) (17.5) has an intuitive interpretation – the equilibrium wage is a linear combination of the output from a match, zt, and the household’s outside option, b. As χ goes to zero, we get wt = zt. In this case, workers capture all the surplus of a match. In contrast, as χ → 1, the ﬁrm has all the bargaining power, so workers are paid the minimum required to get them to search, which equals their outside option of b. The wage outcome can be substituted into the free entry condition to ﬁnd an expression for the job ﬁlling rate, qt: qt = κ χ(zt −
b). (17.19) pins downs down the job-ﬁlling rate. Note two things. First, we require that zt > b. If b > zt, it would never make sense for households to search – the maximum wage they can earn is zt, and if their outside option is greater than this, they would never choose to work. Second, as χ → 0 we have qt → ∞. As χ → 0, a ﬁrm has no bargaining power, and the equilibrium wage is wt = zt. But if this is the wage, the ﬁrm makes a negative proﬁt of κ by posting a vacancy. Hence, no ﬁrms would choose to post any vacancies. If total vacancies are zero, then the vacancy ﬁlling rate is undeﬁned/inﬁnite. (17.19) (17.19) implicitly determines the number of vacancies posted, Vt. How can we see this? First, note that ut = 1 – i.e. all households begin the period unemploymed. Second, combine (17.19) with (17.13). Doing so, we get: κ χ(zt − b) = ψtM ( 1 Vt, 1) (17.20) (17.20) is now one equation in one unknown, Vt. To explicitly solve for Vt we need to make a functional form assumption on the matching function, M (⋅). But once we have Vt, we can determine the job ﬁnding rate, ft, from (17.12): ft = ψtM (1, Vt) (17.21) Once we have the job ﬁnding rate, we can determine the end of period unemployment t. Since all households begin as unemploymed, the end of period unemployment is level, u′ simply one minus the job ﬁnding rate, or: 392 u′ t = 1 − ft (17.22) To gain some additional insights, in the next subsection we specify a particular functional form for M (⋅) which allows us to clearly elucidate several results. 17.4.4 Example We assume that the matching function is Cobb Douglas, similar to our preferred production function speciﬁcation throughout the rest of the book. In particular, with 0 < ρ
< 1, let: Ht = ψtu1−ρ t vρ t (17.23) With this functional form, M ( 1 Vt, 1) = ψtV ρ−1 t. From (17.20), the total number of vacancies is therefore implicitly given by: Or: κ χ(zt − b) = ψtV ρ−1 t Vt = (ψtχ(zt − b) κ 1 1−ρ ) (17.24) (17.25) (17.25) provides a number of intuitive results. First, Vt is increasing in zt. This makes sense – the larger is zt, the bigger the potential gains to a ﬁrm from posting a vacancy. Second, Vt is also increasing in ψt. The bigger is ψt, the more eﬃcient vacancy-posting is at generating matches, and therefore the more vacancies ﬁrms ought to want to post. Third, vacancy posting is increasing in χ, which represents a ﬁrm’s bargaining weight. The bigger is χ, the more a ﬁrm stands to gain from successfully ﬁnding a match. Fourth, Vt is decreasing in the household’s outside option, b. The bigger is b, the higher must be the equilibrium wage, and the lower are the gains to a ﬁrm from posting a vacancy. Finally, the number of vacancies is naturally decreasing in the ﬁxed cost of posting a vacancy, κ. Once we have derived an expression for the equilibrium number of vacancies, we can derive an expression for the job ﬁnding rate. In particular, we have: ft = ψt (ψtχ(zt − b) κ ρ 1−ρ ) (17.26) The job ﬁnding rate, (17.26), is increasing in Vt, and hence depends on exogenous variables and parameters in qualitatively the same way that Vt does. In particular, the job ﬁnding 393 rate is greater the higher is match eﬃciency, ψt; the higher is productivity, zt; the higher is the ﬁrm’s bargaining weight, χ; and the lower are the household’s outside option, b,
