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11+𝑟𝑟1,𝑡𝑡 𝐶𝐶𝑡𝑡+1 𝐶𝐶𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡+1 Original bundle New bundle Hypothetical bundle with new 𝑟𝑟𝑡𝑡 on same indifference curve 𝐶𝐶0,𝑡𝑡 𝐶𝐶0,𝑡𝑡+1 𝐶𝐶1,𝑡𝑡 𝐶𝐶1,𝑡𝑡+1 𝐶𝐶0,𝑡𝑡ℎ 𝐶𝐶0,𝑡𝑡+1ℎ The increase in rt causes the budget line to pivot through the endowment point, and is shown in the diagram in blue. To think about how this impacts the consumption bundle, it is useful to consider a hypothetical budget line which has the same slope as the new budget line (i.e. the slope given by the new, higher rt), but is positioned in such a way that the household would choose to locate on the original indiﬀerence curve. The hypothetical budget line is shown in the diagram in orange. Since this hypothetical budget line is steeper, but allows the household to achieve the same lifetime utility, the household must choose a hypothetical consumption bundle with lower current consumption and higher future consumption. This hypothetical consumption bundle is labeled C h 0,t+1. The movement to this hypothetical budget line represents what we call the substitution eﬀect – it shows how the consumption bundle would change after a change in the interest rate, where the household is compensated with suﬃcient income so as to leave lifetime utility unchanged. The substitution eﬀect has the household substitute away from the relatively more expensive good (period t consumption) and into the relatively cheaper good (period t + 1 consumption). 0,t and C h This is not the only eﬀect at play, however. This hypothetical budget line which allows the household to achieve the same lifetime utility level is unattainable – it lies everywhere outside of the actual new budget line, given in blue. The income e�
��ect is the movement from the hypothetical bundle with a higher rt but unchanged lifetime utility to a new indiﬀerence curve tangent to the new budget line, both shown in blue. The income eﬀect in this diagram looks similar to what was shown above for a change in Yt or Yt+1 – the household reduces, relative to the hypothetical consumption bundle, consumption in both periods. The new consumption bundle is labeled (C1,t, C1,t+1). Since both the substitution and income eﬀects go in the same direction for period t consumption (i.e. reduce it), Ct deﬁnitely falls when rt increases. Though the picture is drawn where Ct+1 rises, in principle this eﬀect is ambiguous – the substitution eﬀect says to increase Ct+1, whereas the income eﬀect is to reduce it. Which dominates is unclear in general. The intuitive way to think about the competing income and substitution eﬀects is as follows. As noted earlier, we can think of rt as the intertemporal price of consumption. When rt goes up, current consumption becomes expensive relative to future consumption. Holding income ﬁxed, the household would shift away from current consumption and into future consumption. But there is an income eﬀect. Since the household is originally borrowing, an increase in rt increases the cost of borrowing. This is like a reduction in its future income – for a given amount of current borrowing, the household will have less future income available after paying oﬀ its debt. A reduction in future income makes the household want to reduce consumption in both periods. Since income and substitution eﬀects go in the same direction for period t consumption, we can conclude that Ct falls when rt increases if the household is 198 originally borrowing. But we cannot say with certainty what the total eﬀect is for a saver. Let’s now consider the case of a saver formally. Figure 9.9 shows the case graphically. Initially the household locates at (C0,t, C0,t+1), where C0,t < Yt. The increase in rt causes the budget line to pivot through the endowment point, shown in blue. Figure 9.9: Increase in rt: Initially a Saver Let’s again use
a hypothetical budget line with the new slope given by the new higher rt but shifted such that the household would locate on the original indiﬀerence curve. This is shown in orange. In this hypothetical situation, the household would reduce Ct but increase Ct+1. To determine the total eﬀect on consumption, we need to think of how the household would move from the hypothetical budget line to the actual new budget line. The hypothetical orange budget line lies everywhere inside of the actual new budget line. This means that the both period t and period t + 1 consumption will increase, relative to the hypothetical case, when moving to the actual new budget line. This is the income eﬀect. The income and substitutions eﬀects go in the same direction for period t + 1 consumption, meaning that we can determine that Ct+1 deﬁnitely increases. The income eﬀect has period t consumption increasing, in contrast to the substitution eﬀect, which features Ct falling. Hence, the total eﬀect on period t consumption is theoretically ambiguous. The intuition for these eﬀects is similar to above. If the household is originally saving, a higher rt means that it will earn a higher return on that saving. For a given amount of current saving, a higher rt would generate more available income to spend in period t + 1 after earning interest. This is like having more future income, which makes the household 199 𝐶𝐶𝑡𝑡+1 𝐶𝐶𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡+1 𝐶𝐶0,𝑡𝑡 𝐶𝐶0,𝑡𝑡+1 𝐶𝐶1,𝑡𝑡+1 𝐶𝐶1,𝑡𝑡 𝐶𝐶0,𝑡𝑡ℎ 𝐶𝐶0,𝑡𝑡+1ℎ Original bundle New bundle Hypothetical bundle with new 𝑟𝑟𝑡𝑡 on same indifference curve want to consume more in both periods. This is the income e�
�ect, which for the case of period t consumption works opposite the substitution eﬀect. Table 9.1 summarizes the qualitative eﬀects of an increase in rt on Ct and Ct+1, broken down by income and substitution eﬀects. “+” means that the variable in question goes up when rt goes up, and “-” signs mean that the variable goes down. The substitution eﬀect does not depend on whether the household is initially borrowing or saving – Ct decreases and Ct+1 increases when rt goes up. The income eﬀect depends on whether the household is originally borrowing or saving. The total eﬀect is the sum of the income and substitution eﬀects. Table 9.1: Income and Substitution Eﬀects of Higher rt Substitution Eﬀect Income Eﬀect Total Eﬀect Ct Borrower Saver Ct+1 Borrower Saver - + + + +?? + From here on out, unless otherwise noted, we will assume that the substitution eﬀect dominates the income eﬀect. This means that the sign of the total eﬀect of an increase in rt is driven by the substitution eﬀect. This seems to be the empirically relevant case. This means that we assume that, when rt increases, Ct goes down while Ct+1 goes up, regardless of whether the household is initially borrowing or saving. From this graphical analysis, we can conclude that there exists a consumption function which maps the things which the household takes as given – Yt, Yt+1, and rt – into the optimal level of current consumption, Ct. We will denote this consumption function by: Ct = C d(Yt +, Yt+1 +, rt − ). (9.32) Here C d(⋅) is a function mapping current and future income and the real interest rate into the current level of consumption. In (9.32), the “+” and “−” signs under each argument of the consumption function denote the signs of the partial derivatives; e.g. ∂Cd > 0. The ∂Yt partial derivative with respect to rt is negative under the assumption that the substitution eﬀect dominates the income e�
�ect. As noted, we refer to the partial derivative with respect to current income as the marginal propensity to consume, or MPC. (⋅) (9.32) is qualitative. To get an explicit expression for the consumption function, we would 200 need to make a functional form on the utility function, u(⋅). We can think about the Euler equation as providing one equation in two unknowns (Ct and Ct+1). The intertemporal budget constraint is another equation in two unknowns. One can combine the Euler equation with the intertemporal budget constraint to solve for an analytic expression for the consumption function. Suppose that the ﬂow utility function is the natural log, so u(Ct) = ln Ct. Then the Euler equation tells us that Ct+1 = β(1 + rt)Ct. Take this expression for Ct+1 and plug it into the intertemporal budget constraint, which leaves just Ct on the left hand side. Simplifying, one gets: Ct = 1 1 + β [Yt + Yt+1 1 + rt ]. (9.33) (9.33) is the consumption function for log utility. We can calculate the partial derivatives of Ct with respect to each argument on the right hand side as follows: ∂Ct ∂Yt ∂Ct ∂Yt+1 ∂Ct ∂rt = = 1 1 + β 1 1 + β = − Yt+1 1 + β 1 1 + rt (1 + rt)−2. (9.34) (9.35) (9.36) The MPC is equal to 1 1+β, which is between 0 and 1 since β is between 0 and 1. The closer β is to zero (i.e. the more impatient the household is), the closer is the MPC to 1. For this particular functional form, the MPC is just a number and is independent of the level of current income. This will not necessarily be true for other utility functions, though throughout the course we will often treat the MPC as a ﬁxed number independent of the level of income for tractability. The partial derivative with respect to future income, (9.35), is positive, as predicted from our indiﬀerence curve / budget line analysis. Note that rt could potentially be negative, but it can never be less than −1.
If it were less than this, saving a unit of goods would entail paying back more goods in the future, which no household would ever take. If the real interest rate is negative but greater than −1, this would mean that saving a unit of goods would entail getting back less than one unit of goods in the future; the household may be willing to accept this if it cannot otherwise store its income across time. For the more likely case in which rt > 0, the partial derivative of period t consumption with respect to future income is positive but less than the partial with respect to current income. Finally, the partial derivative with respect to the real interest rate is less than or equal to zero. It would be equal to zero in the case in which Yt+1 = 0. In this case, with no future income the household would choose to save in the ﬁrst period, and for this particular speciﬁcation of 201 preferences the income and substitution eﬀects would exactly cancel out. Otherwise, as long as Yt+1 > 0, the substitution eﬀect dominates, and current consumption is decreasing in the real interest rate. 1 With the log utility function, the consumption function takes a particularly simple form which has a very intuitive interpretation. In particular, looking at (9.33), one sees current consumption is simply proportional to the present discounted value of the stream of income. The present discounted value of the stream of income is Yt + Yt+1, while the proportionality 1+rt constant is 1+β. An increase in either Yt or Yt+1 increases the present value of the stream of income, the increase in Yt by more than the increase in Yt+1 if rt > 0. An increase in rt reduces the present discounted value of the stream of income so long as Yt+1 > 0, because future income ﬂows get more heavily discounted. Thinking of current consumption as proportional to the present discounted value of the stream of income is a very useful way to think about consumption-saving behavior, even though the consumption function only works out explicitly like this for this particular log utility speciﬁcation. 9.4 Extensions of the Two Period Consumption-Saving Model 9.4.1 Wealth In our baseline analysis, we assumed two things: (i) the household begins period t with no stock of wealth and (ii) there is only one asset with which the household can transfer resources
across time, the stock of which we denote with St. In this subsection we relax both of these assumptions. In particular, suppose that the household begins life with with an exogenous stock of wealth, Ht−1. You could think about this as a quantity of housing or shares of stock. Suppose that the period t price of this asset (denominated in units of goods) is Qt, which the household takes as given. The household can accumulate an additional stock of this wealth to take into period t + 1, which we denote with Ht. The period t budget constraint is: Ct + St + QtHt ≤ Yt + QtHt−1 This budget constraint can equivalently be written: Ct + St + Qt(Ht − Ht−1) ≤ Yt (9.37) (9.38) In (9.38), the household has some exogenous income in period t, Yt. It can consume, Ct, buy bonds, St, or buy/sell some of the other asset at price Qt, where Ht − Ht−1 is the change in the stock of this asset. 202 The period t + 1 budget constraint can be written: Ct+1 + St+1 + Qt+1 (Ht+1 − Ht) ≤ Yt+1 + (1 + rt)St (9.39) In (9.39), the household has exogenous income in period t + 1, Yt+1, and gets interest plus principal on its savings it brought into period t + 1, (1 + rt)St. It can consume, accumulate more savings, or accumulate more of the other asset, Ht+1. Since there is no period t + 2, the terminal conditions will be that St+1 = Ht+1 = 0 – the household will not want to leave any wealth (either in the form of St+1 or Ht+1) over for a period in which it does not live. This means that the second period budget constraint can be re-written: Ct+1 ≤ Yt+1 + (1 + rt)St + Qt+1Ht (9.40) If we assume that (9.40) holds with equality, we can solve for St as: − Yt+1 1 + rt Now, plugging this into (9.38), where we also assume that it holds with equality, yields: St = Ct+1 1
+ rt − Qt+1Ht 1 + rt (9.41) (9.42) Ct + Ct+1 1 + rt + QtHt−1 + Qt+1Ht 1 + rt + QtHt = Yt + Yt+1 1 + rt This is the modiﬁed intertemporal budget constraint. It reduces to (9.7) in the special case that Ht and Ht−1 are set to zero. The left hand side is the present discounted value of the stream of expenditure – Ct + Ct+1 is the present discounted value of the stream of 1+rt consumption, while QtHt is the present discounted value expenditure on the asset, Ht (i.e. purchases in period t of the asset). On the right hand side, we have the present discounted value of the stream of income, Yt + Yt+1, plus the existing value of the asset, QtHt−1, plus 1+rt the present discounted value of the asset in period t + 1, Qt+1Ht+1. Eﬀectively, we can think about the situation like this. The household begins life with Ht−1 of the asset, which it sells at price Qt. It then decides how much of the asset to buy to take into the next period, which is also buys at Qt. Hence, QtHt−1 is income in period t and QtHt is expenditure on the asset in period t. Then, the household sells oﬀ whatever value of the asset it has left in period t + 1 at price Qt+1. The present value of this is Qt+1Ht 1+rt. In general, the household gets to optimally choose how much of the asset to take into period t + 1. In other words, it gets to choose Ht, and there would be a ﬁrst order condition for this, in many ways similar to the Euler equation for consumption. Because we are focused on consumption here, to simplify matters let us assume that the household must choose Ht = 0 – in other words, the household simply sells oﬀ all of its asset in period t, and doesn’t take 1+rt 203 any of the asset into period t + 1. In this case, the modiﬁed intertemporal budget constraint reduces to: Ct + Ct+1 1 +
rt = Yt + Yt+1 1 + rt + QtHt−1 (9.43) In this case, we can think about QtHt−1 as simply representing exogenous income for the household. The consumption Euler equation is unaﬀected by the presence of this term in the intertemporal budget constraint, but it does aﬀect the budget line. In particular, we can think about the endowment point for the budget line as being Yt + QtHt−1 in period t, and Yt+1 in period t + 1. Let us analyze how an increase in Qt ought to impact consumption and saving behavior using an indiﬀerence curve / budget line diagram. Suppose that initially the consumer has income of Y0,t in period t and Y0,t+1 in period t + 1. The consumer is endowed with Ht−1 units of the asset and the original price of the asset is Q0,t. Suppose that the consumer initially chooses a consumption bundle C0,t, C0,t+1, shown at the tangency of the black indiﬀerence curve with the black budget line in Figure 9.10. Figure 9.10: Increase in Qt Suppose that there is an exogenous increase in Qt to Q1,t > Q0,t. This has the eﬀect of increasing the period t endowment point, which causes the entire budget line to shift to the right. The consumer will locate on the new, blue budget line at a consumption bundle C1,t, C1,t+1, where both period t and t + 1 consumption are higher. These eﬀects are similar to what happens after an increase in current income. The household will increase its current saving, St. 204 𝐶𝐶𝑡𝑡+1 𝐶𝐶𝑡𝑡 𝑌𝑌0,𝑡𝑡+𝑄𝑄0,𝑡𝑡𝐻𝐻𝑡𝑡−1 𝑌𝑌0,𝑡𝑡+1 𝑌𝑌0,𝑡𝑡+𝑄𝑄1,𝑡𝑡𝐻𝐻𝑡
𝑡−1 𝐶𝐶0,𝑡𝑡 𝐶𝐶0,𝑡𝑡+1 𝐶𝐶1,𝑡𝑡 𝐶𝐶1,𝑡𝑡+1 Original endowment Original consumption bundle New consumption bundle New endowment Empirical Evidence This situation is similar to the stock market boom of the 1990s. There is an alternative way to think about this model. Suppose that the household enters period t with no stock of the asset, so Ht−1 = 0. Suppose further that the household has to purchase an exogenous amount of the asset to take into the next period, Ht. One can think about this situation as the household being required to purchase a house to live in, which it will sell oﬀ after period t + 1. In this case, the intertemporal budget constraint becomes: Ct + Ct+1 1 + rt + QtHt = Yt + Yt+1 1 + rt + Qt+1Ht 1 + rt This can be re-written: Ct + Ct+1 1 + rt = Yt + Yt+1 1 + rt + Ht ( Qt+1 1 + rt − Qt) (9.44) (9.45) We can think about the current endowment point as being given by Yt − QtHt, since QtHt is required, exogenous expenditure. The future endowment point is Yt+1 + Qt+1Ht. Suppose that initially the household has income Y0,t and Y0,t+1, and the future price of the asset is Q0,t+1. This is shown in Figure 9.11. The household chooses an initial consumption bundle of C0,t, C0,t+1. Suppose that there is an anticipated increase in the future price of the asset, to Q1,t+1 > Q0,t+1. This eﬀectively raises the future endowment of income, which causes the budget line to shift outward, shown in blue. The household will choose a new consumption bundle, C1,t, C1,t+1, where both period t and period t + 1 consumption are higher. Graphically, the eﬀects here are similar to what happens when there
is an exogenous increase in future income. The household will reduce its current saving. 205 Figure 9.11: Increase in Qt+1 Empirical Evidence This setup is similar to the housing boom of the mid-2000s. Households expected future increases in house prices. This caused them to expand consumption and reduce saving. When the future increase in house prices didn’t materialize, consumption collapsed, which helped account for the Great Recession. The analysis in this subsection is greatly simpliﬁed in that we have ignored the fact that the household can choose Ht in reality. When Ht can be chosen, the eﬀects get more complicated. But the general gist is that wealth, broadly deﬁned, is something which ought to impact consumption behavior. We have documented two recent examples where wealth, housing wealth or stock market wealth, have played an important role in driving consumption behavior. 9.4.2 Permanent and Transitory Income Changes In the analysis above, we have examined the partial derivatives of consumption with respect to current and future income. Partial derivatives hold everything else ﬁxed. So when we talk about the partial eﬀect of an increase in Yt on Ct, we are holding Yt+1 ﬁxed. While this is a valid exercise in the context of the model, it does not necessarily correspond with what we know about changes in income in the data. In particular, changes in income empirically tend to be quite persistent in the sense that higher current income tends to be 206 𝐶𝐶𝑡𝑡+1 𝐶𝐶𝑡𝑡 𝑌𝑌0,𝑡𝑡+𝑄𝑄𝑡𝑡𝐻𝐻𝑡𝑡 𝑌𝑌0,𝑡𝑡+1+𝑄𝑄0,𝑡𝑡+1𝐻𝐻𝑡𝑡 𝐶𝐶0,𝑡𝑡 𝐶𝐶1,𝑡𝑡 𝐶𝐶0,𝑡𝑡+1 𝐶𝐶1,𝑡𝑡+1 Original consumption bundle New consumption bundle Original endowment New endowment
𝑌𝑌0,𝑡𝑡+1+𝑄𝑄1,𝑡𝑡+1𝐻𝐻𝑡𝑡 positively correlated with higher future income. A bonus would be an example of a one time change in income. But when you get a raise, this is often factored into future salaries as well. Consider the qualitative consumption function derived above, (9.32). Totally diﬀerentiate this about some point: dCt = ∂C d(⋅) ∂Yt dYt+1 + ∂C d(⋅) ∂rt dYt + ∂C d(⋅) ∂Yt+1 In words, (9.46) says that the total change in consumption is (approximately) the sum of the partial derivatives times the change in each argument. Let’s consider holding the real interest rate ﬁxed, so drt = 0. Consider what we will call a transitory change in income, so that dYt > 0 but dYt+1 = 0 – i.e., income only changes in the current period. Then the change in consumption divided by the change in income is just equal to the MPC, ∂C(⋅) ∂Yt (9.46) drt : dCt dYt = ∂C d(⋅) ∂Yt Next, consider what we will call a permanent change in income, where dYt > 0 and dYt+1 = dYt – i.e. income goes up by the same amount in both periods. The change in consumption divided by the change in income in this case is given by the sum of the partial derivatives of the consumption function with respect to the ﬁrst two arguments: (9.47) dCt dYt = ∂C d(⋅) ∂Yt + ∂C d(⋅) ∂Yt+1 (9.48) Since both of these partial derivatives are positive, (9.48) reveals that consumption will react more to a permanent change in income than to a transitory change in income. This means that saving will increase by less to a permanent change in income than to a transitory change in income. This result is a natural consequence of the household’
s desire to smooth consumption relative to income. If the household gets a one time increase in income in period t, income is relatively non-smooth across time. To smooth consumption relative to income, the household needs to increase its saving in period t, so as to be able to increase consumption as well in the future. But if income goes up in both periods, the household doesn’t need to adjust its saving as much, because it will have extra income in the future to support extra consumption. Hence, saving will go up by less, and consumption more, to a permanent change in income. Example Suppose that the utility function is log, so that the consumption function is given by (9.33). For simplicity, assume that β = 1 and rt = 0. From the Euler equation, this means that the household wants Ct = Ct+1. The intertemporal 207 budget constraint with these restrictions just says that the sum of consumption is equal to the sum of income. Combining these two together, we get: Ct = 1 2 (Yt + Yt+1) (9.49) In other words, with log utility, β = 1, and rt = 0, consumption is just equal to average income across periods. The MPC for this consumption function is 1 2. But if there is a permanent change in income, with Yt and Yt+1 both going up by the same amount, then average income goes up by the same amount, and hence = 1. In consumption will go up by the amount of the increase in income – i.e. dCt dYt other words, with this setup, a household will consume half and save half of a transitory change in income, but it will consume all of a permanent change in income, with no adjustment to its saving behavior. These results about the diﬀerential eﬀects of permanent and transitory changes in income on consumption have important implications for empirical work. For a variety of diﬀerent reasons, the magnitude of the MPC is an object of interest to policy makers. One would be tempted to conclude that one could identify the MPC by looking at how consumption reacts to changes in income. This would deliver the correct value of the MPC, but only in the case that the change in income under consideration is transitory. If the change in income is persistent, consumption will react by more than the MPC. If one isn’t careful, one could
easily over-estimate the MPC. 9.4.3 Taxes Let us augment the the basic model to include a situation where the household must pay taxes to a government. In particular, assume that the household has to pay Tt and Tt+1 to the government in periods t and t + 1, respectively. The two ﬂow budget constraints can be written: Ct + St ≤ Yt − Tt Ct+1 + St+1 ≤ Yt+1 − Tt+1 + (1 + rt)St (9.50) (9.51) Imposing the terminal condition that St+1 = 0, and assuming that these constraints hold with equality, gives the modiﬁed intertemporal budget constraint: 208 Ct + Ct+1 1 + rt = Yt − Tt + Yt+1 − Tt+1 1 + rt (9.52) (9.52) is similar to (9.7), except that income net of taxes, Yt − Tt and Yt+1 − Tt+1, appear on the right hand side. The Euler equation is identical. Functionally, changes in Tt or Tt+1 operate exactly the same as changes in Yt or Yt+1. We can write a modiﬁed consumption function as: Ct = C d(Yt − Tt, Yt+1 − Tt+1, rt) (9.53) Consider two diﬀerent tax changes, one transitory, dTt ≠ 0 and dTt+1 = 0, and one permanent, where dTt ≠ 0 and dTt+1 = dTt. Using the total derivative terminology from Section 9.4.2, we can see that the eﬀect of a transitory change in taxes is: While the eﬀect of a permanent tax change is: dCt dTt = − ∂C d(⋅) dYt dCt dTt = − [ ∂C d(⋅) ∂Yt + ∂C d(⋅) ∂Yt+1 ] (9.54) (9.55) Here, ∂Cd ∂Yt (⋅) is understood to refer to the partial derivative of the consumption function with respect to the ﬁr
st argument, Yt − Tt. Since Tt enters this negatively, this is why the negative signs appear in (9.54) and (9.55). The conclusion here is similar to our conclusion about the eﬀects of permanent and transitory income changes. Consumption ought to react more to a permanent change in taxes than a transitory change in taxes. Empirical Evidence There are two well-known empirical papers on the consumption responses to tax changes. Shapiro and Slemrod (2003) study the responses of planned consumption expenditures to the Bush tax cuts in 2001. Most households received rebate checks of either $300 or $600, depending on ﬁling status. These rebates were perceived to be nearly permanent, being the ﬁrst installment of a ten year plan (which was later extended). Our theory suggests that households should have spent a signiﬁcant fraction of these tax rebates, given their near permanent nature. Shapiro and Slemrod (2003) do not ﬁnd this. In particular, in a survey only 22 percent of households said that they planned to spend their tax rebate checks. This is inconsistent with the basic predictions of the theory as laid out in this chapter. 209 The same authors conducted a follow up study using a similar methodology. Shapiro and Slemrod (2009) study the response of consumption to the tax rebates from 2008, which were part of the stimulus package aimed at combatting the Great Recession. They ﬁnd that only about one-ﬁfth of respondents planned to spend their tax cuts from this stimulus plan. In contrast to the 2001 tax cuts, the 2008 rebate was understood to only be a temporary, one year tax cut. Hence, the low fraction of respondents who planned to spend their rebate checks is broadly in line with the predictions of the theory, which says that a household should save a large chunk of a temporary change in net income. 9.4.4 Uncertainty We have heretofore assumed that future income is known with certainty. In reality, while households may have a good guess of what future income is, future income is nevertheless uncertain from the perspective of period t – a household could get laid oﬀ in period t + 1 (income lower than expected) or it could win the lottery (income higher than expected). Our basic results that consumption ought to be forward-looking carry over into an environment with uncertainty – if a household expects an increase in future income,
even if that increase is uncertain, the household will want to consume more and save less in the present. In this subsection, we explore the speciﬁc role that uncertainty might play for consumption and saving decisions. Let’s consider the simplest possible environment. Suppose that future income can take on two possible values: Y h t+1, where the h and l superscripts stand for high and low. Let t+1 the probability that income is high be given by 0 ≤ p ≤ 1, while the probability of getting low income is 1 − p. The expected value of Yt+1 is the probability-weighted average of possible realizations: > Y l E(Yt+1) = pY h t+1 + (1 − p)Y l t+1 (9.56) Here, E(⋅) is the expectation operator. If p = 1 or p = 0, then there is no uncertainty and we are back in the standard case with which we have been working. The basic optimization problem of the household is the same as before, with the exception that it will want to maximize expected lifetime utility, where utility from future consumption is uncertain because future income is uncertain – if you end up with the low draw of future income, future consumption will be low, and vice-versa. In particular, future consumption will take on two values, given current consumption which is known with certainty: 210 C h t+1 C l t+1 = Y h t+1 = Y l t+1 + (1 + rt)(Yt − Ct) + (1 + rt)(Yt − Ct) (9.57) (9.58) The expected value of future consumption is E(Ct+1) = pC h t+1 t+1. The Euler equation characterizing optimal behavior looks similar, but on the right hand side there is expected marginal utility of future consumption: + (1 − p)C l u′(Ct) = β(1 + rt)E [u′(Ct+1)] (9.59) The key insight to understanding the eﬀects of uncertainty is that the expected value of a function is not in general equal to the function of the expected value. Marginal utility, u′(⋅), is itself a function, and as such in general the expected value of the marginal utility of future consumption is not equal to the marginal
utility of expected consumption. The example below makes this point clear: Example Suppose that the utility function is the natural log, u(⋅) = ln(⋅). Suppose that, = 2 given the choice of current Ct, future consumption can take on two values: C h t+1 = 1. Assume that the probability of the high realization is p = 0.5. The and C l expected value of consumption is: t+1 E(Ct+1) = 0.5 × 2 + 0.5 × 1 = 1.5 (9.60) Given log utility, the marginal utility of future consumption, u′(Ct+1), can take on two values as well: 1 1. Expected marginal utility is: 2 and 1 E [u′(Ct+1)] = 0.5 × 1 2 + 0.5 × 1 1 = 0.75 (9.61) Hence, in this particular example, the expected marginal utility of consumption is 0.75. What is the marginal utility of expected consumption? This is just the inverse of expected consumption, which is 1 3. Note that this is less than the 1.5 expected marginal utility of consumption. = 2 Let us consider the case in which the third derivative of the ﬂow utility function is strictly positive, u′′′(⋅) > 0. This is satisﬁed in the case of log utility used in the example above. 211 Figure 9.12 plots u′(Ct+1) as a function of Ct+1. The second derivative of the utility function being negative means that this plot is downward-sloping (i.e. the second derivative, u′′(⋅), is the derivative of the ﬁrst derivative, u′(⋅), and is hence the slope of u′(⋅) against Ct+1). The third derivative being positive means that the slope gets ﬂatter (i.e. closer to zero, so less negative) the bigger is Ct+1. In other words, the plot of marginal utility has a “bowed-in” shape in a way similar to an indiﬀerence curve. On the horizontal axis, we draw in the low and high realizations of future consumption. We can evaluate marginal utility at these consumption values by reading oﬀ the curve on the vertical axis. Marginal utility will be
high when consumption is low and low when consumption is high. The expected value of future consumption, E[Ct+1], lies in between the high and low realizations of consumption. The marginal utility of expected consumption, u′(E[Ct+1]), can be determined by reading oﬀ the curve at this point on the vertical axis. The expected marginal utility of consumption can be determined by drawing a straight line between marginal utility evaluated in the low draw of consumption and marginal utility when consumption is high. We then determine expected marginal utility of consumption by reading oﬀ of the line (not the curve) at the expected value of future consumption. Given the bowed-in shape of the plot of marginal utility, the line lies everywhere above the curve (i.e. marginal utility is convex, given a positive third derivative). This means that E[u′(Ct+1)] > u′(E[Ct+1]) – i.e. expected marginal utility is higher than the marginal utility of expected consumption. This is shown in the ﬁgure below and a formal proof follows. Figure 9.12: Expected Marginal Utility and Marginal Utility of Expected Consumption Mathematical Diversion 212 𝐶𝐶𝑡𝑡+1 𝑢𝑢′(𝐶𝐶𝑡𝑡+1) 𝐶𝐶𝑡𝑡+1ℎ 𝐶𝐶𝑡𝑡+1𝑙𝑙 𝐸𝐸[𝐶𝐶𝑡𝑡+1] 𝑢𝑢′(𝐶𝐶𝑡𝑡+1𝑙𝑙) 𝑢𝑢′(𝐶𝐶𝑡𝑡+1ℎ) 𝐸𝐸[𝑢𝑢′(𝐶𝐶𝑡𝑡+1)] 𝑢𝑢′(𝐸𝐸[𝐶𝐶𝑡𝑡+1]) We want to prove that expected marginal utility of future consumption can be evaluated at the line connecting the marginal utilities of consumption in the low and high states. The slope of the line connecting these points is
simply “rise over run,” or: slope = u′(C h t+1 C h ) − u′(C l − C l t+1 t+1 t+1 ) (9.62) Since we are dealing with a line, the slope at any point must be the same. Hence, the slope at the point E[Ct+1] must be equal to the expression found above. Let’s treat the value of marginal utility of consumption evaluated at E[Ct+1] as an unknown, call it x. The slope of the line at this point is equal to: slope = x − u′(C l ) t+1 E[Ct+1] − C l t+1 Because the slope is everywhere the same, we have: ) x − u′(C l t+1 E[Ct+1] − C l t+1 = u′(C h t+1 C h ) − u′(C l − C l t+1 t+1 t+1 (9.63) (9.64) ) Note that E[Ct+1] = pC h t+1 C l ) + C l t+1 t+1. Hence, we can write (9.64) as: + (1 − p)C l t+1, which can be written: E[Ct+1] = p(C h t+1 − ) x − u′(C l t+1 − C l p(C h t+1 t+1 ) = u′(C h t+1 C h ) − u′(C l − C l t+1 t+1 t+1 ) (9.65) Let’s work through this expression to solve for x. First: x − u′(C l t+1 ) = p(C h t+1 − C l t+1 This simpliﬁes to: )u′(C h t+1 C h ) − u′(C l − C l t+1 t+1 t+1 ) (9.66) x − u′(C l t+1 ) = pu′(C h t+1 ) − pu′(C l t+1 ) (9.67) Which further simpliﬁes to: x = pu′(C h t+1 ) + (1 − p)u′
(C l t+1 ) = E[u′(Ct+1)] (9.68) 213 In other words, the line evaluated at E[Ct+1] is the expected marginal utility of consumption. Having now shown how we can graphically determine the expected marginal utility of consumption, let us graphically analyze what happens to the expected marginal utility of consumption when there is an increase in uncertainty. To be precise, let us consider what is called a mean-preserving spread. In particular, suppose that the high realization of income gets bigger, Y h 0,t+1, in such a way that there is no change in the expected realization of income (holding the probabilities ﬁxed). In particular: 0,t+1, and the low realization of income gets smaller,t+1 1,t+1 pY h 1,t+1 + (1 − p)Y l 1,t+1 = pY h 0,t+1 + (1 − p)Y l 0,t+1 (9.69) The higher and lower possible realizations of income in the next period translate into higher and lower possible realizations of future consumption without aﬀecting the expected value of future consumption. We can graphically characterize how this increase in uncertainty impacts the expected marginal utility of consumption. This is shown below in Figure 9.13. We can see that the increase in uncertainty raises the expected marginal utility of consumption, even though expected consumption, and hence also the marginal utility of expected consumption, are both unaﬀected. Figure 9.13: An Increase in Uncertainty An intuitive way to think about this graph is as follows. If the third derivative of the utility function is positive, so that marginal utility is convex in consumption, the heightened 214 𝐶𝐶𝑡𝑡+1 𝑢𝑢′(𝐶𝐶𝑡𝑡+1) 𝐶𝐶0,𝑡𝑡+1ℎ 𝐶𝐶0,𝑡𝑡+1𝑙𝑙 𝐸𝐸[𝐶𝐶𝑡𝑡+1] 𝑢𝑢′(𝐶𝐶0,𝑡𝑡+1𝑙𝑙)
𝑢𝑢′(𝐶𝐶0,𝑡𝑡+1ℎ) 𝐸𝐸[𝑢𝑢′(𝐶𝐶0,𝑡𝑡+1)] 𝑢𝑢′(𝐸𝐸[𝐶𝐶𝑡𝑡+1]) 𝐶𝐶1,𝑡𝑡+1𝑙𝑙 𝐶𝐶1,𝑡𝑡+1ℎ 𝑢𝑢′(𝐶𝐶1,𝑡𝑡+1ℎ) 𝑢𝑢′(𝐶𝐶1,𝑡𝑡+1𝑙𝑙) 𝐸𝐸[𝑢𝑢′(𝐶𝐶1,𝑡𝑡+1)] ↑ uncertainty → ↑𝐸𝐸[𝑢𝑢′(𝐶𝐶𝑡𝑡+1)] bad state raises marginal utility of consumption more than the improved good state lowers marginal utility, so on net expected marginal utility increases. This means that a meanpreserving increase in uncertainty will raise the expected marginal utility of consumption. How will this impact optimal consumer behavior? To be optimizing, a household must choose a consumption allocation such that (9.59) holds. If the increase in uncertainty drives up the expected marginal utility of consumption, the household must alter its behavior in such a way to as make the Euler equation hold. The household can do this by reducing Ct, which drives up u′(Ct) and drives down E[u′(Ct+1)] (because reducing Ct raises expected future consumption via (9.57)). In other words, a household ought to react to an increase in uncertainty by increasing its saving. We call this precautionary saving. The motive for saving in the work of most of this chapter is that saving allows a household to smooth its consumption relative to its income. Our discussion in this subsection highlights an additional motivation for saving behavior. In this uncertainty example, saving is essentially a form of self-insurance. When you purchase conventional insurance products, you are giving up some current consumption (i.e
. paying a premium) so that, in the event that something bad happens to you in the future, you get a payout that keeps your consumption from falling too much. That’s kind of what is going on in the precuationary saving example, although diﬀerently from an explicit insurance product the payout is not contingent on the realization of a bad state. You give up some consumption in the present (i.e. you save), which gives you more of a cushion in the future should you receive a low draw of income. Empirical Evidence 9.4.5 Consumption and Predictable Changes in Income Let us continue with the idea that the realization of future income is uncertain. While there is a degree of uncertainty in the realization of future income, some changes in income are predictable (e.g. you sign a contract to start a new job with a higher salary starting next year). Our analysis above shows that current consumption ought to react to anticipated changes in future income – i.e. > 0. ∂C(⋅) ∂E[Yt+1] Suppose, for simplicity, that β(1 + rt) = 1. This means that the Euler equation under uncertainty reduces to: u′(Ct) = E[u′(Ct+1)] (9.70) In other words, an optimizing household will choose a consumption bundle so as to equate the marginal utility of consumption today with the expected marginal utility of consumption in the future. If there were no uncertainty, (9.70) would implies that Ct = Ct+1 – i.e. the 215 household would desire equal consumption across time. This will not in general be true if we assume that the future is uncertain via the arguments above that the marginal utility of expected future consumption does not in general equal the expected value of the marginal utility of future consumption. If we are willing to assume, however and in contrast to what we did above in discussing precautionary saving, that the third derivative of the utility function is zero, as would be the case with the quadratic utility function given above in (9.10), it would be the case that Ct = E[Ct+1] even if there is uncertainty over the future. In other words, the household would expect consumption to be constant across time, even though it may not be after the fact given that the realization of future income is uncertain. Hall (1978) assumes a utility function satisfying these properties and derives the implication that Ct
= E[Ct+1]. He refers to this property of consumption as the “random walk.” It implies that expected future consumption equals current consumption, and that changes in consumption ought to be unpredictable. Hall (1978) and others refer to the theory underlying this implication as the “life cycle - permanent income hypothesis.” The permanent income hypothesis is often abbreviated as PIH. While this only strictly holds if (i) β(1 + rt) = 1 and (ii) the third derivative of the utility function is zero, so that there is no precautionary saving, something close to the random walk implication that changes in consumption ought to be unpredictable holds more generally in an approximate sense. There have been many empirical tests of the PIH. Suppose that a household becomes aware at time t that its income will go up in the future. From our earlier analysis, this ought to result in an increase in Ct and a reduction in St. If the random walk implication holds, then in expectation Ct+1 should go up by the same amount as the increase in Ct. This has a stark implication: consumption ought not to change (relative to its period t value) in period t + 1 when income changes. This is because E[Ct+1] − Ct = 0 if the assumptions underlying the random walk model hold. In other words, consumption ought not to react in the period that income changes, because this anticipated change in income has already been worked into the consumption plan of an optimizing household. Below we describe two empirical studies of this prediction of the model. Empirical Evidence Social Security taxes are about seven percent of your gross income, and your employer withholds these payroll taxes from your paycheck and remits them directly to the government. However, there is a cap on the amount of income that is subject to Social Security taxes. In 2016, the maximum amount of taxable income subject to the Social Security tax was $118,500. Suppose that a household earns double this amount per year, or $237,000. Suppose this worker is paid 216 monthly. For each of the ﬁrst six months of the year, about 7 percent of your monthly income is withheld. But starting in July, there is no 7 percent withheld, because you have exceeded the annual cap. Hence, a worker with this level of income will experience an increase in his/her take-home pay starting in the second half of the year. Since this increase in take-home pay is perfectly predictable from the beginning of
the year, the household’s consumption behavior should not change when its after-tax paycheck goes up in the second half of the year. In other words, consumption in the ﬁrst half of the year ought to incorporate the knowledge that take-home pay will increase in the second half of the year, and there should be no change in consumption in the second half of the year relative to the ﬁrst half of the year. Parker (1999) studies the reaction of consumption of households who have withholding of Social Security phased out at some point in the calendar year. He ﬁnds that consumption increases when take-home pay predictably increases after the Social Security tax withholding phases out. This is inconsistent with the predictions of the theory. As another example of an empirical test of the PIH, there is a well-known monthly mortality cycle. In particular, deaths tend to decline immediately before the ﬁrst day of a new month but spike immediately thereafter. Evans and Moore (2012) provide a novel explanation for this mortality cycle. They argue, and provide evidence, that the mortality cycle is tied to physical activity, which is in turn tied to receipt of income. In particular, greater physical activity is correlated with higher (short run) mortality rates – e.g. you can’t get in a car accident if you aren’t driving, you can’t die of an overdose if you are not taking drugs, etc.. They document that physical activity is correlated with receipt of paychecks. Many workers are paid on or around the ﬁrst of the month. They argue that the receipt of a paycheck (which is predictable), leads to a consumption boom, which triggers higher mortality. That consumption would react to a predictable change in income (such as receipt of paycheck) is inconsistent with the predictions of the theory. 9.4.6 Borrowing Constraints In our baseline analysis, we have assumed that a household can freely borrow or save at interest rate rt. In reality, many households face imperfect (or no) access to credit markets. Households may not be able to borrow at all, or the interest rate on borrowing may exceed the 217 interest that can be earned on saving. We refer to such situations as borrowing constraints. Consider ﬁrst an extreme form of a borrowing constraint. In particular, it is required that St ≥ 0. In other words, a household cannot borrow in period t, although it can freely save at