and the ﬁxed cost of posting a vacancy, κ. As noted above, the end of period unemployment level, u′ t, is simply one minus the job ﬁnding rate, or: u′ t = 1 − ft = 1 − ψt ( ψtχ(zt − b) κ ρ 1−ρ ) (17.27) (17.27) gives an expression for the equilibrium (end of period) unemployment level (also equivalent to the rate). It is quite intuitive. For example, the more productive is the economy (i.e. zt is bigger), the lower will be the unemployment rate. The bigger ψt (i.e. the more eﬃcient is the matching process), the lower will be the unemployment rate. The greater is the cost of posting a vacancy, the higher will be the unemployment rate. One can use (17.27) to draw a number of useful inferences as pertains economic policy. Policies which increase unemployment beneﬁts, modeled here as b, ought to increase the unemployment rate. This could help explain why, for example, European countries, which typically have fairly generous unemployment beneﬁts, have higher unemployment rates on average than does the US. Similarly, policies which make the matching process less eﬃcient result in smaller values of ψt and higher unemployment. Otherwise well-intentioned policies, such as the well-known Ban the Box initiative, could have unintended consequences that could result in higher unemployment. “Ban the Box” prohibits employers from asking potential employees about criminal history. The motivation behind such a policy is to prevent discrimination against those who have a criminal history. But if a ﬁrm cannot access such information, it might be wary to hire at all. The lack of information could reduce matching eﬃciency and actually result in higher unemployment, particularly among the groups such initiatives are intended to help. Indeed, this is in fact what some recent research suggests.9 17.4.5 Eﬃciency Is the equilibrium of the search and matching model laid out in the previous sections eﬃcient? To answer this question, we need to consider a social planner’s problem like we did in Chapter 15 and examine whether the equilibrium allocations coincide with what a benevolent planner would choose. We can deﬁne the planner’s objective as the total output from consum
mated matches, which will be ztM (Vt, 1); plus the unemployment beneﬁts accruing to unmatched households, (1 − M (Vt, 1))b; minus the total cost of posting vacancies, κVt. This problem can be written 9See Doleac and Hansen (2016). 394 as one in which the planner simply chooses how many vacancies to post. Once Vt is known, all other allocations follow. Formally, the planner’s problem is: U = max Vt M (Vt, 1)zt + (1 − M (Vt, 1))b − κVt. The ﬁrst-order condition is ∂U ∂Vt This can be rearranged as = 0 ⇔ Mv(Vt, 1)zt − Mv(Vt, 1)b = κ. (17.28) Recall, the free-entry condition in equilibrium is Mv(Vt, 1) = κ zt − b. (17.29) κ χ(zt − b). For the equilibrium allocation to be eﬃcient, from (17.29) and (17.30) it evidently must be the case that Mv(Vt, 1) = χqt. To make better sense of this, let us return to the Cobb-Douglas matching function example. With this matching function, we would have: (17.30) qt = From (17.13), we have: Mv(Vt, 1) = ρψtV ρ−1 t qt = ψtV ρ−1 t (17.31) (17.32) Comparing (17.31)-(17.32), we can only have Mv(Vt, 1) = χqt if χ = ρ. In other words, for the equilibrium to be eﬃcient it must be the case that the bargaining weight of ﬁrms, χ, equals the elasticity of the matching function with respect to vacancies, ρ. In general there is no reason to expect this to be the case. If the number of vacancies increases by one percent, the number of matches increases by ρ percent. χ represents ﬁrms’ bargaining power. If ﬁrms have too much bargaining
power (i.e. χ is higher than ρ), they post “too many” vacancies in equilibrium. If ﬁrms do not have enough bargaining power, they post “too few” vacancies relative to the eﬃcient benchmark. The reason why the equilibrium often fails to be optimal is because of an externality and, in particular, a congestion externality. The congestion externality here is that ﬁrms do not take into account how entering the market reduces the job ﬁlling rate for all the other ﬁrms. If their bargaining power is high, too many ﬁrms enter the market. If their bargaining power is too low, not enough ﬁrms enter into the 395 market relative to what a benevolent social planner would desire. 