rt. This is depicted graphically below in Figure 9.14. The strict borrowing constraint introduces a vertical kink into the budget line at the endowment point. Points where Ct > Yt are no longer feasible. The hypothetical budget line absent the borrowing constraint is depicted in the dashed line. Figure 9.14: Borrowing Constraint: St ≥ 0 A less extreme version of a borrowing constraint is a situation in which the interest rate on borrowing exceeds the interest rate on saving, i.e. rb t is the borrowing rate t and rs t the saving rate. This introduces a kink in the budget constraint at the endowment point, but it is not a completely vertical kink – the budget line is simply steeper in the borrowing region in comparison to the saving region. The strict constraint with St ≥ 0 is a special case of this, where rb t t, where rb = ∞. > rs 218 𝐶𝐶𝑡𝑡+1 𝐶𝐶𝑡𝑡 (1+𝑟𝑟𝑡𝑡)𝑌𝑌𝑡𝑡+𝑌𝑌𝑡𝑡+1 𝑌𝑌𝑡𝑡+𝑌𝑌𝑡𝑡+11+𝑟𝑟𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡+1 Infeasible if 𝑆𝑆𝑡𝑡≥0 Figure 9.15: Borrowing Constraint: rb t > rs t For the remainder of this subsection, let’s continue with the strict borrowing constraint in which St ≥ 0. Figure 9.16 shows a case where the borrowing constraint is binding: by this we mean a situation in which the household would like to choose a consumption bundle where it borrows in the ﬁrst period (shown with the dashed indiﬀerence curve, and labeled C0,d,t, C0,d,t+1). Since this point is unattainable, the household will locate on the closest possible indiﬀerence curve, which is shown with the solid line. This point will occur at the kink in the budget constraint
– in other words, if the household would like to borrow in the absence of the borrowing constraint, the best it can do is to consume its endowment each period, with C0,t = Y0,t and C0,t+1 = Y0,t+1. Note, because the budget line is kinked at this point, the Euler equation will not hold – the slope of the indiﬀerence curve is not tangent to the budget line at this point. The borrowing constraint would not bind if the household would prefer to save – if it prefers to save, the fact that it cannot borrow is irrelevant. We would say this is a non-binding borrowing constraint. 219 𝐶𝐶𝑡𝑡+1 𝐶𝐶𝑡𝑡 (1+𝑟𝑟𝑡𝑡𝑠𝑠)𝑌𝑌𝑡𝑡+𝑌𝑌𝑡𝑡+1 𝑌𝑌𝑡𝑡+𝑌𝑌𝑡𝑡+11+𝑟𝑟𝑡𝑡𝑏𝑏 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡+1 𝑟𝑟𝑡𝑡𝑏𝑏>𝑟𝑟𝑡𝑡𝑠𝑠 Figure 9.16: A Binding Borrowing Constraint Let’s examine what happens to consumption in response to changes in current and future income when a household faces a binding borrowing constraint. Suppose that there is an increase in current income, from Y0,t to Y1,t. This shifts the endowment point (and hence the kink in the budget line) out to the right, as shown in Figure 9.17. In the absence of the borrowing constraint, the household would move from C0,d,t, C0,d,t+1 to C1,d,t, C1,d,t+1, shown with the blue dashed indiﬀerence curve. As long as the increase in current income is not so big that the borrowing constraint ceases to bind, this point remains unattainable
to the household. The best the household will be able to do is to locate at the new kink, which occurs along the dashed orange indiﬀerence curve. Current consumption increases by the full amount of the increase in current income and there is no change in future consumption. Intuitively, if the household would like to consume more than its current income in the absence of the constraint, giving it some more income it just going to induce it to spend all of the additional income. 220 𝐶𝐶𝑡𝑡+1 𝐶𝐶𝑡𝑡 𝐶𝐶0,𝑡𝑡=𝑌𝑌0,𝑡𝑡 𝐶𝐶0,𝑡𝑡+1=𝑌𝑌0,𝑡𝑡+1 𝐶𝐶0,𝑑𝑑,𝑡𝑡 𝐶𝐶0,𝑑𝑑,𝑡𝑡+1 Figure 9.17: A Binding Borrowing Constraint, Increase in Yt Next, consider a case in which the household anticipates an increase in future income, from Y0,t+1 to Y1,t+1. This causes the endowment point to shift up. Absent the borrowing constraint, the household would like to increase both current and future consumption to C1,d,t and C1,d,t+1. This point is unattainable. The best the household can do is to locate at the new kink point, which puts it on the orange indiﬀerence curve. In this new bundle, current consumption is unchanged, and future consumption increases by the amount of the (anticipated) change in future income. 221 𝐶𝐶𝑡𝑡+1 𝐶𝐶𝑡𝑡 𝐶𝐶0,𝑡𝑡=𝑌𝑌0,𝑡𝑡 𝐶𝐶1,𝑡𝑡+1=𝐶𝐶0,𝑡𝑡+1=𝑌𝑌0,𝑡𝑡+1 𝐶�
��0,𝑑𝑑,𝑡𝑡 𝐶𝐶0,𝑑𝑑,𝑡𝑡+1 𝐶𝐶1,𝑑𝑑,𝑡𝑡 𝐶𝐶1,𝑑𝑑,𝑡𝑡+1 𝐶𝐶1,𝑡𝑡=𝑌𝑌1,𝑡𝑡 Figure 9.18: A Binding Borrowing Constraint, Increase in Yt+1 A binding borrowing constraint will signiﬁcantly alter the implications of the basic two period consumption model. In particular, the MPC out of current income will be one, as opposed to less than one, and consumption will not be forward looking. There is an important implication of this result for policy. Above we argued that transitory tax cuts would have smaller eﬀects than permanent tax cuts if the household is trying to smooth its consumption relative to its income. This will not be the case if the household is facing a binding borrowing constraint – the eﬀect on consumption will be independent of the perceived persistence of the tax cut. For political reasons, very persistent tax cuts are often unpopular because of eﬀects these might have on the national debt. If some fraction of the population is borrowing constrained, our analysis suggests that “targeted and temporary” tax cuts may nevertheless have a big stimulative eﬀect on consumption even if they are temporary, so long as the tax cuts are targeted at those likely to be borrowing constrained (typically members of the population with little wealth and low income). Another implication of binding borrowing constraints is that it might help provide resolution to some of the documented empirical failures of the random walk model. Absent a borrowing constraint, consumption ought to be forward-looking, and anticipated future changes in income ought to already be incorporated into current consumption. But with a binding borrowing constraint, consumption ceases to be forward-looking, and consumption cannot react to an anticipated change in income until that change in income is realized. 222 𝐶𝐶𝑡𝑡+1 𝐶𝐶𝑡𝑡 𝐶𝐶0,𝑡𝑡=𝐶𝐶1,�
��𝑡=𝑌𝑌0,𝑡𝑡 𝐶𝐶0,𝑡𝑡+1=𝑌𝑌0,𝑡𝑡+1 𝐶𝐶0,𝑑𝑑,𝑡𝑡 𝐶𝐶0,𝑑𝑑,𝑡𝑡+1 𝐶𝐶1,𝑑𝑑,𝑡𝑡 𝐶𝐶1,𝑑𝑑,𝑡𝑡+1 𝐶𝐶1,𝑡𝑡+1=𝑌𝑌1,𝑡𝑡+1 9.5 Summary • The consumption-savings problem is dynamic. Given its lifetime resources, the house- hold chooses consumption and saving to maximize lifetime utility. • The household faces a sequence of period budget constraints which can be combined into a lifetime budget constraint which says the present discounted value of consumption equals the present discounted value of income. • The opportunity cost of one unit of current consumption is 1 + rt future units of consumption. rt must be greater than −1 because a consumer would never exchange one unit of consumption today for fewer than zero units of future consumption; rt could be negative if income is otherwise non-storable. • The key optimality condition coming out of the household’s optimization problem is the Euler equation. The Euler equation equates the marginal rate of substitution of consumption today for consumption tomorrow equal to one plus the real interest rate. • Graphically, the optimal consumption bundle is where the indiﬀerence curve just “kisses” the budget constraint. • Current consumption increases with current income, but less than one for one. The reason is that diminishing marginal returns of consumption leads the household to smooth consumption. • Similarly, current consumption increases with increases in future income. This implies that favorable news about future income should be reﬂected in today’s consumption. • An increase in the real interest rate makes current consumption more expensive which is known as the substitution eﬀect. The increase in the real interest rate also has either a positive or negative income eﬀect depending on if the consumer is a borrower or saver. When the income
eﬀect is negative, current consumption unambiguously falls in response to an increase in the real interest rate. When the income and substitution eﬀects move in diﬀerent directions, the response of consumption in period t to an increase in the real interest rate is theoretically ambiguous. • Permanent changes in income ought to have larger eﬀects on current consumption than temporary changes. A corollary of this is that a permanent tax cut will stimulate consumption more than a temporary tax cut. 223 • Fluctuations in asset prices and wealth can impact consumption in ways similar to changes in current or future income. • If the future is uncertain, there may be an additional motivation saving other than smoothing consumption relative to income. This is known as precautionary saving and emerges if the third derivative of the utility function is positive. If that is the case, the expected value of the marginal utility of future consumption exceeds the marginal utility of the expected value of future consumption. • If a household takes all information about current and future income into its current consumption choice, changes in future consumption should be unpredictable provided β(1 + rt) = 1 and the third derivative of the utility function is zero. This is known as the random walk hypothesis. • Binding borrowing constraints make consumption less forward-looking and less smooth relative to income. Key Terms • Marginal utility of consumption • Diminishing marginal utility • Euler equation • Consumption function • Substitution eﬀect • Income eﬀect • Marginal propensity to consume • Permanent income hypothesis • Borrowing constraints Questions for Review 1. Explain why St+1 = 0. 2. Starting with 9.4 and 9.5, mathematically derive the lifetime budget con- straint. 3. Suppose you win a lottery and you are given the option between $10 today and $0 tomorrow or $5 today and $5 tomorrow. Which would you choose? 224 4. Write down the Euler equation in general terms and describe its economic intuition. 5. Graphically depict the solution to the consumption-saving problem. Clearly state why you know it is the solution. 6. Graphically show the eﬀects of an increase in Yt. Does consumption unam- biguously go up in both periods? Why or why not? 7. Graphically show the eﬀects of an increase in rt. Does consumption unam- biguously go up in both periods? Why or
why not? 8. Suppose β(1 + rt) < 1. Is the growth rate of consumption positive, negative, or zero? Exercises 1. Consumption-Savings Consider a consumer with a lifetime utility function U = u(Ct) + βu(Ct+1) that satisﬁes all the standard assumptions listed in the book. The period t and t + 1 budget constraints are Ct + St = Yt Ct+1 + St+1 = Yt+1 + (1 + r)St (a) What is the optimal value of St+1? Impose this optimal value and derive the lifetime budget constraint. (b) Derive the Euler equation. Explain the economic intuition of the equa- tion. (c) Graphically depict the optimality condition. Carefully label the intercepts of the budget constraint. What is the slope of the indiﬀerence curve at the optimal consumption basket, (C ∗ )? t, C ∗ t+1 (d) Graphically depict the eﬀects of an increase in Yt+1. Carefully label the intercepts of the budget constraint. Is the slope of the indiﬀerence curve ), diﬀerent than in part c? at the optimal consumption basket, (C ∗ (e) Now suppose Ct is taxed at rate τ so consumers pay 1 + τ for one unit of period t consumption. Redo parts a-c under these new assumptions. t, C ∗ t+1 225 (f) Suppose the tax rate increases from τ to τ ′. Graphically depict this. Carefully label the intercepts of the budget constraint. Is the slope of the indiﬀerence curve at the optimal consumption basket, (C ∗ ), diﬀerent than in part e? Intuitively describe the roles played by the substitution and income eﬀects. Using this intuition, can you deﬁnitively ∂τ and ∂C∗ prove the sign of ∂C∗ ∂τ? It is not necessary to use math for this. Describing it in words is ﬁne. t, C ∗ t+1 t+1 t 2. Consumption with Borrowing Constraints Consider the following consumption- savings problem. The consumer maximizes max Ct,Ct+1,St ln Ct + β ln
Ct+1 subject to the lifetime budget constraint Ct + Ct+1 1 + rt = Yt + Yt+1 1 + rt and the borrowing constraint Ct ≤ Yt. This last constraint says that savings cannot be negative in the ﬁrst period. Equivalently, this is saying consumers cannot borrow in the ﬁrst period. (a) Draw the budget constraint. (b) Assuming the constraint does not bind, what is the Euler equation? (c) Using the Euler equation, lifetime budget constraint and borrowing constraint, solve for the period t consumption function. Clearly state under what circumstances the borrowing constraint binds. (d) Suppose Yt = 3, Yt+1 = 10, β = 0.95 and r = 0.1. Show the borrowing constraint binds. (e) Suppose there is a one time tax rebate that increases Yt to 4. Leave Yt+1 = 10, β = 0.95 and r = 0.1. What is the marginal propensity to consume out of this tax rebate? 3. [Excel Problem] Suppose we have a household with the following lifetime utility function: U = ln Ct + β ln Ct+1 (a) Create an Excel ﬁle to compute indiﬀerence curves numerically. Suppose 226 that β = 0.95. Create range of values of Ct+1 from 0.5 to 1.5, with a space of 0.01 between (i.e. create a column of potential Ct+1 values ranging from 0.5, 0.51, 0.52, and so on). For each value of Ct+1, solve for the value of Ct which would yield a lifetime utility level of U = 0. Plot this. (b) Re-do part (a), but for values of lifetime utility of U = −0.5 and U = 0.5. (c) Verify that “northeast” is the direction of increasing preference and that the indiﬀerence curves associated with diﬀerent levels of utility do not cross. 4. Suppose we have a household with the following (non-diﬀerentiable) utility function: U = min(Ct, Ct+1) With this utility function, utility equals the minimum of period t and t + 1 consumption. For example, if Ct = 3 and Ct+1 = 4, then U = 3. If
Ct = 3 and Ct+1 = 6, then U = 3. If Ct = 5 and Ct+1 = 4, then U = 4. (a) Since this utility function is non-diﬀerentiable, you cannot use calculus to characterize optimal behavior. Instead, think about it a little bit without doing any math. What must be true about Ct and Ct+1 if a household with this utility function is behaving optimally? (b) The period t and t + 1 budget constraints are 9.4 and 9.5 respectively. Use the condition from (a) and the intertemporal budget constraint to derive the consumption function. (c) Is the MPC between 0 and 1? Is consumption decreasing in the real interest rate? 5. Some Numbers. A consumer’s income in the current period is Yt = 250 and income in the future period is Yt+1 = 300. The real interest rate rt is 0.05, or 5%, per period. Assume there are no taxes. (a) Determine the consumer’s lifetime wealth (present discounted value of lifetime income). (b) As in the previous problem, assume that the consumer’s preferences are such that the current and future consumption are perfect complements, so that he or she always wants to have equal consumption in the current and future periods. Draw the consumer’s indiﬀerence curves. (c) Solve for the consumer’s optimal current-period and future-period con- 227 sumption, and for optimal saving as well. Is the consumer a lender or a borrower? Show this situation in a diagram with the consumer’s budget constraint and indiﬀerence curves. (d) Now suppose that instead of Yt = 250, the consumer has Yt = 320. Again, determine optimal consumption in the current and future periods and optimal saving, and show this in a diagram. Is the consumer a lender or a borrower? (e) Explain the diﬀerences in your results between parts 5c and 5d. 228 Chapter 10 A Multi-Period Consumption-Saving Model In this chapter we consider an extension of the two period consumption model from Chapter 9 to more than two periods. The basic intuition from the two period model carries over to the multi-period setting. The addition of more than two periods makes the distinction between permanent and transitory changes in income more stark. It also allows us to
think about consumption-saving behavior over the life cycle. 10.1 Multi-Period Generalization Suppose that the household lives for the current period, period t, and T subsequent periods, to period t + T. This means that the household lives for a total of T + 1 periods – the current period plus T additional periods. For simplicity, assume that there is no uncertainty. The household begins its life with no wealth. Each period there is a potentially diﬀerent interest rate, rt+j, for j = 0,... T − 1, which determines the rate of return on saving taken from period t + j to t + j + 1. The household faces a sequence of ﬂow budget constraints (one in each period) as follows: Ct + St ≤ Yt Ct+1 + St+1 ≤ Yt+1 + (1 + rt)St Ct+2 + St+2 ≤ Yt+2 + (1 + rt+1)St+1 ⋮ Ct+T + St+T ≤ Yt+T + (1 + rt+T −1)St+T −1 (10.1) (10.2) (10.3) (10.4) In the multi-period framework the distinction between the stock of savings and ﬂow saving is starker than in the two period model. In the ﬂow budget constraints, St+j, for j = 0,..., T, denotes the stock of savings (with an s at the end) that the household takes from period t + j into period t + j + 1. Flow saving in each period is the change in the stock of savings, or St+j − St+j−1 is ﬂow saving in period t + j. Only when j = 0 (i.e. the ﬁrst period) will ﬂow saving and the stock of savings the household takes into the next period be the same. 229 As in the two period model, St+T denotes the stock of savings which the household takes from period t + T into t + T + 1. Since the household isn’t around for period t + T + 1, and since no lender will allow the household to die in debt, it must be the case that St+T = 0. This is a terminal condition, similar to the idea that St+1 = 0 in the two
period framework. So as to simplify matters, let us assume that rt+j = r for all j = 0,..., T − 1. In other words, let us assume that the interest rate is constant across time. This simpliﬁes the analysis when collapsing the sequence of ﬂow budget constraints into one intertemporal budget constraint, but does not fundamentally aﬀect the analysis. Making use of our terminal condition, St+T = 0, plus the assumption that each period’s budget constraint holds with equality, we can iteratively eliminate the savings terms from the ﬂow budget constraints starting from the end of the household’s life. This is conceptually similar to what we did in the two period model. For example, one can solve for St+T −1 = 1+r in the ﬁnal period. Then one can plug this in to the t + T − 1 budget constraint, and Ct+T 1+r then solve for St+T −2. One can keep going. Doing so, one arrives at a generalized intertemporal budget constraint given by (10.5): − Yt+T Ct + Ct+1 1 + r + Ct+2 (1 + r)2 + ⋅ ⋅ ⋅ + Ct+T (1 + r)T = Yt + Yt+1 1 + r + Yt+2 (1 + r)2 + ⋅ ⋅ ⋅ + Yt+T (1 + r)T (10.5) In words, this says that the present discounted value of the stream of consumption must equal the present discounted value of the stream of income. To discount a future value back to period t, you multiply by (1+r)j for a value j periods from now. This denotes how much current value you’d need to have an equivalent future value, given a ﬁxed interest rate of r. Expression (10.5) is a straightforward generalization of the intertemporal budget constraint in the two period setup. 1 Household preferences are an extension of the two period case. In particular, lifetime utility is: U = u(Ct) + βu(Ct+1) + β2u(Ct+2) + β3u(Ct+3) +... βT u(Ct+T ) (10.6) This setup
embeds what is called geometric discounting. In any given period, you discount the next period’s utility ﬂow by a ﬁxed factor, 0 < β < 1. Put diﬀerently, relative to period t, you value period t + 1 utility at β. Relative to period t + 2, you also value period t + 3 utility at β. This means that, relative to period t, you value period t + 2 utility at β2. And so on. Eﬀectively, your discount factor between utility ﬂows depends only on the number of periods away a future utility ﬂow is. Since β < 1, if T is suﬃciently big, then the relative weight on utility in the ﬁnal period relative to the ﬁrst period can be quite low. 230 The household problem can be cast as choosing a sequence of consumption, Ct, Ct+1, Ct+2,..., Ct+T to maximize lifetime utility subject to the intertemporal budget constraint: max Ct,Ct+1,...,Ct+T U = u(Ct) + βu(Ct+1) + β2u(Ct+2) + β3u(Ct+3) +... βT u(Ct+T ) (10.7) Ct + Ct+1 1 + r + Ct+2 (1 + r)2 + ⋅ ⋅ ⋅ + Ct+T (1 + r)T s.t. = Yt + Yt+1 1 + r + Yt+2 (1 + r)2 + ⋅ ⋅ ⋅ + Yt+T (1 + r)T (10.8) One can ﬁnd the ﬁrst order optimality conditions for this problem in an analogous way to the two period case – one solves for one of the consumption values from the intertemporal solve for Ct+T in the budget constraint in terms of the other consumption levels (e.g. intertemporal budget constraint), plugs this into the objective function, and this turns the problem into an unconstrained problem of choosing the other consumption values. Because this is somewhat laborious, we will not work through the optimization, although we do so in the example below for log utility with three total periods. The optimality conditions are a sequence of
T Euler equations for each two adjacent periods of time. These can be written: u′(Ct) = β(1 + r)u′(Ct+1) u′(Ct+1) = β(1 + r)u′(Ct+2) u′(Ct+2) = β(1 + r)u′(Ct+3) ⋮ u′(Ct+T −1) = β(1 + r)u′(Ct+T ) (10.9) (10.10) (10.11) (10.12) (10.13) These Euler equations look exactly like the Euler equation for the two period problem. Since there are T + 1 total periods, there are T sets of adjacent periods, and hence T Euler equations. Note that one can write the Euler equations in diﬀerent ways. For example, one could plug (10.10) into (10.9) to get: u′(Ct) = β2(1 + r)2u′(Ct+2). The intuition for why these Euler equations must hold at an optimum is exactly the same as in a two period model. Consider increasing Ct+1 by a small amount. The marginal beneﬁt of this is the marginal utility of period t + 1 consumption, which is βu′(Ct+1) (it is multiplied by β to discount this utility ﬂow back to period t). The marginal cost of doing this is saving one fewer unit in period t, which means reducing consumption in the next period by 1 + r units. The marginal cost is thus β2(1 + r)u′(Ct+2). Equating marginal beneﬁt to marginal 231 cost gives (10.10). One can think about there being a separate indiﬀerence curve / budget line diagram for each two adjacent periods of time. The example below works all this out for a three period case (so T = 2) with log utility. Example Suppose that a household lives for a total of three periods, so T = 2 (two future periods plus the present period). Suppose that the ﬂow utility function is the natural log, so that lifetime utility is: U = ln Ct + β ln Ct+1 + β2 ln Ct+2 (10.14) Assume that the interest rate is constant across time at r. The
sequence of ﬂow budget constraints, assumed to hold with equality and imposing the terminal condition, are: Ct + St = Yt Ct+1 + St+1 = Yt+1 + (1 + r)St Ct+2 = Yt+2 + (1 + r)St+1 In (10.17), solve for St+1: St+1 = Ct+2 1 + r − Yt+2 1 + r Now, plug (10.18) into (10.16): Ct+1 + Ct+2 1 + r = Yt+1 + Yt+2 1 + r + (1 + r)St Now, solve for St in (10.19): St = Ct+1 1 + r + Ct+2 (1 + r)2 − Yt+1 1 + r − Yt+2 (1 + r)2 (10.15) (10.16) (10.17) (10.18) (10.19) (10.20) Now, plug (10.20) into the period t ﬂow budget constraint for St. Re-arranging terms yields: Ct + Ct+1 1 + r + Ct+2 (1 + r)2 = Yt + Yt+1 1 + r + Yt+2 (1 + r)2 (10.21) 232 This is the intertemporal budget constraint when T = 2, a special case of the more general case presented above, (10.5). The household’s problem is then: max Ct,Ct+1,Ct+2 U = ln Ct + β ln Ct+1 + β2 ln Ct+2 (10.22) s.t. Ct + Ct+1 1 + r + Ct+2 (1 + r)2 = Yt + Yt+1 1 + r + Yt+2 (1 + r)2 (10.23) To solve this constraint problem, solve the intertemporal budget constraint for one of the choice variables. In particular, let’s solve for Ct+2: Ct+2 = (1 + r)2Yt + (1 + r)Yt+1 + Yt+2 − (1 + r)2Ct − (1 + r)Ct+1 (10.24) Now, we can plug this into (10.22), which transforms the problem
into an unconstrained one of choosing Ct and Ct+1: max Ct,Ct+1 U = ln Ct + β ln Ct+1 +... + β2 ln [(1 + r)2Yt + (1 + r)Yt+1 + Yt+2 − (1 + r)2Ct − (1 + r)Ct+1] The partial derivatives with respect to Ct and Ct+1 are: ∂U ∂Ct ∂U ∂Ct+1 = 1 Ct = β − β2(1 + r)2 1 Ct+2 − β2(1 + r) 1 Ct+2 1 Ct+1 (10.25) (10.26) In these ﬁrst order conditions, we have noted that the argument inside the third period ﬂow utility function is just Ct+2. Setting each of these derivatives to zero and simplifying: 233 1 Ct 1 Ct+1 = β2(1 + r)2 1 Ct+2 = β(1 + r) 1 Ct+2 (10.27) (10.28) From (10.28), we can see that 1 Ct+2 = 1 β(1+r) 1 Ct+1. Plugging this into (10.27) gives: 1 Ct = β(1 + r) 1 Ct+1 (10.29) Expressions (10.29) and (10.28) are the two Euler equations for the two sets of adjacent periods. We can use these conditions to solve for Ct+2 and Ct+1 in terms of Ct. In particular, from (10.29), we have: From (10.27), we have: Ct+1 = β(1 + r)Ct Ct+2 = β2(1 + r)2Ct (10.30) (10.31) Now, we can plug (10.31) and (10.30) into the intertemporal budget constraint, (10.23), leaving only Ct on the left hand side: Ct + β(1 + r)Ct 1 + r + β2(1 + r)2Ct (1 + r)2 = Yt + Yt+1 1 + r + Yt+2 (1 + r)2 We can then solve for Ct as: Ct = 1 1 + β + β2 [Yt + Yt+1 1 + r
+ Yt+2 (1 + r)2 ] (10.32) (10.33) Here, Ct is proportional to the present discounted value of lifetime income in a way that looks very similar to the consumption function with log utility in the two period case. What is diﬀerent is that the MPC is smaller than in the two period case due to the addition of the extra β2 term in the denominator of the proportionality constant. To go from Euler equations to consumption function, one can combine the Euler equations with the intertemporal budget constraint to solve for Ct alone as a function of the stream 234 of income and the ﬁxed interest rate. This is done in the case of log utility in the example above. To do this in the more general case, one either needs to make an assumption on the utility function, or an assumption on the relative magnitudes of β and 1 + r. Let’s go with the latter. In particular, let’s assume that β(1 + r) = 1. If this is the case, this implies that the household wants to equate the marginal utilities of consumption across time. But if there is no uncertainty, this then implies equating consumption across time. In other words, if β(1 + r) = 1, then the household will desire constant consumption across time – it will want Ct = Ct+1 = Ct+2 =... Ct+T. Let’s denote this ﬁxed value of consumption by ¯C. One can think about the intuition for the constant desired level of consumption under this restriction as follows. β < 1 incentivizes the household to consume in the present at the expense of the future – this would lead to declining consumption over time. r > 0 works in the opposite direction – it incentivizes the household to defer consumption to the future, because the return on saving is high. If β(1 + r) > 1, then the household will want the marginal utility of consumption to decline over time, which means that consumption will be increasing. In other words, with this restriction the beneﬁt to deferring consumption (i.e. r) is smaller than the cost of deferring consumption, which is governed by β. In contrast, if β(1 + r) < 1, then the household will want the marginal utility of consumption to increase over time, which means that consumption will be declining. If β(1 + r
) > 1, it means that the household is either suﬃciently patient (β big) and/or the return to saving is suﬃciently big (r high) that it pays to defer consumption to the future. If β(1 + r) < 1, then the household is either suﬃciently impatient (β small) and/or the return to saving is suﬃciently low (r low) that the household would prefer to frontload consumption. If β(1 + r) = 1, then the incentive to consume now at the expense of the present (β < 1) is oﬀset by the incentive to defer consumption to the future (r > 0), and the household desires a constant level of consumption. If consumption is constant across time, at ¯C, then the level of consumption can be factored out of the intertemporal budget constraint, (10.5) leaving: ¯C [1 + 1 1 + r + 1 (1 + r)2 +... 1 (1 + r)T ] = Yt + Yt+1 1 + rt + Yt+2 (1 + r)2 + ⋅ ⋅ ⋅ + Yt+T (1 + r)T (10.34) Since we have assumed that β(1 + r) = 1, then 1 1+r = β. This means that we can write (10.34) as: ¯C [1 + β + β2 + ⋅ ⋅ ⋅ + βT ] = Yt + βYt+1 + β2Yt+2 + ⋅ ⋅ ⋅ + βT Yt+T (10.35) A useful mathematical fact, derived in the Mathematical Diversion below, is that: 235 1 + β + β2 + ⋅ ⋅ ⋅ + βT = 1 − βT +1 1 − β (10.36) works out to simply 1 If T is suﬃciently big (or β suﬃciently small), then βT +1 is approximately zero, and this 1−β. This means that we solve for the constant level of consumption as: ¯C = 1 − β 1 − βT +1 [Yt + βYt+1 + β2Yt+2 + �
� ⋅ ⋅ + βT Yt+T ] (10.37) In this expression, consumption is constant across time and is proportional to the present discounted value of the stream of income, where the constant of proportionality is given by 1−β 1−βT +1. Mathematical Diversion Suppose you have a discounted sum, where 0 < β < 1: S = 1 + β + β2 +... βT (10.38) Multiply both sides of (10.38) by β: Sβ = β + β2 + β3 +... βT +1 (10.39) Subtract (10.39) from (10.38): S(1 − β) = 1 − βT +1 (10.40) In doing this subtraction, all but the ﬁrst term of S and the negative of the last term of Sβ cancel out. We can then solve for S as: S = 1 − βT +1 1 − β (10.41) This is the same expression presented in the main text, (10.36). 10.2 The MPC and Permanent vs. Transitory Changes in Income Suppose that β(1 + r) = 1 so that the consumption function is given by (10.37). We can calculate the marginal propensity to consume, ∂ ¯C ∂Yt, as: 236 ∂ ¯C ∂Yt = 1 − β 1 − βT +1 This MPC is positive and less than one, because βT +1 < β. Only if T = 0 (i.e. the household only lives for one period) would the MPC be equal to one. Further, the MPC will be smaller the bigger is T – the bigger is T, the smaller is βT +1, and hence the bigger is 1 − βT +1. In other words, the longer the household expects to live, the smaller ought to be its MPC. This has obvious policy implications in a world where you have households who are diﬀerent ages (i.e. have more or less remaining periods of life). We sometimes call such setups overlapping generations models. We would expect younger people (i.e. people with bigger T ) to have larger MPCs than older folks (i.e. people with smaller T ). (10.42) The intuition for why the MPC is decreasing in
T is straightforward. If the household desires constant consumption, it has to increase its saving in a period where income is high in order to increase consumption in other periods where income is not higher. The more periods where consumption needs to increase when income is not higher, the more the household has to increase its saving in the period where income increases. Hence, in response to a one period increase in income the household increases its consumption by less the bigger is T. The partial derivative of the constant value of consumption with respect to income received j periods from now is: ∂ ¯C ∂Yt+j = βj 1 − β 1 − βT +1 (10.43) If j = 0, then this reduces to (10.42). The bigger is j, the smaller is βj. In (10.37), consumption in each period is equal to a proportion of the present discounted value of ﬂow utility. The further oﬀ in time extra income is going to accrue, the smaller the eﬀect this has on the present discounted value of the stream of income, since β < 1. This means that the household adjusts its consumption less to an anticipated change in future income the further out into the future is that anticipated change in income. If T is suﬃciently large (i.e. the household lives for a suﬃciently long period of time), then βT +1 ≈ 0, and we can approximate the MPC with 1 − β. If β is large (i.e. relatively close to 1), then the MPC can be quite small. For example, if β = 0.95 and T is suﬃciently big, then the MPC is only 0.05. In other words, when the household lives for many periods, the MPC is not only less than 1, it ought to be quite close to 0. A household ought to save the majority of any extra income in period t, which is necessary to ﬁnance higher consumption in the future. To think about the distinction between permanent and transitory changes in income, we can take the total derivative of the consumption function, (10.37). This is: 237 d ¯C = 1 − βT +1 1 − β [dYt + βdYt+1 + β2dYt+2 + ⋅ ⋅ ⋅ + βT dYt+T ]
(10.44) For a transitory change in income, we have dYt > 0 and dYt+j = 0, for j > 0. Then the eﬀect of a transitory change in income on the ﬁxed level of consumption is just the MPC: = 1 − βT +1 1 − β Next, consider a permanent change in income, where dYt > 0 and dYt+j = dYt for j > 0. If income changes by the same amount in all future periods, then we can factor this out, which means we can write (10.44) as: (10.45) d ¯C dYt d ¯C = 1 − βT +1 1 − β dYt [1 + β + β2 + ⋅ ⋅ ⋅ + βT ] From (10.36), we know that the term remaining in brackets in (10.46) is then cancels with the ﬁrst term, leaving: d ¯C dYt = 1 (10.46) 1−β 1−βT +1. This (10.47) In other words, if β(1 + r) = 1, then a household ought to spend all of a permanent change in income, with no adjustment in saving behavior. Intuitively, the household wants a constant level of consumption across time. If income increases by the same amount in all periods, the household can simply increase its consumption in all periods by the same amount without adjusting its saving behavior. The analysis here makes the distinction between transitory and permanent changes in income from the two period model even starker. The MPC out of a transitory change in income ought to be very small, while consumption ought to react one-to-one to a permanent change in income. One can see this distinction between transitory and permanent changes in income even more cleanly if, in addition to assuming that β(1 + r) = 1, we further assume that β = 1 (and hence r = 0). In this case, the household still desires a constant level of consumption across time. But the intertemoral budget constraint just works out to the sum of consumption being equal to the sum of income. Hence, the consumption function becomes:1 ¯C = 1 T + 1 [Yt + Yt+1 + Yt+2 +... Yt+T ] (10
.48) 1One would be tempted to look at (10.37), plug in β = 1, and conclude that it is undeﬁned, since 1 − β = 0 and 1 − βT +1 = 0 if β = 1. 0/0 is undeﬁned, but one can use L’Hopital’s rule to determine that limβ→1 1−β 1−βT +1 = 1 T +1. 238 In other words, in (10.48) consumption is simply equal to average lifetime income – T + 1 is the number of periods the household lives, and the term in brackets is the sum of income across time. If T is suﬃciently large, then a transitory change in income has only a small eﬀect on average lifetime income, and so consumption reacts little. If there is a permanent change in income, then average income increases by the increase in income, and so the household consumes all of the extra income. Example Suppose that the household lives for 100 periods, so T = 99. Suppose it initially has income of 1 in each period. This means that average lifetime income is 1, so consumption is equal to 1 and is constant across time. Suppose that income in period t goes up to 2. This raises average lifetime income to 101 100, or 1.01. So consumption will increase by 0.01 in period t and all subsequent periods. The household increases its saving in period t by 0.99 (2 - 1.01). This extra saving is what allows the household to achieve higher consumption in the future. In contrast, suppose that income goes up from 1 to 2 in each and every period of life. This raises average lifetime income to 2. Hence, consumption in each period goes up by 1, from 1 to 2. There is no change in saving behavior. 10.3 The Life Cycle We can use our analysis based on the assumption that β(1 + r) = 1, which gives rise to constant consumption across time given by (10.37), to think about consumption and saving behavior over the life-cycle. In particular, suppose that a household enters adulthood in period t. It expects to retire after period t + R (so R is the retirement date) and expects to die after period t + T (so T is the death date). Suppose that it begins its working life with Yt units of income. Up until the retirement date, it expects its income to
grow each period by gross growth rate GY ≥ 0. In terms of the net growth rate, we have GY = 1 + gY, so GY = 1 would correspond to the case of a ﬂat income proﬁle. After date t + R, it expects to earn income level Y R, which we can think about as a retirement beneﬁt. This beneﬁt is expected to remain constant throughout retirement. Under these assumptions, we can solve for the constant level of consumption across time, ¯C, as follows: 239 ¯C = 1 − β 1 − βT +1 [Yt + βGY Yt + β2G2 Y Yt + ⋅ ⋅ ⋅ + βRGR Y Yt + βR+1Y R + βR+2Y R + ⋅ ⋅ ⋅ + βT Y R] (10.49) can be simpliﬁed as follows: ¯C = 1 − β 1 − βT +1 Yt [1 + βGY + (βGY )2 + ⋅ ⋅ ⋅ + (βGY )R] + 1 − β 1 − βT +1 This can be simpliﬁed further by noting that: (10.49) βR+1Y R [1 + β + ⋅ ⋅ ⋅ + βT −R−1] (10.50) 1 + βGY + (βGY )2 + ⋅ ⋅ ⋅ + (βGY )R = 1 − (βGY )R+1 1 − βGY 1 + β + ⋅ ⋅ ⋅ + βT −R−1 = 1 − βT −R 1 − β We can these plug these expressions in to get: ¯C = 1 − β 1 − βT +1 1 − (βGY )R+1 1 − βGY Yt + 1 − β 1 − βT +1 βR+1 1 − βT −R 1 − β Y R (10.51) (10.52) (10.53) (10.53) is the consumption function. Note that if T = R (i.e. if the household retires the same period it dies, so that there is no retirement period), then βT −R = β0 = 1
, so the last term drops out. We can see that consumption is clearly increasing in (i) initial income, Yt, and (ii) the retirement beneﬁt, Y R. It is not as straightforward to see, but consumption is also increasing in the growth rate of income during working years, gY. As long as the retirement beneﬁt is not too big relative to income during lifetime (i.e. Y R isn’t too large), consumption will be increasing in R (i.e. a household will consume more the longer it plans to work). Figure 10.1 plots hypothetical time paths for consumption and income across the life cycle. We assume that income starts out low, but then grows steadily up until the retirement date. Income drops substantially at retirement to Y R. The consumption proﬁle is ﬂat across time – this is a consequence of the assumption that β(1 + r) = 1. 240 Figure 10.1: Consumption and Income: The Life Cycle Within the ﬁgure, we indicate how the household’s saving behavior should look over the life cycle. Early in life, income is low relative to future income. Eﬀectively, one can think that current income is less than average income. This means that consumption ought to be greater than income, which means that the household is borrowing. During “prime working years,” which occur during the middle of life when income is high, the household ought to be saving. At ﬁrst, this saving pays oﬀ the accumulated debt from early in life. Then, this saving builds up a stock of savings, which will be used to ﬁnance consumption during retirement when income falls. Figure 10.2 plots a hypothetical stock of savings over the life cycle which accords with the consumption and income proﬁles shown in Figure 10.1. The stock of savings begins at zero, by assumption that the household begins its life with no wealth. The stock of savings grows negative (i.e. the household goes into debt for a number of periods). Then the stock of savings starts to grow, but remains negative. During this period, the household is paying down its accumulated debt. During the middle of life, the stock of savings turns positive and grows, reaching a peak at the retirement date. The stock of savings then declines as the household draws down its savings during retirement years. The household dies with a
zero stock of savings – this is simply a graphical representation of the terminal condition which we used to get the intertemporal budget constraint. 241 𝐶𝐶𝑡𝑡+𝑗𝑗,𝑌𝑌𝑡𝑡+𝑗𝑗 𝑌𝑌𝑡𝑡+𝑗𝑗 𝐶𝐶𝑡𝑡+𝑗𝑗 𝑗𝑗 0 𝑅𝑅 𝑇𝑇 borrow save dissave 𝑌𝑌𝑅𝑅 Figure 10.2: The Stock of Savings over The Life Cycle It is important to note that the time paths of income, consumption, and savings in Figures 10.1 and 10.2 are hypothetical. The only general conclusion is that the consumption proﬁle ought to be ﬂat (under the assumption that β(1 + r) = 1). Whether the household ever borrows or not depends on the income proﬁle – if income grows slowly enough, or if the retirement period is long enough, the household may immediately begin life by doing positive saving. The key points are that consumption ought to be ﬂat and savings ought to peak at retirement. Empirical Evidence The basic life cycle model laid out above predicts that consumption ought not to drop at retirement. This is an implication of the PIH which we studied in Chapter 9. Retirement is (more or less) predictable, and therefore consumption plans ought to incorporate the predictable drop in income at retirement well in advance. This prediction does not depend on the assumption that β(1 + r) = 1 and that the consumption proﬁle is ﬂat. One could have β(1 + r) > 1, in which case consumption would be steadily growing over time, or β(1 + r) < 1, in which case consumption would be declining over time. Relative to its trend (ﬂat, increasing, or decreasing) consumption should not react to a predictable change in income like at retirement. 242 𝑗𝑗 𝑇𝑇 𝑅𝑅 0 𝑆𝑆𝑡𝑡+𝑗𝑗 0 The so-called “retirement consumption puzzle”
documents that consumption expenditure drops signiﬁcantly at retirement. This is not consistent with the implications of the basic theory. Aguiar and Hurst (2005) note that there is a potentially important distinction between consumption and expenditure. In the data, we measure total dollar expenditure on goods. We typically call this consumption. But it seems plausible that people who are more eﬃcient consumers (for example, they ﬁnd better deals at stores) might have lower expenditure than a less eﬃcient consumer, even if consumption is the same. At retirement, the opportunity cost of one’s time goes down signiﬁcantly. Relative to those actively working, retired persons spend more time shopping (e.g. clipping coupons, searching for better deals), cook more meals at home relative to eating out, etc.. All of these things suggest that their expenditure likely goes down relative to their actual consumption. Aguiar and Hurst (2005) use a novel data set to measure caloric intake for individuals, and ﬁnd that there is no drop in caloric intake at retirement, though expenditure on food drops. They interpret this evidence as being consistent with the predictions of the life cycle model. 10.4 Summary • In a multi-period context the diﬀerence between “savings” and “saving” becomes important. The former is a stock variable, whereas the latter is a ﬂow variable. If the consumer’s current consumption is less than their current income, then saving is positive and adds to their savings. • The lifetime budget constraint is derived by combining all the sequential budget constraints. Like the two-period case, the present discounted value of consumption equals the present discounted value of income. • The marginal propensity to consume out of current income is decreasing the longer individuals are expected to live. This has the implication that the marginal propensity to consume out of tax cuts should be higher for older workers. • The marginal propensity to consume out of future income is decreasing in the number of periods before the income gain is realized. • The life cycle model predicts that consumers will borrow early in life when current earnings are below average lifetime earnings, save in midlife when current earnings are above average lifetime earnings, and dissave during retirement. 243 Key Terms • Saving and savings Questions for Review 1. What is the terminal condition in a multi-period model? Explain why this terminal condition makes sense. 2. Write down the Euler equation.