17.5 Wage Posting and Directed Search If you scan job advertisements you will notice one thing often appears in the job ad that does not show up in the random search model: the wage. In the DMP model wages are determined after a potential worker and ﬁrm meet. In the real world, people often decide whether to submit their application to a job based on the wage the job is oﬀering. All else equal, an unemployed worker would prefer to apply to higher wage jobs. At the same time, higher wage jobs will attract more applicants and lower the probability that any one applicant gets the job. The tradeoﬀ between the probability of a job oﬀer and the wage is at the heart of directed search models pioneered by Moen (1997), Shimer (1996), and Acemoglu and Shimer (1999). The survey by Rogerson et al. (2005) also compares the directed search model to the DMP model. Our treatment of the directed search model is similar to theirs. As in the DMP model, all workers start oﬀ as unemployed. We assume the worker can apply to one job per period and that it is free to apply to a job. The expected utility to applying to a job at ﬁrm i is Ui = b + fi,t(wi,t − b) where wi,t is the wage oﬀered by ﬁrm i and fi,t is the probability of a successful match with ﬁrm i. It must be the case that the expected utility of applying to one job equals the expected utility of
applying to all other jobs. Why is this so? Suppose the expected utility of applying to ﬁrm i is higher than ﬁrm j, i.e. fi,t(wi,t − b) > fj,twj,t − b). Then, more workers would apply to ﬁrm i pushing down the job ﬁnding probability, fi,t. This would continue until the expected utility for applying to any job is the same as applying to any other. Thus, we can write the expected utility as U = b + fi,t(wi,t − b) (17.33) We can already observe that ﬁrms paying higher wages necessarily have more job applicants (and hence lower job ﬁnding rates). Again, this is the key tradeoﬀ in the model. If a ﬁrm successfully matches with a job candidate, they produce zt units of output. As before, the cost to posting a vacancy is κ. Expected proﬁts for the ﬁrm are Πt = −κ + qi,t(yt − wi,t) (17.34) where qi,t is the probability the ﬁrm ﬁlls their vacancy. q and f are linked by the constant returns to scale matching function Hi,t = ψtM (ui,t, Vi,t). 396 The job ﬁnding and job ﬁlling rates are given by fi,t = Hi,t ui,t = ψtM (ui,t, Vi,t) ui,t = ψtM (1, ) 1 Li,t qi,t = Hi,t Vi,t = ψtM (ui,t, Vi,t) Vi,t = ψtM (Li,t, 1) (17.35) (17.36) where Li,t = ui,t is the “queue length” for ﬁrms oﬀering a wage of wi,t. This is just the inverse Vi,t of labor market tightness from the last section. A ﬁrm that oﬀers higher wages will attract more applicants which drives down fi,t and increases qi,t. Note that, qi,t fi,t = Li,t. and that
fi,t and qi,t are functions of Li,t. Firms choose Li,t and wi,t to maximize ( 17.34) subject to (17.33). In other words, ﬁrms their “queue” length and wage Again, the tradeoﬀ is to oﬀer a high wage to get a lot of applicants or oﬀer a low wage and get few applicants. Solve for wi,t in (17.33) wi,t = U fi,t − 1 − fi,t fi,t b. Substitute this into proﬁts for ﬁrm. Π = −κ + q(Li,t)(zt − wi,t) = −κ + qi,t (zt − U fi,t + 1 − fi,t fi,t b) = −κ + qi,t(zt − b) − Li,t(U − b) The maximization problem is then, Πt = −κ + q(Li,t)(zt − b) − Li,t(U − b) max Li,t The ﬁrst order condition is ∂Πt ∂Li,t = q′(Li,t)(zt − b) − (U − b) = 0. (17.37) The only thing involving i is Li,t. Under our assumptions of the matching function, q′ is decreasing in L. Intuitively, increasing the queue length raises the probability of a ﬁrm ﬁlling a vacancy but at a decreasing rate. From (17.37), we can infer all ﬁrms choose the same 397 queue length. This also means they must choose the same wage. Thus, we can drop the i subscripts. Now, let’s solve for the wage. Substitute U out of (17.33) with the ﬁrst order condition. q′(Lt)(zt − b) = f (Lt)(wt − b) Since q f = L we can write q′(Lt)(zt − b) = qt Lt (wt − b) ⇔ wt = b + q′(Lt)Lt qt = b + q,L(zt − b) (zt − b) (17.38) (17.39)

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