What is the economic interpretation on this equation? 3. An old person and young person both win a lottery worth the same dollar value. According to the life cycle model, whose current consumption will increase by more? How do you know? 4. Describe why a permanent change in taxes has a larger eﬀect on consumption than a one-time change in taxes. 5. Suppose the generosity of social security beneﬁts increase. How would this aﬀect the consumption of someone in their prime working years? Exercises 1. Suppose that a household lives for three periods. Its lifetime utility is: U = ln Ct + β ln Ct+1 + β2 ln Ct+2 It faces the following sequence of ﬂow budget constraints (which we assume hold with equality): Ct + St = Yt Ct+1 + St+1 = Yt+1 + (1 + rt)St Ct+2 + St+2 = Yt+2 + (1 + rt+1)St+1 Note that we are not imposing that the interest rate on saving/borrowing between period t and t + 1 (i.e. rt) is the same as the rate between t + 1 and t + 2 (i.e. rt+1). (a) What will the terminal condition on savings be? In other words, what value should St+2 take? Why? 244 (b) Use this terminal condition to collapse the three ﬂow budget constraints into one intertemporal budget constraint. Argue that the intertemporal budget constraint has the same intuitive interpretation as in the two period model. (c) Solve for Ct+2 from the intertemporal budget constraint, transforming the problem into an unconstrained one. Derive two Euler equations, one relating Ct and Ct+1, and the other relating Ct+1 and Ct+2. (d) Use these Euler equations in conjunction with the intertemporal budget constraint to solve for Ct as a function of Yt, Yt+1, Yt+2, r. (e) Derive an expression for the marginal propensity to consume, i.e. ∂Ct. ∂Yt Is this larger or smaller than in the two period case with the same consumption function? What is the intuition for your answer? (f) Derive an expression for the eﬀect of r on Ct – i.
e. derive an expression ∂Ct ∂r. Under what condition is this negative? 2. Life Cycle / Permanent Income Consumption Model [Excel Problem] Suppose that we have a household that lives for T + 1 periods, from period 0 to period T. Its lifetime utility is: U = u(C0) + +βu(C1) + β2u(C2) + ⋅ ⋅ ⋅ + βT u(CT ) U = T ∑ t=0 βtu(Ct) The household has a sequence of income, Y0, Y1,..., YT, which it takes as given. The household can borrow or lend at constant real interest rate r, with r > 0. The household faces a sequence of period budget constraints: C0 + S0 = Y0 C1 + S1 = Y1 + (1 + r)S0 C2 + S2 = Y2 + (1 + r)S1 ⋮ CT = YT + (1 + r)ST −1 Here St, t = 0, 1,..., T is the stock of savings that the household takes from period t to period t+1. The ﬂow, saving, is deﬁned as the change in the stock, 245 or St − St−1 (hence, in period 0, the ﬂow and the stock are the same thing). The sequence of budget constraints can be combined into the intertemporal budget constraint: C0 + C1 1 + r + C2 (1 + r)2 + ⋅ ⋅ ⋅ + CT (1 + r)T = Y0 + Y1 1 + r + Y2 (1 + r)2 + ⋅ ⋅ ⋅ + YT (1 + r)T T ∑ t=0 Ct (1 + r)t = T ∑ t=0 Yt (1 + r)t Once can show that there are T diﬀerent optimality conditions, satisfying: u′(Ct) = β(1 + r)u′(Ct+1) for t = 0, 1,..., T − 1 (a) Provide some intuition for this sequence of optimality conditions. (b) Assume that β(1 + r) = 1. What does this imply about consumption across time? Explain.
(c) Assume that r = 0.05. What must β be for the restriction in (b) to be satisﬁed? (d) Using your answer from (b), solve for an analytic expression for con- sumption as a function of r and the stream of income. (e) Now create an Excel ﬁle to numerically analyze this problem. Suppose that income grows over time. In particular, let Yt = (1 + gy)tY0 for t = 0, 1,..., T. Suppose that gy = 0.02 and that Y0 = 10. Assume that T = 50. Use this, in conjunction with the value of r from (c), to numerically solve for the time path of consumption. Create a graph plotting consumption and income against time. (f) Given your time series of consumption and income, create a time series of savings (stock) and saving (ﬂow). In period t, t = 0, 1,..., T, your savings should be the stock of savings that the household leaves that period with (they enter period 0 with nothing, but leave with something, either positive or negative). Create a graph plotting the time series of 246 savings. What is true about the stock of savings that the household leaves over after period T? (g) Are there periods in which your ﬂow saving variable is negative/positive but consumption is less than/greater than income? If so, what accounts for this? Explain. (h) Now modify the basic problem such that the household retires at date R < T. In particular, assume that the income process is the same as before, but goes to zero at date R + 1: Yt = (1 + gy)tY0 for t = 0, 1,..., R. Re-do the Excel exercise assuming that R = 39, so that income goes to 0 in period 40. Show the plot of consumption and income against time, and also plot the time series behavior of the stock of savings. Comment on how the life cycle of savings is aﬀected by retirement. (i) One popular proposal ﬂoating around right now is to raise the retirement age in the hope of making Social Security solvent. Suppose that the retirement age were increased by ﬁve years, from R = 39 to R = 44. What eﬀect would this have on consumption
? Other things being equal, do you think this change would be good or bad for the economy in the short run? 247 Chapter 11 Equilibrium in an Endowment Economy In Chapter 9, we studied optimal consumption-saving behavior in a two period framework. In this framework, the household takes the real interest rate as given. In this Chapter, we introduce an equilibrium concept in a world in which the household behaves optimally, but its income is exogenously given. We refer to this setup as equilibrium in an endowment economy. We call it an endowment economy to diﬀerentiate it from a production economy, where the total amount of income available for a household to consume and/or save is endogenously determined. 11.1 Model Setup Suppose that there are many households in an economy. We index these households by j and suppose that the total number of households is L. For example, consumption in period t of the jth household is denoted Ct(j). We assume that L is suﬃciently large that these households behave as price-takers. These households live for two periods, t (the present) and t + 1 (the future). Each period, they earn an exogenous amount of income, Yt(j) and Yt+1(j). For simplicity, assume that there is no uncertainty. They begin life with no wealth. They can save or borrow in period t at a common real interest rate, rt. Because they behave as price-takers, they take rt as given. All households have the same preferences. Lifetime utility for household j is: U (j) = u(Ct(j)) + βu(Ct+1(j)) (11.1) The period utility function has the same properties outlined in Chapter 9. Household j faces the sequence of period budget constraints: Ct(j) + St(j) ≤ Yt(j) Ct+1(j) + St+1(j) ≤ Yt+1(j) + (1 + rt)St(j) (11.2) (11.3) Imposing that these budget constraints both hold with equality, and imposing the terminal 248 condition that St+1(j) = 0, we arrive at the intertemporal budget constraint: Ct(j) + Ct+1(j) 1 + rt = Yt(j) + Yt+1(j) 1 + rt (11.
4) The household’s problem is to choose Ct(j) and Ct+1(j) to maximize (11.1) subject to (11.4). Since this is the same setup encountered in Chapter 9, the optimality condition is the familiar Euler equation: u′(Ct(j)) = β(1 + rt)u′(Ct+1(j)) (11.5) Since all agents in the economy face the same real interest rate, rt, and the Euler equation (Ct(j)) must hold for all agents, it follows that must be the same for all agents. Eﬀectively, (Ct+1(j)) this means that all agents will have the same expected growth rate of consumption, but the levels of consumption need not necessarily be the same across agents. Qualitatively, the Euler equation can be combined with the intertemporal budget constraint to yield a qualitative consumption function of the sort: u′ βu′ Ct(j) = C d(Yt(j), Yt+1(j), rt) (11.6) Consumption is increasing in current and future income, and decreasing in the real interest rate. The partial derivative of the consumption function with respect to the ﬁrst argument, ∂Cd, is positive but less than one. We continue to refer to this as the marginal propensity ∂Yt (⋅) to consume, or MPC. 11.2 Competitive Equilibrium Though each agent takes the real interest rate as given, in the aggregate the real interest rate is an endogenous variable determined as a consequence of equilibrium. We will deﬁne an important concept called a competitive equilibrium as follows: a competitive equilibrium is a set of prices and quantities for which all agents are behaving optimally and all markets simultaneously clear. The price in this economy is rt, the real interest rate. We can interpret this as an intertemporal price of goods – rt tells you how much future consumption one can acquire by foregoing some current consumption. The quantities are values of Ct(j) and Ct+1(j). One could also think of saving, St(j), as an equilibrium outcome. What does it mean for “markets to clear” in this context? Loosely speaking, you can think about markets clearing as supply equaling demand. The one market in this economy is the market for bonds – a household decides how much it wants to
save, St(j) > 0, or borrow, St(j) < 0, given rt. In the aggregate, saving must be zero in this economy. Mathematically, 249 the sum of St(j) across households must equal zero: L ∑ j=1 St(j) = 0 (11.7) Why must this be the case? Consider a world where L = 2. Suppose that one household wants to borrow, with St(1) = −1. Where are the funds for this loan to come from? They must come from the second household, who must have St(2) = 1. If St(2) ≠ 1, then there would either be too much (or too little) saving for household 1 to borrow one unit. Hence, it must be the case that aggregate saving is equal to zero in this economy. This would not hold if the model featured capital (like in the Solow model), where it is possible to transfer resources across time through the accumulation of capital. Suppose that the ﬁrst period budget constraint holds with equality for all agents. Then, summing (11.2) across all L agents, we get: Ct(j) + L L ∑ ∑ j=1 j=1 j=1 Yt(j) as aggregate consumption and income, j=1 Ct(j) and Yt = ∑L respectively. Imposing the market-clearing condition that aggregate saving equals zero yields the aggregate resource constraint: Now, deﬁne Ct = ∑L St(j) = L ∑ j=1 Yt(j) (11.8) Ct = Yt (11.9) In other words, in the aggregate, consumption must equal income in this economy. This is again an artifact of the assumption that there is no production in this economy, and hence no way to transfer resources across time through investment (i.e. It = 0). Eﬀectively, one can think about equilibrium in this economy as follows. Given exogenous values of Yt(j) and Yt+1(j), and given an interest rate, rt, each household determines its consumption via (11.6). The real interest rate, rt, must adjust so that each household setting its consumption according to its consumption function is consistent with aggregate consumption equaling aggregate income. 11.3 Identical Agents and Graphical Analysis of the Equilibrium We
have already assumed that all agents have identical preferences (i.e. they all have the same β and same ﬂow utility function). In addition, let us further assume that they all face the same income stream – i.e. Yt(j) and Yt+1(j) are the same for all j. To simplify matters even further, let us normalize the total number of households to L = 1. This means that 250 Yt(j) = Yt and Ct(j) = Ct for all agents. This may seem a little odd. If L > 1, consumption and income of each type of agent would equal average aggregate consumption and income. But since we have normalized L = 1, the average of the aggregates is equal to what each individual household does. With all agents the same, optimality requires that: Ct = C d(Yt, Yt+1, rt) (11.10) Market-clearing requires that St = 0. Since all agents are the same, this means that, in equilibrium, no household can borrow or save. Intuitively, the reason for this is straightforward. If one agent wanted to borrow, then all agents would want to borrow (since they are all the same). But this can’t be, since one agent’s borrowing must be another’s saving. Hence, in equilibrium, agents cannot borrow or save. St = 0 implies the aggregate resource constraint: Ct = Yt (11.11) Expressions (11.10) and (11.11) are two equations in two unknowns (since Yt and Yt+1 are taken to be exogenous). The two unknowns are Ct and rt (the quantity and the price). Eﬀectively, the competitive equilibrium is a value of rt such that both of these equations hold (the ﬁrst requires that agents behave optimally, while the second says that markets clear). Mathematically, combining these two equations yields one equation in one unknown: Yt = C d(Yt, Yt+1, rt) (11.12) In equilibrium, rt must adjust to make this expression hold, given exogenous values of Yt and Yt+1. We can analyze the equilibrium of this economy graphically using the familiar tools of supply and demand. Let us focus ﬁrst on the demand side, which is more interesting since there is no
production in this economy. Let us introduce an auxiliary term which we will call desired aggregate expenditure, Y d t. Desired aggregate expenditure is simply the consumption function: Y d t = C d(Yt, Yt+1, rt) (11.13) Desired aggregate expenditure is a function of current income, Yt, future income, Yt+1, and the real interest rate, rt. We can graph this in a plot with Y d t on the vertical axis and Yt on the horizontal axis. This means that in drawing this graph we are taking the values of Yt+1 and rt as given. We assume that desired expenditure is positive even with zero current 251 income; that is, C d(0, Yt+1, rt) > 0. The level of desired expenditure for zero current income is often times called “autonomous expenditure” (the “autonomous” refers to the fact that this represents expenditure which is autonomous, i.e. independent, of current income). As current income rises, desired expenditure rises, but at a less than one-for-one rate (since the MPC is less than one). Hence, a graph of desired expenditure against current income starts with a positive vertical intercept and is upward-sloping with slope less than one. For simplicity, we will draw this “expenditure line” as a straight line (i.e. we assume a constant MPC), though it could have curvature more generally. The expenditure line is depicted in Figure 11.1 below. Figure 11.1: Expenditure and Income In Figure 11.1, we have drawn in a 45 degree line, which splits the plane in half, starts = Yt. In equilibrium, total in the origin, has slope of 1, and shows all points where Y d t expenditure must equal total income. So, the equilibrium value of Yt must be a point where the expenditure line crosses the 45 degree line. Given that we have assumed that autonomous expenditure is positive and that the MPC is less than 1, graphically one can easily see that the expenditure line must cross the 45 degree line exactly once. In the graph, this point is labeled Y0,t. The amount of autonomous expenditure depends on the expected amount of future income and the real interest rate, rt. A higher value of rt reduces autonomous expenditure, and therefore shifts the expenditure line down. This results in a lower level of Y
t where income equals expenditure. The converse is true for a lower real interest rate. We deﬁne a curve called the IS curve as the set of (rt, Yt) pairs where income equals expenditure and the household behaves optimally. In words, the IS curve traces out the 252 𝑌𝑌𝑡𝑡𝑑𝑑 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝐶𝐶𝑡𝑡=𝐶𝐶𝑑𝑑(𝑌𝑌𝑡𝑡,𝑌𝑌𝑡𝑡+1,𝑟𝑟𝑡𝑡) 𝑌𝑌0,𝑡𝑡 𝑌𝑌0,𝑡𝑡𝑑𝑑 Slope = MPC < 1 Slope = 1 combinations of rt and Yt for which the expenditure line crosses the 45 degree line. IS stands for investment equals saving. There is no investment and no saving in an endowment economy, so investment equaling saving is equivalent to consumption equaling income. We will also use a curve called the IS curve that looks very much like this in a more complicated economy with endogenous production and non-zero saving and investment later in the book. We can derive the IS curve graphically as follows. Draw two graphs on top of one another, both with Yt on the horizontal axis. The upper graph is the same as Figure 11.1, while the lower graph has rt on the vertical axis. Start with some value of the real interest rate, call it r0,t. Given a value of Yt+1, this determines the level of autonomous expenditure (i.e. the vertical axis intercept of the expenditure line). Find the level of income where expenditure equals income, call this Y0,t. “Bring this down” to the lower graph, giving you a pair, (r0,t, Y0,t). Then, consider a lower interest rate, r1,t. This raises autonomous expenditure, shifting
the expenditure line up. This results in a higher level of income where income equals expenditure, call it Y1,t. Bring this point down, and you have another pair, (r1,t, Y1,t). Next, consider a higher value of the interest rate, r2,t. This lowers autonomous expenditure, resulting in a lower value of current income where income equals expenditure. This gives you a pair (r2,t, Y2,t). In the lower graph with rt on the vertical axis, if you connect these pairs, you get a downward-sloping curve which we call the IS curve. In general, it need not be a straight line, though that is how we have drawn it here. This is shown below in Figure 11.2. 253 Figure 11.2: Derivation of the IS Curve The IS curve is drawn holding Yt+1 ﬁxed. Hence, changes in Yt+1 will cause the entire IS curve to shift to the right or to the left. Suppose that initially we have Y0,t+1. Suppose that we are initially at a point (r0,t, Y0,t) where income equals expenditure. Suppose that future income increases to Y1,t+1 > Y0,t+1. Holding the real interest rate ﬁxed at r0,t, the increase in future income raises autonomous expenditure, shifting the expenditure line up. This is shown in the upper panel of Figure 11.3. This upward shift of the expenditure line means that the level of current income where income equals expenditure is higher for a given real interest rate. Bringing this down to the lower graph, this means that the entire IS curve must shift to the right. A reduction in future income would have the opposite eﬀect, with the IS curve shifting in. 254 𝑌𝑌𝑡𝑡𝑑𝑑=𝐶𝐶𝑑𝑑(𝑌𝑌𝑡𝑡,𝑌𝑌𝑡𝑡+1,𝑟𝑟0,𝑡𝑡) 𝑌𝑌𝑡𝑡𝑑𝑑=𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑
𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑟𝑟𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝐶𝐶𝑑𝑑(𝑌𝑌𝑡𝑡,𝑌𝑌𝑡𝑡+1,𝑟𝑟2,𝑡𝑡) 𝑌𝑌𝑡𝑡𝑑𝑑=𝐶𝐶𝑑𝑑(𝑌𝑌𝑡𝑡,𝑌𝑌𝑡𝑡+1,𝑟𝑟1,𝑡𝑡) 𝑟𝑟0,𝑡𝑡 𝑟𝑟2,𝑡𝑡 𝑟𝑟1,𝑡𝑡 𝑟𝑟2,𝑡𝑡>𝑟𝑟0,𝑡𝑡>𝑟𝑟1,𝑡𝑡 𝐼𝐼𝐼𝐼 Figure 11.3: IS Curve Shift: ↑ Yt+1 The IS curve summarizes the demand side of the economy, showing all (rt, Yt) points where income equals expenditure. The supply side of the economy summarizes production, which must equal both income and expenditure in equilibrium. Since we are dealing with an endowment economy where there is no production, this is particularly simple. Generically, deﬁne the Y s curve as the set of (rt, Yt) pairs where agents are behaving optimally, consistent with the production technology in the economy. Since income is exogenous in an endowment economy, the Y s curve is just a vertical line at the exogenously given level of current income, Y0,t. This is shown below in Figure 11.4. 255 𝑌𝑌𝑡𝑡𝑑𝑑=𝐶𝐶𝑑𝑑(𝑌𝑌𝑡𝑡,𝑌𝑌0
,𝑡𝑡+1,𝑟𝑟0,𝑡𝑡) 𝑌𝑌𝑡𝑡𝑑𝑑=𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑟𝑟𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝐶𝐶𝑑𝑑(𝑌𝑌𝑡𝑡,𝑌𝑌1,𝑡𝑡+1,𝑟𝑟0,𝑡𝑡) 𝑟𝑟0,𝑡𝑡 𝑌𝑌1,𝑡𝑡+1>𝑌𝑌0,𝑡𝑡+1 𝐼𝐼𝐼𝐼(𝑌𝑌0,𝑡𝑡+1) 𝐼𝐼𝐼𝐼(𝑌𝑌1,𝑡𝑡+1) Figure 11.4: The Y s Curve In equilibrium, the economy must be on both the Y s and Y d curves. This is shown in Figure 11.5 below. 256 𝑌𝑌𝑡𝑡 𝑟𝑟𝑡𝑡 𝑌𝑌𝑠𝑠 𝑌𝑌0,𝑡𝑡 Figure 11.5: Equilibrium We can use the graphs in Figure 11.5 to analyze how rt will react to changes in current and future income in equilibrium. We do so in the subsections below. 11.3.1 Supply Shock: Increase in Yt Suppose that there is an exogenous increase in current income, from Y0,t to Y1,t. This results in the Y s curve shifting out to the right. There is no shift of the IS curve since a change in Yt does not aﬀect autonomous expenditure.
These eﬀects are shown in Figure 11.6 below: 257 𝑌𝑌𝑡𝑡𝑑𝑑 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝐶𝐶𝑡𝑡=𝐶𝐶𝑑𝑑(𝑌𝑌𝑡𝑡,𝑌𝑌𝑡𝑡+1,𝑟𝑟0,𝑡𝑡) 𝑌𝑌0,𝑡𝑡 𝑌𝑌0,𝑡𝑡𝑑𝑑 𝑟𝑟0,𝑡𝑡 𝐼𝐼𝐼𝐼 𝑌𝑌𝑠𝑠 𝑟𝑟𝑡𝑡 𝑌𝑌𝑡𝑡 Figure 11.6: Supply Shock: Increase in Yt The rightward shift of the Y s curve results in the real interest rate declining in equilibrium, to r1,t. The lower real interest rate raises autonomous expenditure, so the expenditure line shifts up (shown in green) in such a way that income equals expenditure at the new level of Yt. Intuitively, one can think about the change in the interest rate as working to “undo” the consumption smoothing which we highlighted in Chapter 9. When current income increases, the household (though there are many households, because they are all the same and we have normalized the total number to one, we can talk of there being one, representative household) would like to increase its current consumption but by less than the increase in current income. It would like to save what is leftover. But in equilibrium, this is impossible since there is no one who wants to borrow. Hence, the real interest rate must fall to dissuade the household from increasing its saving. The real interest rate has to fall suﬃciently so that the household is behaving according to its consumption function, but where its consumption simply
equals its income. 258 𝑌𝑌𝑡𝑡𝑑𝑑 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝐶𝐶𝑡𝑡=𝐶𝐶𝑑𝑑(𝑌𝑌𝑡𝑡,𝑌𝑌𝑡𝑡+1,𝑟𝑟0,𝑡𝑡) 𝑌𝑌0,𝑡𝑡 𝑌𝑌0,𝑡𝑡𝑑𝑑 𝑟𝑟0,𝑡𝑡 𝐼𝐼𝐼𝐼 𝑌𝑌𝑠𝑠 𝑟𝑟𝑡𝑡 𝑌𝑌𝑠𝑠′ 𝑌𝑌1,𝑡𝑡 𝑟𝑟1,𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝐶𝐶𝑡𝑡=𝐶𝐶𝑑𝑑(𝑌𝑌𝑡𝑡,𝑌𝑌𝑡𝑡+1,𝑟𝑟1,𝑡𝑡) 𝑌𝑌𝑡𝑡 11.3.2 Demand Shock: Increase in Yt+1 Next, suppose that agents anticipate an increase in future income, from Y0,t+1 to Y1,t+1. This aﬀects the current demand for goods, not the current supply. As shown in Figure 11.3, a higher Yt+1 causes the IS curve to shift out to the right. This is shown in blue in Figure 11.7 below. Figure 11.7: Demand Shock: Increase in Yt+1 The rightward shift of the IS
curve, combined with no shift in the Y s curve, means that in equilibrium Yt is unchanged while rt rises. The higher rt reduces autonomous expenditure back to its original level, so that the expenditure line shifts back down so as to intersect the 45 degree line at the ﬁxed level of current income. Why does rt rise when the household expects more future income? When future income is expected to increase, to smooth consumption the household would like to increase its current consumption by borrowing. But, in equilibrium, the household cannot increase its borrowing. Hence, rt must rise so as to dissuade the 259 𝑌𝑌𝑡𝑡𝑑𝑑 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝐶𝐶𝑡𝑡=𝐶𝐶𝑑𝑑�𝑌𝑌𝑡𝑡,𝑌𝑌0,𝑡𝑡+1,𝑟𝑟0,𝑡𝑡� = 𝐶𝐶𝑑𝑑�𝑌𝑌𝑡𝑡,𝑌𝑌1,𝑡𝑡+1,𝑟𝑟1,𝑡𝑡� 𝑌𝑌0,𝑡𝑡 𝑌𝑌0,𝑡𝑡𝑑𝑑 𝑟𝑟0,𝑡𝑡 𝐼𝐼𝐼𝐼 𝑌𝑌𝑠𝑠 𝑟𝑟𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝐶𝐶𝑡𝑡=𝐶𝐶𝑑𝑑(𝑌𝑌𝑡𝑡,𝑌𝑌1,𝑡𝑡+1,𝑟
𝑟0,𝑡𝑡) 𝑟𝑟1,𝑡𝑡 𝐼𝐼𝐼𝐼′ 𝑌𝑌𝑡𝑡 household from increasing its borrowing. In the new equilibrium, the consumption function must hold with the higher value of Yt+1 where Ct is unchanged. This necessitates an increase in rt. The exercises of examining how the real interest rate reacts to a change Yt or a change in Yt+1 reveal a useful insight. In particular, in equilibrium the real interest rate is a measure of how plentiful the future is expected to be relative to the present. If Yt+1 is expected to rise relative to Yt, then rt rises. In contrast, if Yt rises relative to Yt+1, then rt falls. As such, rt is a measure of how plentiful the future is expected to be relative to the present. This is because rt must adjust so as to undo the consumption smoothing that a household would like to do for a given rt. While it is only true that Ct = Yt in equilibrium in an endowment economy, this insight will also carry over into a more complicated model with capital accumulation, saving, and investment. Does the idea that the real interest rate conveys information about the plentifulness of the future relative to the present hold in the data? It does. In Figure 11.8, we show a scatter plot of the real interest rate (on the vertical axis) against a survey measure of expected real GDP growth in the US over the next ten years.1 This is only based on twenty-ﬁve years of annual data but the relationship between the two series is clearly positive, with a correlation of about 0.3. The correlation is not as strong as might be predicted by our simple model, but the model is in fact too simple – the real world features a number of complicating factors, like capital accumulation and endogenous production. But nevertheless the simple insight that the equilibrium real interest rate tells you something about how good the future is expected to be relative to the present still seems to hold in the data. 1The expected 10 year real GDP growth forecast is from the Survey of Professional Forecasters. The real interest rate series is the 10 year Treasury interest rate less than the ten year expected CPI inﬂation rate, also from the SPF. 260 Figure 11.8
: Real Interest Rates and Expected Output Growth 11.3.3 An Algebraic Example Continue with the setup outlined in this section – agents are all identical and the total number of households is normalized to one. Suppose that the ﬂow utility function is the natural log. This means that the Euler equation can be written: The consumption function is: Total desired expenditure is: Ct+1 Ct = β(1 + rt) Ct = 1 1 + β Yt + 1 1 + β Yt+1 1 + rt Y d t = 1 1 + β Yt + 1 1 + β Yt+1 1 + rt Equating expenditure with income gives an expression for the IS curve: (11.14) (11.15) (11.16) Yt+1 1 + rt (11.17) is a mathematical expression for the IS curve. It is decreasing in rt and shifts out if Yt+1 increases. Given an exogenous amount of current output, the equilibrium real interest rate can then be solved for as: Yt = 1 β (11.17) 261 -1.000.001.002.003.004.005.002.002.202.402.602.803.003.203.403.60Real RateExpected 10 Yr Output Growth 1 + rt = 1 β Yt+1 Yt (11.18) In (11.18), we observe that the equilibrium real interest rate is simply proportional to the expected gross growth rate of output. This makes it very clear that the equilibrium real interest rate is a measure of how plentiful the future is expected to be relative to the present. Note that nothing prohibits the real interest rate from being negative – if Yt+1 is suﬃciently small relative to Yt, and β is suﬃciently close to one, then we could have rt < 0. 11.4 Agents with Diﬀerent Endowments Now, let us suppose that agents have identical preferences, but potentially have diﬀerent endowments of income. Each type of agent has the Euler equation given by (11.5) and corresponding consumption function given by (11.6). The aggregate market-clearing condition is the same as in the setup where all households were identical. For simplicity, suppose that there are two types of agents, 1 and 2. Households of the same type are identical. Assume
that there are L1 of type 1 agents, and L2 of type 2 agents, with L1 + L2 = L being the total number of households in the economy. Let’s suppose that agents of type 1 receive income of Yt(1) = 1 in the ﬁrst period, but Yt+1(1) = 0 in the second. Agents of type 2 have the reverse pattern: Yt(2) = 0 and Yt+1(2) = 1. Suppose that agents have log utility. This means that the generic consumption function for any agent of any type is given by: Ct(j) = 1 1 + β [Yt(j) + Yt+1(j) 1 + rt ] for j = 1, 2 (11.19) Plugging in the speciﬁed endowment patterns for each type of agent yields the agent speciﬁc consumption functions: Ct(1) = Ct(2 + rt (11.20) (11.21) The aggregate market-clearing condition is that aggregate consumption equals aggregate income, or L1Ct(1) + L2Ct(2) = L1 (the aggregate endowment is L1 because there are L1 of type 1 agents who each receive one unit of the endowment). Plug in the consumption functions for each type of agent: 262 Now, use this to solve for rt: 1 1 + β [L1 + L2 1 + rt ] = L1 1 + rt = 1 β L2 L1 (11.22) (11.23) You will note that (11.23) is identical to (11.18) when all agents are identical, since L2 = Yt+1 (i.e. this is the aggregate level of future income) while L1 = Yt (i.e. this is the aggregate level of current income). In other words, introducing income heterogeneity among households does not fundamentally alter the information conveyed by equilibrium real interest rate. This setup is, however, more interesting in that there will be borrowing and saving going on at the micro level, even though in aggregate there is no borrowing or saving. We can plug in the expression for the equilibrium real interest rate into the consumption functions for each type, yielding: 1 Ct(1) = 1 + β Ct(2) = β 1 + β L1 L2 (11.24) (
11.25) We can use this to see how much agents of each type borrow or save in equilibrium. The saving function for a generic household is St(j) = Yt(j) − Ct(j), or: = β 1 + β St(1) = 1 − 1 1 + β St(2) = − β L1 1 + β L2 (11.26) (11.27) Here, we see that St(1) > 0 (households of type 1 save), while St(2) < 0 (households of type 2 borrow). It is straightforward to verify that aggregate saving is zero: St = L1St(1) + L2St(2) St = L1 β 1 + β − L2 β 1 + β L1 L2 = 0 (11.28) (11.29) In this setup, while aggregate saving is zero, individual saving and borrowing is not. 263 Agents of type 1 save, while agents of type 2 borrow. This makes sense – type 1 households have all their income in the ﬁrst period, while type 2 agents have all their income in the second period. These households would like to smooth their consumption relative to their income – type 1 households are natural savers, while type 2 agents are natural borrowers. Since these agents are diﬀerent, there is a mutually beneﬁcial exchange available to them. These agents eﬀectively engage in intertemporal trade, wherein type 1 households lend to type 2 households in the ﬁrst period, and then type 2 households pay back some of their income to type 1 households in the second period. This mutually beneﬁcial exchange arises from diﬀerences across agents. Nevertheless, these diﬀerences do not matter for the equilibrium value of rt, which depends only on the aggregate endowment pattern. Now let’s change things up a bit. Continue to assume two types of agents with identical preferences. There are L1 and L2 of each type of agent, with L = L1 + L2 total agents. Let’s change the endowment patterns a little bit. In particular, suppose that type 1 agents have Yt(1) = 0.75 and Yt+1(1) = 0.25 L2 and Yt+1(2) = 0.75. L1 Relative to the example worked out
above, the aggregate endowments in each period here are the same:. The type 2 agents have Yt(2) = 0.25 L1 L2 Yt = 0.75L1 + 0.25L2 Yt+1 = 0.25L1 L2 L1 = L1 L1 L2 + 0.75L2 = L2 (11.30) (11.31) Plug in these new endowment patterns to derive the consumption functions for each type of agent: + Ct(1) = 0.75 1 + β 0.25 L1 L2 1 + β Ct(2) = 0.25 L2 L1 (1 + β)(1 + rt) + 0.75 (1 + β)(1 + rt) (11.32) (11.33) Aggregate consumption is: 264 Ct = L1Ct(1) + L2Ct(2) 0.25L2 Ct = 0.75L1 (1 + β)(1 + rt) 1 + β Ct = L1 1 + β L2 (1 + β)(1 + rt) + + + 0.25L1 1 + β + 0.75L2 (1 + β)(1 + rt) (11.34) (11.35) (11.36) Now, equate aggregate consumption to the aggregate endowment (i.e. impose the market- clearing condition): Now solve for rt: L1 1 + β + L2 (1 + β)(1 + rt) = L1 1 + rt = 1 β L2 L1 (11.37) (11.38) Note that the expression for the equilibrium real interest rate here, (11.38), is identical to what we had earlier, (11.23). In particular, rt depends only on the aggregate endowments across time, in both setups Yt+1 = L2 and Yt = L1, not how those endowments are split across diﬀerent types of households. Similarly, the aggregate level of consumption depends only on the aggregate endowments. We did this example with particular endowment patterns, but you can split up the endowment patterns however you like (so long as the aggregate endowments are the same) and you will keep getting the same expression for the equilibrium real interest rate. These examples reveal a crucial point, a point which motivates the
use of representative agents in macroeconomics. In particular, so long as agents can freely borrow and lend with one another (through a ﬁnancial intermediary), the distribution of endowments is irrelevant for the equilibrium values of aggregate prices and quantities. It is often said that this is an example of complete markets – as long as agents can freely trade with one another, microeconomic distributions of income do not matter for the evolution of aggregate quantities and prices. Markets would not be complete if there were borrowing constraints, for example, because then agents could not freely trade with one another. In such a case, equilibrium quantities and prices would depend on the distribution of resources across agents. 11.5 Summary • In this chapter the real interest rate is an endogenous object. 265 • Endogenizing prices and allocations requires an equilibrium concept. We use a competitive equilibrium which is deﬁned as a set of prices and allocations such that all individuals optimize and markets clear. • In equilibrium, some individuals can save and others can borrow, but in aggregate there is no saving. • The IS curve is deﬁned as the set of (rt, Yt) points where total desired expenditure equals income. • The Y s curve is deﬁned as the set of all (rt, Yt) points such that individuals are behaving optimally and is consistent with the production technology of the economy. Since output is exogenously supplied in the endowment economy, the aggregate supply curve is a simple vertical line. • An increase in the current endowment shifts the aggregate supply curve to the right and lowers the equilibrium real interest rate. • An increase in the future endowment can be thought of as a “demand” shock. In this case the equilibrium real interest rate rises. • Provided all individuals are free to borrow and lend, the aggregate real interest rate is invariant to the distribution of endowments. Key Terms • Market clearing • Competitive equilibrium • Desired aggregate expenditure • Autonomous aggregate expenditure • IS curve • Y s curve • Complete markets Questions for Review 1. Write down the equations for a competitive equilibrium in a representative agent economy and describe what each one represents. 266 2. How do changes in rt provide information about the scarcity of resources today relative to tomorrow? 3. Graphically derive the IS curve. 4. Graphically depict the equilibrium in the IS - Y s graph. 5. Show graphically the eﬀect of an increase in Yt+1 on consumption and the
interest rate. Clearly explain the intuition. 6. Under what circumstances does the distribution of endowments become irrelevant for determining aggregate quantities? Exercises 1. General Equilibrium in an Endowment Economy Suppose the economy is populated by many identical agents. These agents act as price takers and take current and future income as given. They live for two periods: t and t + 1. They solve a standard consumption-savings problem which yields a consumption function Ct = C(Yt, Yt+1, rt). (a) What are the signs of the partial derivative of the consumption function? Explain the economic intuition. (b) Suppose there is an increase in Yt holding Yt+1 and rt ﬁxed. How does the consumer want to adjust its consumption and saving? Explain the economic intuition. (c) Suppose there is an increase in Yt+1 holding Yt and rt ﬁxed. How does the consumer want to adjust its consumption and saving? Explain the economic intuition. (d) Now let’s go to equilibrium. What is the generic deﬁnition of a competi- tive equilibrium? (e) Deﬁne the IS curve and graphically derive it. (f) Graph the Y s curve with the IS curve and show how you determine the real interest rate. (g) Suppose there is an increase in Yt. Show how this aﬀects the equilibrium real interest rate. Explain the economic intuition for this. (h) Now let’s tell a story. Remember we are thinking about this one good as fruit. Let’s say that meteorologists in period t anticipate a hurricane 267 in t + 1 that will wipe out most of the fruit in t + 1. How is this forecast going to be reﬂected in rt? Show this in your IS − Y s graph and explain the economic intuition. (i) Generalizing your answer from the last question, what might the equilibrium interest rate tell you about the expectations of Yt+1 relative to Yt? 2. Equilibrium with linear utility: Suppose that there exist many identical households in an economy. The representative household has the following lifetime utility function: U = Ct + βCt+1 It faces a sequence of period budget constraints which can be combined into one intertemporal budget constraint: Ct + Ct+1 1 + rt = Yt + Yt+1 1 +
rt The endowment, Yt and Yt+1, is exogenous, and the household takes the real interest rate as given. (a) Derive the consumption function for the representative household (note that it will be piecewise). (b) Derive a saving function for this household, where saving is deﬁned as St = Yt − Ct (plug in your consumption function and simplify). (c) Solve for expressions for the equilibrium values of rt. (d) How does rt react to changes in Yt and Yt+1. What is the economic intuition for this? (e) If j indexes the people in this economy, does Sj,t have to equal 0 for all j? How is this diﬀerent from the more standard case? 3. The Yield Curve Suppose you have an economy with one type of agent, but that time lasts for three periods instead of two. Lifetime utility for the household is: U = ln Ct + β ln Ct+1 + β2 ln Ct+2 The intertemporal budget constraint is: 268 Ct + Ct+1 1 + rt + Ct+2 (1 + rt)(1 + rt+1) = Yt + Yt+1 1 + rt + Yt+2 (1 + rt)(1 + rt+1) rt is the interest rate on saving / borrowing between t and t + 1, while rt+1 is the interest rate on saving / borrowing between t + 1 and t + 2. (a) Solve for Ct+2 in the intertemporal budget constraint, and plug this into lifetime utility. This transforms the problem into one of choosing Ct and Ct+1. Use calculus to derive two Euler equations – one relating Ct to Ct+1, and the other relating Ct+1 to Ct+2. (b) In equilibrium, we must have Ct = Yt, Ct+1 = Yt+1, and Ct+2 = Yt+2. Derive expressions for rt and rt+1 in terms of the exogenous endowment path and β. (c) One could deﬁne the “long” interest rate as the product of one period interest rates. In particular, deﬁne (1 + r2,t)2 = (1 + rt)(1 + rt+1) (the squared term
on 1 + r2,t reﬂects the fact that if you save for two periods you get some compounding). If there were a savings vehicle with a two period maturity, this condition would have to be satisﬁed (intuitively, because a household would be indiﬀerent between saving twice in one period bonds or once in a two period bond). Derive an expression for r2,t. (d) The yield curve plots interest rates as a function of time maturity. In this simple problem, one would plot rt against 1 (there is a one period maturity) and r2,t against 2 (there is a two period maturity). If Yt = Yt+1 = Yt+2, what is the sign of slope of the yield curve (i.e. if r2,t > r1,t, then the yield curve is upward-sloping). (e) It is often claimed that an “inverted yield curve” is a predictor of a recession. If Yt+2 is suﬃciently low relative to Yt and Yt+1, could the yield curve in this simple model be “inverted” (i.e. opposite sign) from what you found in the above part? Explain. 4. Heterogeneity in an endowment economy Suppose we have two types of households: A and B. The utility maximization problem for a consumer of type i is subject to max Ct,Ct+1 ln Ci,t + β ln Ci,t+1 Ci,t + Ci,t+1 1 + rt = Yi,t + Yi,t+1 1 + rt 269 Note that the A and B households have the same discount rate and the same utility function. The only thing that is possibly diﬀerent is their endowments. (a) Write down the Euler equation for households A and B. (b) Solve for the time t and t + 1 consumption functions for households A and B. (c) Suppose (YA,t, YA,t+1) = (1, 2) and (YB,t, YB,t+1) = (2, 1). Solve for the equilibrium interest rate. (d) Substitute this market clearing interest rate back into your consumption functions for type A and B households and solve for the equilibrium allocations. Which household
is borrowing in the ﬁrst period and which household is saving? What is the economic intuition for this? (e) Describe why borrowing and savings occur in this economy, but not the representative household economy. Why does household B have higher consumption in each period? (f) Assuming β = 0.9 compare the lifetime utility of each type of household when they consume their endowment versus when they consume their equilibrium allocation. That is calculate household A’s utility when it consumes its endowment and compare it to when household A consumes its equilibrium allocation. Which utility is higher? Do the same thing for household B. What is the economic intuition for this result? 270 Chapter 12 Production, Labor Demand, Investment, and Labor Supply In this chapter, we analyze the microeconomic underpinnings of the ﬁrm problem. In particular, we derive expressions for labor and investment demand. We also augment the household side of the model to include an endogenous labor choice. The work done in this chapter serves as the backbone of the neoclassical and Keynesian models to come. 12.1 Firm We assume that there exists a representative ﬁrm. This representative ﬁrm produces output, Yt, using capital, Kt, and labor, Nt, as inputs. There is an exogenous productivity term, At, which the ﬁrm takes as given. Inputs are turned into outputs via: Yt = AtF (Kt, Nt) (12.1) This is the same production assumed throughout Part II. We do not model growth in labor augmenting technology. In the terminology of Chapter 6, one can think about ﬁxing Zt = 1. Somewhat diﬀerently than in the chapters studying the long run, we are going to keep the time subscript on At. We want to entertain the consequences of both transitory changes in productivity (i.e. At changes but At+1 does not) as well as anticipated changes in productivity (i.e. At+1 changes, and agents are aware of this in period t, but At is unaﬀected). The production function has the same properties as assumed earlier. It is increasing in both arguments – FK > 0 and FN > 0, so that the marginal products of capital and labor are both positive. It is concave in both arguments – FKK < 0 and FN N < 0, so that there are diminishing marginal products of capital and
labor. The cross-partial derivative between capital and labor is positive, FKN > 0. This means that more capital raises the marginal product of labor (and vice-versa). We also assume that both inputs are necessary to produce anything, so F (0, Nt) = F (Kt, 0) = 0. Finally, we assume that the production function has constant returns to scale. This means F (γKt, γNt) = γF (Kt, Nt). In words, this means that if you double both capital and labor, you double output. The Cobb-Douglas production 271 function is a popular functional form satisfying these assumptions: F (Kt, Nt) = K α t N 1−α t, 0 < α < 1 (12.2) Figure 12.1 plots a hypothetical production function. In particular, we plot Yt as a function of Nt, holding Kt and At ﬁxed. The plot starts in the origin (labor is necessary to produce output), and is increasing, but at a decreasing rate. If either At or Kt were to increase, the production function would shift up (it would also become steeper at each value of Nt). This is shown with the hypothetical blue production function in the graph. Figure 12.1: The Production Function There is a representative household who owns the representative ﬁrm, but management is separated from ownership (i.e. the household and ﬁrm are separate decision-making entities). Both the household and ﬁrm live for two periods – period t (the present) and period t + 1 (the future). The ﬁrm is endowed with some existing capital, Kt, and hires labor, Nt, at real wage rate, wt. Capital is predetermined (and hence exogenous) in period t – the only variable factor of production for the ﬁrm in period t is labor. Investment constitutes expenditure by the ﬁrm on new capital which will be available for production in the future. The capital accumulation equation is the same as in the Solow model in Chapter 5: Kt+1 = It + (1 − δ)Kt (12.3) We assume that the ﬁrm must borrow funds from a ﬁnancial intermediary to fund investment. The cost of borrowing is rt, which is the same interest rate faced by the 272 �
�𝑌𝑡𝑡 𝑁𝑁𝑡𝑡 𝑌𝑌𝑡𝑡=𝐴𝐴𝑡𝑡𝐹𝐹(𝐾𝐾𝑡𝑡,𝑁𝑁𝑡𝑡) ↑𝐴𝐴𝑡𝑡 or ↑𝐾𝐾𝑡𝑡 household.1 Let BI BI t t denote borrowing by the ﬁrm to ﬁnance its investment. We assume that = It, so that all of investment expenditure must be ﬁnanced by borrowing. In period t, the ﬁrm’s proﬁt is the diﬀerence between its revenue (equal to its output, Yt) and its payments to labor, wtNt. This proﬁt is returned to the household as a dividend, Dt: Dt = Yt − wtNt (12.4) In period t + 1, the ﬁrm faces the same capital accumulation equation (12.3), but will not want to leave any capital over for period t + 2 (since the ﬁrm ceases to exist after period t + 1). Since the ﬁrm desires Kt+2 = 0, this implies that It+1 = −(1 − δ)Kt+1. This is analogous to the household not wanting to die with a positive stock of savings. In other words, in period t + 1 the ﬁrm does negative investment, which amounts to selling oﬀ its remaining capital in a sort of “liquidation sale.” After production in t + 1 takes place, there are (1 − δ)Kt+1 units of capital remaining (some of the capital brought into t + 1 is lost due to depreciation). This is sold oﬀ and is a component of ﬁrm revenue in period t + 1. Firm expenses in t + 1 include the labor bill, wt+1Nt+1, as well as paying oﬀ the interest and principal on the loan taken out in period t to ﬁnance investment, (1 + rt)BI t. The dividend returned to the household in t + 1 is then:
Dt+1 = Yt+1 + (1 − δ)Kt+1 − wt+1Nt+1 − (1 + rt)BI t The value of the ﬁrm is the present discounted value of dividends: Vt = Dt + 1 1 + rt Dt+1 (12.5) (12.6) Future dividends are discounted by 1 1+rt, where rt is the interest rate relevant for household saving/borrowing decisions. Why is this the value of the ﬁrm? Ownership in the ﬁrm is a claim to its dividends. The amount of goods that a household would be willing to give up to purchase the ﬁrm is equal to the present discounted value of its dividends, where the present discounted value is calculated using the interest rate relevant to the household. If we plug in the production function as well as the expressions for period t and t + 1 dividends, the value of the ﬁrm can be written: 1In a previous edition of the book we introduced an exogenous credit spread into the model at this point, with rI t = rt + ft, where ft is an exogenous spread and can be interpreted as a return to ﬁnancial intermediation. We now omit the credit spread here but return it later in the book when discussing ﬁnancial crises. 273 Vt = AtF (Kt, Nt) − wtNt+ 1 1 + rt [At+1F (Kt+1, Nt+1) + (1 − δ)Kt+1 − wt+1Nt+1 − (1 + rt)BI t ] (12.7) From the perspective of period t, the ﬁrm’s objective is to choose its labor input, Nt, and investment, It, to maximize its value, (12.7). This maximization problem is subject to two constraints – the capital accumulation restriction, (12.3), and the requirement that investment be ﬁnanced by borrowing, It = BI t. The ﬁrm’s constrained optimization problem can therefore be written: max Nt,It Vt = AtF (Kt, Nt) − wtNt+ 1 1 + rt [At+1F (Kt+1, Nt+1) + (1 − δ
)Kt+1 − wt+1Nt+1 − (1 + rt)BI t ] (12.8) s.t. Kt+1 = It + (1 − δ)Kt It = BI t If we combine the two constraints with one another, we can write: BI t = Kt+1 − (1 − δ)Kt (12.9) (12.10) (12.11) Substituting (12.11) in to eliminate BI t, we can then re-write the optimization problem, (12.8), as an unconstrained problem of choosing Nt and Kt+1: max Nt,Kt+1 1 1 + rt Vt = AtF (Kt, Nt) − wtNt+ [At+1F (Kt+1, Nt+1) + (1 − δ)Kt+1 − wt+1Nt+1 − (1 + rt) (Kt+1 − (1 − δ)Kt)] (12.12) To ﬁnd the value-maximizing levels of Nt and Kt+1, take the partial derivatives of Vt with respect to each: ∂Vt ∂Nt = AtFN (Kt, Nt) − wt 274 (12.13) ∂Vt ∂Kt+1 = 1 1 + rt [At+1FK(Kt+1, Nt+1) + (1 − δ) − (1 + rt)] (12.14) Setting these partial derivatives equal to zero and simplifying yields: wt = AtFN (Kt, Nt) 1 + rt = At+1FK(Kt+1, Nt+1) + (1 − δ) (12.15) (12.16) Expression (12.15) implicitly deﬁnes a demand for labor.2 In particular, a ﬁrm wants to hire labor up until the point at which the marginal product of labor, FN (Kt, Nt), equals the real wage. The intuition for this condition is simply that the ﬁrm wants to hire labor up until the point at which marginal beneﬁt equals marginal cost. The marginal beneﬁt of an additional unit of labor is the
marginal product of labor. The marginal cost of an additional unit of labor is the real wage. At an optimum, marginal beneﬁt and cost must be equal. If wt > FN (Kt, Nt), the ﬁrm could increase its value by hiring less labor; if wt < FN (Kt, Nt), the ﬁrm could increase its value by hiring more labor. Figure 12.2 plots a hypothetical labor demand function. Since FN N < 0, the marginal product of labor is decreasing in Nt. Hence, the labor demand curve slopes down. It could be curved or a straight line depending on the nature of the production function; for simplicity we have here drawn it is a straight line. Labor demand will increase if either At or Kt increase. For a given wage, if At is higher, the ﬁrm needs a higher level of Nt for the wage to equal the marginal product. Similarly, since we assume that FKN > 0, if Kt were higher, the ﬁrm would need more Nt for a given wt to equate the marginal product of labor with the wage. 2Note that one could also ﬁnd a ﬁrst order condition with respect to future labor, Nt+1, and would arrive at the same ﬁrst order condition, only dated t + 1 instead of t. 275 Figure 12.2: Labor Demand (12.15) implicitly deﬁnes the optimal Nt as a function of At and Kt. We will use the following to qualitatively denote the labor demand function: Nt = N d(wt − Labor demand is a function of the wage, productivity, and capital. The + and − signs denote the qualitative signs of the partial derivatives. Labor demand is decreasing in the real wage, increasing in At, and increasing in the capital stock. (12.17), Kt +, At + ) Next, let us focus on the ﬁrst order condition for the choice of Kt+1, (12.16). First, what is intuition for why this condition must hold? Suppose that the ﬁrm wants to do one additional unit of investment in period t. The marginal cost of doing an additional unit of investment is the interest plus principal that will be owed to the ﬁnancial intermediary in period t + 1, 1 + rt. This represents the marginal cost of
investment and it is not borne until period t + 1. What is the marginal beneﬁt of doing additional investment in period t? One additional unit of investment in period t generates one additional unit of capital in period t + 1. This raises future revenue by the marginal product of future capital, At+1FK(Kt+1, Nt+1). In addition, more investment in t generates some additional liquidation of future capital of amount (1 − δ). Hence, At+1FK(Kt+1, Nt+1) + (1 − δ) represents the marginal beneﬁt of an additional unit of investment in period t. (12.16) simply says to invest up until the point at which the marginal beneﬁt of investment equals the marginal cost. The marginal beneﬁt and marginal cost are both received in the future, and hence the optimality condition needs no discounting. We can re-write (12.16) as: 276 𝑤𝑤𝑡𝑡 𝑁𝑁𝑡𝑡 𝐴𝐴𝑡𝑡𝐹𝐹𝑁𝑁(𝐾𝐾𝑡𝑡,𝑁𝑁𝑡𝑡) ↑𝐴𝐴𝑡𝑡 or ↑𝐾𝐾𝑡𝑡 rt + δ = At+1FK(Kt+1, Nt+1) (12.18) Let us now focus on (12.18) and walk through how changes in things which the ﬁrm takes as given will aﬀect its optimal choice of Kt+1. Suppose that rt increases. This makes the left hand side larger. For (12.18) to hold, the ﬁrm must adjust Kt+1 in such a way to make the marginal product of future capital go up. Suppose that the ﬁrm anticipates an increase in future productivity, At+1. Since there would be no change in the left hand side, the ﬁrm would need to adjust Kt+1 to keep the marginal product of future capital ﬁxed. This requires increasing Kt+1. From these exercises, we can deduce that the period t demand
for future capital is a function of the sort: Kt+1 = K d(rt, At+1 + ) (12.19) − In other words, the demand for future capital is decreasing in the real interest rate and increasing in future productivity. Importantly, relative to labor demand, capital demand is forward-looking – it depends not on current productivity, but rather future productivity. Now, let us use (12.19) to think about the demand for investment. We can do this by combining (12.19) with the capital accumulation equation, (12.3). Taking Kt as given, if the ﬁrm wants more Kt+1, it needs to do more It. Hence, we can deduce that the demand for investment is decreasing in the real interest rate and increasing in the future level of productivity. We can also think about how the ﬁrm’s exogenously given current level of capital, Kt, inﬂuences its desired investment. The current level of capital does not inﬂuence the desired future level of capital, which can be seen clearly in (12.18). But if Kt is relatively high, then the ﬁrm needs to do relatively little It to hit a given target level of Kt+1. Hence, the demand for investment ought to be decreasing in the current level of capital, Kt. Hence, we can deduce that investment demand is qualitatively characterized by: It = I d(rt −, At+1 + Figure 12.3 plots a hypothetical investment demand function. We have drawn it as a line for simplicity, but in principle this investment demand function would have some curvature. Investment demand is decreasing in the real interest rate, so the curve slopes down. It would shift out to the right if At+1 increased, or if ft or Kt decreased. (12.20), Kt − ) 277 Figure 12.3: Investment Demand Expressions (12.1), (12.17), and (12.20) qualitatively summarize the solution to the ﬁrm problem. 12.1.1 Diversion on Debt vs. Equity Finance In the setup currently employed, we have assumed that the ﬁrm ﬁnances its accumulation of capital via debt. By this, we mean that the ﬁrm ﬁnances purchases of new capital by borrowing from a ﬁnancial intermediary.
Bank loans constitute a substantial fraction of ﬁrm investment outlays in the US and other developed countries, particularly so for medium and small sized ﬁrms. An alternative assumption we could make is that the ﬁrm ﬁnances its purchases of new capital via equity. By equity we mean that the ﬁrm purchases new capital by reducing its current dividend, which is equivalent to issuing new shares of stock. In the setup we have described, it turns out that there is no diﬀerence between debt and equity ﬁnance – the resulting optimality conditions will be identical. This is a statement of the Modigliani-Miller theorem in economics/ﬁnance – see Modigliani and Miller (1958). Basically, the theorem states that under certain conditions, how the ﬁrm ﬁnances its investment is irrelevant, which is exactly what we see here. The theorem only holds in special cases and is unlikely to fully characterize reality. In particular, the theory assumes no taxes, no bankruptcy cost, and no asymmetric information between borrowers and lenders, none of which are likely hold in the real world. By focusing on the setup in which ﬁrms must borrow to ﬁnance investment, we are laying the groundwork for a later extension where we introduce time-varying credit spreads. Credit spreads can be interpreted as returns to 278 𝑟𝑟𝑡𝑡 𝐼𝐼𝑡𝑡 𝐼𝐼𝑡𝑡=𝐼𝐼𝑑𝑑(𝑟𝑟𝑡𝑡,𝐴𝐴𝑡𝑡+1,𝐾𝐾𝑡𝑡) ↑𝐴𝐴𝑡𝑡+1 or ↓𝐾𝐾𝑡𝑡 ﬁnancial intermediation and emerge because of things like asymmetric information, to be discussed later. Let q ∈ [0, 1] denote the fraction of the ﬁrm’s period t investment that is ﬁnanced via equity, while 1 − q is the fraction of investment ﬁnanced by debt. If a ﬁrm wants to raise one unit of new capital via equity, it reduces its period t dividend by
this amount. Hence, the period t and t + 1 dividends for the ﬁrm are: Dt = AtF (Kt, Nt) − wtNt − qIt Dt+1 = At+1F (Kt+1, Nt+1) + (1 − δ)Kt+1 − wt+1Nt+1 − (1 + rt)(1 − q)It (12.21) (12.22) In (12.21), the ﬁrm spends qIt to purchase new capital, which reduces the dividend payout. = (1 − q)It. Hence it faces an expense of interest plus principal of The ﬁrm borrows BI t (1 + rt)(1 − q)It in t + 1, which is reﬂected in (12.22). By re-writing the problem as one of choosing Kt+1 instead of It from the capital accumulation equation, we can express the ﬁrm’s optimization problem as an unconstrained maximization problem as before: max Nt,Kt+1 Vt = AtF (Kt, Nt) − wtNt − q(Kt+1 − (1 − δ)Kt)+ At+1F (Kt+1, Nt+1) + (1 − δ)Kt+1 − wt+1Nt+1 − (1 + rt)(1 − q)(Kt+1 − (1 − δ)Kt) 1 + rt (12.23) Take the partial derivatives with respect to the remaining choice variables: ∂Vt ∂Nt ∂Vt ∂Kt+1 = AtFN (Kt, Nt) − wt = −q + 1 1 + rt [At+1FK(Kt+1, Nt+1) + (1 − δ) − (1 + rt)(1 − q)] Setting these derivatives equal to zero yields: AtFN (Kt, Nt) = wt (1 + rt)q + (1 + rt)(1 − q) = At+1FK(Kt+1, Nt+1) + (1 − δ) (12.24) (12.25) (12.26) (12.27) (12
.26) is identical to (12.15). The left hand side of (12.27) reduces to 1 + rt regardless of the value of q. This is the same as the optimality condition derived above, (12.16). This is 279 the Modigliani-Miller theorem in action – it does not matter whether the ﬁrm ﬁnances itself via debt or equity; the implied investment demand function is the same. 12.2 Household Let us now think about the household problem. In many ways, this is identical to the setup from Chapter 9, with the main exception that we now endogenize the choice of labor supply. Generically, let household ﬂow utility now be a function of both consumption, Ct, as well as leisure, Lt = 1 − Nt. Here, we normalize the total endowment of time to 1; Nt denotes time spent working, so 1 − Nt is leisure time. Denote this utility function by u(Ct, 1 − Nt). We assume that uC > 0 and uCC < 0. This means that the marginal utility of consumption is positive, but decreases as consumption gets higher. In addition, we assume that UL > 0 and ULL < 0, where UL is the derivative with respect to the second argument, leisure. This means that more leisure increases utility, but at a diminishing rate. In other words, one can just think of leisure as another “good.” Since utility is increasing in leisure, and leisure is decreasing in labor, utility is decreasing in labor. Lifetime utility is the weighted sum of ﬂow utility from periods t and t + 1, where period t + 1 ﬂow utility gets discounted by 0 < β < 1: U = u(Ct, 1 − Nt) + βu(Ct+1, 1 − Nt+1) (12.28) Example Let’s consider a couple of diﬀerent potential speciﬁcations for the ﬂow utility function. First, suppose that utility is given by: u(Ct, 1 − Nt) = ln Ct + θt ln(1 − Nt) (12.29) In (12.29), we say that utility is “additively separable” in consumption and leisure. Technically, this means that UCL = 0 – i.e. the level of leisure (
or labor) has no inﬂuence on the marginal utility of consumption, and vice versa. θt is an exogenous variable which we will refer to as a preference shock. An increase in θt means that the household values leisure more relative to consumption. For this utility function, utility is increasing and concave in both consumption and leisure. The partial derivatives of this utility function are: 280 > 0 uC = 1 Ct uL = θt uCC = − 1 C 2 t 1 1 − Nt < 0 > 0 uLL = −θt 1 (1 − Nt)2 < 0 uCL = 0 (12.30) (12.31) (12.32) (12.33) (12.34) Next, consider another utility function that is not additively separable. particular, suppose: In u(Ct, 1 − Nt) = ln (Ct + θt ln(1 − Nt)) (12.35) Here, we need to assume that θt is such that Ct + θt ln(1 − Nt) is always positive, so that the log of this term is always deﬁned. Here, utility is non-separable in consumption and leisure in that the cross-partial derivative will not be zero. We can see this below: uC = uL = > 0 1 Ct + θt ln(1 − Nt) 1 Ct + θt ln(1 − Nt) θt 1 − Nt > 0 uCC = − uLL = −θt < 0 1 (Ct + θt ln(1 − Nt))2 1 Ct + θt ln(1 − Nt) 1 (1 − Nt)2 (12.36) (12.37) (12.38) [ θt Ct + θt ln(1 − Nt) + 1] < 0 (12.39) uCL = − θt 1 − Nt 1 (Ct + θt ln(1 − Nt))2 < 0 (12.40) With the ﬂow utility function given by (12.35), we see that consumption and leisure are utility substitutes in the sense that uCL < 0. In other words, this means that, when leisure is high (so labor is low), the marginal
utility of consumption is relatively low. Conversely, when leisure is low (so labor is high), the marginal utility of consumption is high. Put another way, labor and consumption are utility 281 complements. Intuitively, if you’re working a lot, the marginal utility of a beer (more consumption) is higher than if you’re not working very much. As before, the household begins life with no stock of wealth (for simplicity). It faces a sequence of two ﬂow budget constraints. The only complication relative to Chapter 9 is that income is now endogenous rather than exogenous, since the household can decide how much it wants to work. The two ﬂow budget constraints are: Ct + St ≤ wtNt + Dt Ct+1 + St+1 ≤ wt+1Nt+1 + Dt+1 + DI t+1 + (1 + rt)St (12.41) (12.42) In (12.41) the household earns income from two sources – labor income, wtNt, and dividend income from its ownership of the ﬁrm, Dt. In (12.42), the household has four distinct sources of income – labor income and dividend income as in period t, but also interest plus principal from savings brought from t to t + 1, (1 + rt)St, as well as a dividend payout from the ﬁnancial intermediary. We label this dividend payout as DI t+1 and discuss it further below. The ﬁnancial intermediary earns nothing in period t, and hence there is no dividend from the intermediary in (12.41). As before, the household will not want to die with a positive stock of savings, and the ﬁnancial intermediary will not allow the household to die in debt. Hence, St+1 = 0. Imposing that each ﬂow budget constraint hold with equality, one can derive an intertemporal budget constraint: Ct + Ct+1 1 + rt = wtNt + Dt + wt+1Nt+1 + Dt+1 + DI t+1 1 + rt (12.43) (12.43) has the same meaning as the intertemporal budget constraint encountered earlier – the present discounted value of the stream of consumption must equal the present discounted value of the stream of income. The income side is just a bit more complicated. Note that the household
takes Dt, Dt+1, and DI t+1 as given – it technically owns the ﬁrm and the ﬁnancial intermediary, but ownership is distinct from management. The household’s objective is to pick Ct, Ct+1, Nt, and Nt+1 to maximize lifetime utility, (12.28), subject to the intertemporal budget constraint, (12.43): max Ct,Ct+1,Nt,Nt+1 U = u(Ct, 1 − Nt) + βu(Ct+1, 1 − Nt+1) s.t. Ct + Ct+1 1 + rt = wtNt + Dt + wt+1Nt+1 + Dt+1 + DI t+1 1 + rt (12.44) (12.45) 282 We can handle this optimization problem by solving for one of the choice variables in terms of the others from the budget constraint. Let’s solve for Ct+1: Ct+1 = (1 + rt) [wtNt + Dt − Ct] + wt+1Nt+1 + Dt+1 + DI t+1 (12.46) Now, plug (12.46) into (12.44), which transforms this into an unconstrained optimization problem: max Ct,Nt,Nt+1 U = u(Ct, 1 − Nt) + βu ((1 + rt) [wtNt + Dt − Ct] + wt+1Nt+1 + Dt+1 + DI t+1, 1 − Nt+1) (12.47) Now, ﬁnd the partial derivatives with respect to the variables the household gets to choose: ∂U ∂Ct ∂U ∂Nt ∂U ∂Nt+1 = uC(Ct, 1 − Nt) − (1 + rt)βuC(Ct+1, 1 − Nt+1) = −uL(Ct, 1 − Nt) + β(1 + rt)wtuC(Ct+1, 1 − Nt+1) = −βuL(Ct, 1 − Nt) + βwt+1uC(Ct+1, 1 − Nt+1) (12.48) (12.49) (
12.50) In writing these derivatives, we have taken the liberty of noting that the argument in the ﬂow utility function for period t + 1 is in fact Ct+1. Setting these derivatives equal to zero yields: uC(Ct, 1 − Nt) = β(1 + rt)uC(Ct+1, 1 − Nt+1) uL(Ct, 1 − Nt) = β(1 + rt)wtuC(Ct+1, 1 − Nt+1) uL(Ct+1, 1 − Nt+1) = wt+1uC(Ct+1, 1 − Nt+1) (12.51) (12.52) (12.53) If one combines (12.51) with (12.52), (12.52) can be written in a way that looks identical to (12.53), only dated period t instead of period t + 1: uL(Ct, 1 − Nt) = wtuC(Ct, 1 − Nt) (12.54) Let us now stop to take stock of these optimality conditions and develop some intuition for why they must hold. First, note that (12.51) is the same Euler equation as we had in the two period model where income was taken to be exogenous. The marginal utility of current consumption ought to equal 1 + rt times the marginal utility of future consumption. It only 283 looks more complicated in that the marginal utility of consumption could depend on the level of leisure (equivalently the amount of labor supplied). If the household decides to consume one additional unit of goods in period t, then the marginal beneﬁt is uC(Ct, 1 − Nt) – i.e. this is by how much lifetime utility goes up. The cost of consuming an additional unit of goods in period t is saving one fewer unit, which leaves the household with 1+rt fewer units of available resources in the next period. This reduces lifetime utility by βuC(Ct+1, 1 − Nt+1)(1 + rt) – βuC(Ct+1, 1 − Nt+1) is the marginal utility of t + 1 consumption, while 1 + rt is the drop in t + 1 consumption. At an optimum, the marginal beneﬁt of consuming more must equal the marginal cost of doing so. Next
, turn to the ﬁrst order conditions for labor supply. Suppose that the household takes an additional unit of leisure (i.e. works a little less). The marginal beneﬁt of this is the marginal utility of leisure, uL(Ct, 1 − Nt). What is the marginal cost? Taking more leisure means working less, which means foregoing wt units of income. This reduces available consumption by wt units, which lowers utility by this times the marginal utility of consumption. Hence, wtuC(Ct, 1 − Nt) is the marginal utility cost of additional leisure. At an optimum, the marginal utility beneﬁt of leisure must equal the marginal utility cost. The ﬁrst order condition for Nt+1 looks exactly the same (and has the same interpretation) as the period t optimality condition. This is analogous to the ﬁrm’s ﬁrst order conditions for Nt and Nt+1 – these conditions are static in the sense of only depending on current period values of variables. Example Consider the two ﬂow utility functions described in the above example. For the separable case, the ﬁrst order conditions work out to: 1 Ct θt 1 1 − Nt 1 1 − Nt+1 θt = β(1 + rt) 1 Ct+1 1 Ct = wt+1 = wt 1 Ct+1 (12.55) (12.56) (12.57) Next, consider the non-separable utility speciﬁcation. The ﬁrst order conditions work out to: 284 1 Ct+1 + θt ln(1 − Nt+1) 1 Ct + θt ln(1 − Nt) = β(1 + rt) θt 1 1 − Nt 1 1 − Nt+1 θt = wt = wt+1 (12.58) (12.59) (12.60) For these utility speciﬁcations, the ﬁrst order conditions for the choice of labor look similar to one another, with the exception that in the non-separable case Ct drops out altogether. Having derived these optimality conditions, let us now think about how consumption and labor supply ought to react to changes in the things which the household takes as given. One can use an
indiﬀerence curve-budget line diagram to think about the choice of consumption in period t and t + 1. Though income is now endogenous because of the choice of labor, treating income as given when thinking about how much to consume gives rise to exactly the same kind of indiﬀerence curve budget-line diagram which we encountered in Chapter 9. Consumption will increase if current income increases, but by less than the increase in current income. In other words, the MPC is positive but less than one. Consumption will also increase if the household anticipates an increase in future income. There are competing income and substitution eﬀects at work with regard to the real interest rate. As before, we assume that the substitution eﬀect dominates, so that consumption is decreasing in the real interest rate. Therefore, the qualitative consumption function which we will use is the same as in the earlier model: Ct = C d(Yt +, Yt+1 +, rt − ) (12.61) One might note some incongruity here. What appears in the household’s intertemporal budget constraint is wtNt + Dt as income each period, not Yt. In equilibrium, as we discuss at the end of this chapter, we will see that wtNt + Dt = Yt − It. So writing the consumption function in terms of Yt is not quite correct. But doing so does not miss out on any important feature of the model, and is consistent with our previous work in Chapter 9. It is also very common to express the consumption function in terms of aggregate income. Next, let us think about how Nt and Ct ought to react to a change in the wage. To do this, we need to build a new indiﬀerence curve - budget line diagram. To ﬁx ideas, suppose that there is only one period (so that the household does not do any saving). The budget constraint facing the household would be: 285 Ct = wtNt + Dt (12.62) Nt is restricted to lie between 0 and 1. When Nt = 0 (so Lt = 1), then the household’s consumption is simply the dividend it receives from the ﬁrm, Ct = Dt. If the household takes no leisure, so Nt = 1, then consumption is the wage plus the dividend. In a graph with Ct on the vertical axis and
Lt = 1 − Nt on the horizontal axis, we can plot the budget constraint. The vertical intercept (when Lt = 0) is Ct = wt + Dt. The maximum value leisure can take on is 1, at which point Ct = Dt. Assume that Dt > 0. Between these two points, the budget line slopes down – as leisure goes up, consumption falls at rate wt. Since leisure cannot go above 1 and we assume Dt > 0, this means that there is a kink in the budget constraint at this point. Figure 12.4 plots the hypothetical budget line below: Figure 12.4: The Consumption-Leisure Budget Line An indiﬀerence curve in this setup is a combination of Ct, Lt values which yield a ﬁxed overall level of utility. The slope of the indiﬀerence is the ratio of the marginal utilities – − uL. uC Because of the assumed concavity of preferences, the indiﬀerence curve has a bowed-in shape, just like in the dynamic consumption-saving model. A higher indiﬀerence curve represents a higher overall level of utility. Hence, we can think about the household’s problem as trying to pick Ct, Lt to get on the highest indiﬀerence curve possible which does not violate the budget constraint. For this exercise, we rule out the “corner” solutions in which the household would choose either no work or no leisure. As in the two period consumption model, getting on 286 𝐶𝐶𝑡𝑡 𝐿𝐿𝑡𝑡 1 𝐷𝐷𝑡𝑡 𝑤𝑤𝑡𝑡+𝐷𝐷𝑡𝑡 Feasible Infeasible the highest indiﬀerence curve possible subject to the budget line requires that the slope of = wt, which is nothing more the indiﬀerence curve equals the slope of budget line, or uL uC than restatement of (12.54). Figure 12.5 below shows a hypothetical situation in which the household chooses C0,t, L0,t (equivalently, N0,t) where the indiﬀerence curve is tangent to the budget line. Figure 12.5:
Optimal Consumption-Leisure Choice Now, let’s consider graphically the eﬀects of an increase in wt. This has the eﬀect of making the budget line steeper (and increasing the vertical axis intercept). This is shown with the blue line Figure 12.6. To think about how this impacts the choice of consumption and leisure, let’s use the tool of isolating income and substitution eﬀects as we did for the eﬀects of a change in rt in the two period consumption-saving model. In particular, draw in a hypothetical budget line, with slope given by the new wt, where the household would optimally locate on the original indiﬀerence curve, labeled U0 in the graph. The substitution eﬀect is to substitute away from leisure and into consumption. When wt goes up, leisure is relatively more expensive (you are foregoing more earnings), and so it seems natural that Nt should rise. But there is also an income eﬀect, which is shown from the change from the hypothetical allocation where U0 is tangent to the hypothetical budget line to the new indiﬀerence curve. Because the original bundle now lies inside the new budget line, there is an income eﬀect wherein the household can get to a higher indiﬀerence curve. This income eﬀect involves increasing both Ct and Lt, which means reducing Nt. Eﬀectively, for a given 287 𝐶𝐶𝑡𝑡 𝐿𝐿𝑡𝑡 1 𝐷𝐷𝑡𝑡 𝑤𝑤𝑡𝑡+𝐷𝐷𝑡𝑡 𝑈𝑈0 𝐿𝐿0,𝑡𝑡 𝐶𝐶0,𝑡𝑡 𝑢𝑢𝐿𝐿𝑢𝑢𝐶𝐶=𝑤𝑤𝑡𝑡 amount of labor input, a household earns more income, which leads it to desire more leisure and more consumption. The net eﬀect is for consumption to increase, whereas the net eﬀect on
Lt (and hence Nt) is ambiguous because of the competing income and substitution eﬀects. The picture has been drawn where the substitution eﬀect dominates, so that Lt falls (and hence Nt rises). This is the empirically plausible case, and unless otherwise noted we shall assume that the substitution eﬀect dominates, so that Nt is increasing in wt. Figure 12.6: Optimal Consumption-Leisure Choice, Increase in wt Mathematically, one can see the income and substitution eﬀects at work by focusing on the ﬁrst order condition for labor supply: uL(Ct, 1 − Nt) uC(Ct, 1 − Nt) = wt (12.63) In (12.63), when wt increases, the ratio of marginal utilities must increase. Since we know that consumption will increase, we know that uC ought to go down, which on its own makes the ratio of the marginal utilities increase. Depending on how much uC decreases, one could need Nt to increase (Lt to decrease, which would drive uL up) or decrease (Lt to increase, 288 𝐶𝐶𝑡𝑡 𝐿𝐿𝑡𝑡 1 𝐷𝐷𝑡𝑡 𝑤𝑤0,𝑡𝑡+𝐷𝐷𝑡𝑡 𝐿𝐿0,𝑡𝑡 𝐶𝐶0,𝑡𝑡 𝑤𝑤1,𝑡𝑡+𝐷𝐷𝑡𝑡 𝑈𝑈1 𝑈𝑈0 𝐶𝐶0,𝑡𝑡ℎ 𝐿𝐿0,𝑡𝑡ℎ 𝐶𝐶1,𝑡𝑡 𝐿𝐿1,𝑡𝑡 Original bundle New bundle Hypothetical bundle with new 𝑤𝑤𝑡𝑡 but fixed utility which would drive uL down). We shall assume that the substitution eﬀect always dominates, in a way analogous to how we assumed that
the substitution eﬀect of a change in the real interest rate dominates for consumption. By assuming that the substitution eﬀect dominates, we are implicity assuming that uC falls by suﬃciently little that uL needs to increase, so Lt needs to fall and Nt needs to rise, whenever wt goes up. From looking at (12.63), it becomes clear that anything which might impact consumption (other than wt) ought to also impact Nt. In terms of the graphs, these are things which would inﬂuence the point where the kink in the budget line occurs. In the static setup, this is solely governed by Dt (which one can think of as a stand-in for non-wage income). In a dynamic case, this point would also be inﬂuence by rt and expectations of future income and wages. We will assume that these other eﬀects are suﬃciently small so that they can be ignored. If we were to use the non-separable preference speciﬁcation discussed in the two examples above, this assumption would be valid. In particular, with that preference speciﬁcation, (12.63) becomes: θt 1 1 − Nt = wt (12.64) 1 1−Nt Under this preference speciﬁcation, Ct drops out altogether, and Nt is solely a function of wt. With these preferences, there is no income eﬀect of a change in the wage. If wt goes up, Nt must go up to make go down. In the background, we can think of using this preference speciﬁcation to motivate our assumptions of labor supply. In addition to the real wage, we will allow for an exogenous source of variation in labor. We will denote this via the exogenous variable θt, which appears in (12.64) as a parameter inﬂuencing the utility ﬂow from leisure. More generally, one can think about θt as measuring anything other than the wage which might impact labor supply. We shall assume that when θt goes up, Nt goes down for a given wage. The strict interpretation of this is that a higher value of θ means that people value leisure more. Our generic labor supply speciﬁcation can therefore be written: N
t = N s(wt + Figure 12.7 plots a hypothetical labor supply function. Nt is increasing in the wage. We have drawn this supply function as a straight line, but more generally it could be a curve. The labor supply curve would shift out to the right (i.e. the household would supply more Nt for a given wt) if θt were to decline. (12.65), θt − ) 289 Figure 12.7: Labor Supply 12.3 Financial Intermediary In our model, there is a ﬁnancial intermediary (e.g. a bank) operating in the background. The bank intermediates between households, who borrow/save via St, and the ﬁrm, which needs funding for its capital investment. The ﬁnancial intermediary takes funds from the household, St, and lends them to the ﬁrm, BI t. The ﬁnancial intermediary charges the same interest rate on borrowing and saving, rt, so that the dividend it earns in t + 1 is: DI t+1 = rtBI t − rtSt (12.66) The ﬁnancial intermediary here is not very interesting – it does not get to choose anything here, and just passively earns a dividend that is remitted to the owner (the household). As we shall see below, in equilibrium this dividend is zero anyway. Later in the book, we will augment the model wherein the ﬁnancial intermediary can earn a non-zero proﬁt. 12.4 Equilibrium As in the endowment economy discussed in Chapter 11, equilibrium is deﬁned as a set of prices and allocations under which all agents are behaving optimally and all markets simultaneously clear. Let us be speciﬁc about what this means in the context of the model laid out in this chapter. Agents behaving optimally means that the household behaves according to its consumption function, (12.61), and its labor supply function, (12.65). The ﬁrm produces 290 𝑁𝑁𝑡𝑡=𝑁𝑁𝑠𝑠(𝑤𝑤𝑡𝑡,𝜃𝜃𝑡𝑡) 𝑁𝑁𝑡𝑡 𝑤𝑤𝑡𝑡 �
�𝜃𝜃 output according to its production function, (12.1), and demands labor according to (12.17) and investment according to (12.20). Market-clearing for labor follows naturally from being on both the labor supply and demand curves. Another market here which must clear is the market for savings and investment. In particular, we must have household savings, St, equal ﬁrm borrowing, BI t, which is in turn equal to investment: St = BI t = It (12.67) If this is combined with the household’s period t budget constraint and the deﬁnition of period t dividends, one gets an aggregate resource constraint, which looks similar to the NIPA expenditure deﬁnition of GDP (without government spending or the rest of the world): Yt = Ct + It (12.68) One can then show that the same aggregate resource constraint holds in the future as well. From the household’s budget constraint, we have: Ct+1 = wt+1Nt+1 + (1 + rt)St + Dt+1 + DI t+1 (12.69) Plugging in the deﬁnitions of dividends, this can be written: Ct+1 = wt+1Nt+1 + (1 + rt)St + Yt+1 + (1 − δ)Kt+1 − wt+1Nt+1 − (1 + rt)BI t + rtBI t − rtSt (12.70) Imposing that St = BI t and making some other simpliﬁcations, this becomes: Ct+1 − (1 − δ)Kt+1 = Yt+1 (12.71) Since the capital accumulation equation in t + 1 is Kt+2 = It+1 + (1 − δ)Kt+1, and Kt+2 = 0 is a terminal condition, we see that It+1 = −(1 − δ)Kt+1. Hence, (12.71) is: Yt+1 = Ct+1 + It+1 (12.72) (12.72) is the same as (12.68), only dated t + 1 instead of t. From the perspective of period t, we can think of (12.68) as summarizing loan market
clearing – we do not need to keep track of BI t or St separately. In summary, then, the equilibrium is characterized by the following equations holding: 291 Ct = C d(Yt, Yt+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 (12.73) (12.74) (12.75) (12.76) (12.77) (12.78) Expressions (12.73)-(12.78) comprise six equations in six endogenous variables – wt and rt are endogenous prices, while Ct, It, Nt, and Yt are endogenous quantities. At, At+1, θt, and Kt are exogenous variables. Yt+1 is a future endogenous variable; we will talk a bit more in terms of how to deal with that when we study the equilibrium of the economy graphically in Chapter 18. 12.5 Summary • Firms choose labor and capital to maximize the present discounted value of dividends. These dividends are rebated to households. • The ﬁrm’s demand for labor is increasing in productivity and capital and decreasing in the real wage. • The ﬁrm’s demand for capital is forward looking. It depends positively on future productivity and negatively on the real interest rate and the current capital stock. • The household chooses leisure and consumption to maximize utility. Labor supply may increase or decrease after a change in the real wage as there are oﬀsetting income and substitution eﬀects. Unless otherwise stated, we assume that the substitution eﬀect dominates, and that labor supply is therefore increasing in the real wage. Key Terms • Modigliani-Miller theorem • Dividends Questions for Review 1. State the ﬁve assumptions on the production function we use in this chapter. 292 2. What is the terminal condition for the ﬁrm? Explain the economic logic. 3. Why is investment increasing in future productivity but not aﬀected by current productivity? 4. Paraphrase the Modigliani-Miller theorem. 5. Explain how an increase in the real wage may actually lead the household to supply less labor. 6. Write down the deﬁ
nition of competitive equilibrium in this economy. What equations characterize the equilibrium? Exercises 1. Suppose that the household only lives for one period. The household’s optimization problem is: max Ct,Nt U = ln Ct + θt ln(1 − Nt) s.t. Ct = wtNt In this problem, the household receives no dividend from the ﬁrm. (a) Solve for the optimality condition characterizing the household problem. (b) From this optimality condition, what can you say about the eﬀect of wt on Nt? What is your explanation for this ﬁnding? 2. Excel Problem. Suppose that you have a ﬁrm with a Cobb-Douglas production function for production in period t: Yt = AtK α t N 1−α t The only twist relative to our setup in the main text is that the ﬁrm does not use labor to produce output in period t + 1. The production function in that period is: Yt+1 = At+1K α t+1 (a) Write down the optimization problem for the ﬁrm in this setup. It has to pay labor in period t, wt, and it discounts future dividends by 1. It 1+rt must borrow to ﬁnance its investment at rt. The capital accumulation equation is standard. 293 (b) Using this speciﬁcation of production, derive the ﬁrst order optimality conditions for the optimal choices of Nt and Kt+1. (c) Re-arrange the ﬁrst order optimality conditions to derive explicit expres- sion for the demand for Nt and the demand for Kt+1. (d) Re-arrange your answer from the previous part, making use of the capital accumulation equation, to solve for an expression for It. (e) Create an Excel ﬁle. Assume the following values for exogenous parameters: α = 1/3, δ = 0.1, At = 1, At+1 = 1, and Kt = 2. Create column of possible values of wt, ranging from a low of 1 to a high of 1.5, with a step of 0.01 between entries (i.e. create a column going from 1 to 1.01 to 1.02 all the way to 1
.5). For each possible value of wt, solve for a numeric value of Nt. Plot wt against the optimal value of Nt. Does the resulting demand curve for labor qualitatively look like Figure 12.2? (f) Suppose that At increases to 1.1. Re-calculate the optimal value of Nt for each value of wt. Plot the resulting Nt values against wt in the same plot as what you did on the previous part. What does the increase in At do to the position of the labor demand curve? (g) Go back to assuming the parameter and exogenous values we started with. Create a grid of values of rt ranging from a low 0.02 to a high of 0.1, with a space of 0.001 between (i.e. create a column going from 0.020, to 0.021, to 0.022, and so on). For each value of rt, solve for the optimal level of It. Create a graph with rt on the vertical axis and It on the horizontal axis. Plot this graph. Does it qualitatively look like Figure 12.3? (h) Suppose that At+1 increases to 1.1. For each value of rt, solve for the new optimal It. Plot this in the same ﬁgure as on the previous part. What does the increase in At+1 do to the position of the investment demand curve? 294 Chapter 13 Fiscal Policy In this chapter we augment the model from Chapter 12 to include a government. This government consumes some of the economy’s output each period. We do not formally model the usefulness of this government expenditure. In reality, government spending is motivated for the provision of public goods. Public goods are goods which are non-exclusionary in nature, by which is meant that once the good has been produced, it is impossible (or nearly so) for the producer to exclude individuals from consuming it. An example is military defense. All citizens in a country beneﬁt from the defense its military provides, whether they want to or not. A private military provider would be fraught with problems, because it would be diﬃcult or impossible for the private provider to entice individuals to pay for military services. As such, military services would be under-provided left to the private market. Other examples of public goods include roads, bridges, schools, and parks. We will assume that the government can ﬁnance
its expenditure with a mix of taxes and debt. We will assume that taxes are lump sum, in the sense that the amount of tax an agent pays is independent of any actions taken by that agent. This is an unrealistic description of reality but nevertheless greatly simpliﬁes the analysis and will provide some important insights. 13.1 The Government The model is the same as in Chapter 12, with time lasting for two periods, t (the present) and t + 1 (the future). The government does an exogenous amount of expenditure in each period, Gt and Gt+1. As noted above, we do not model the usefulness of this expenditure, nor do we endogenize the government’s choice of its expenditure. The government faces budget constraints each period in a similar way to the household. These are: Gt ≤ Tt + BG t Gt+1 + rtBG t ≤ Tt+1 + BG t+1 − BG t (13.1) (13.2) 295 > 0 is debt, while BG t In these budget constraints, Tt and Tt+1 denote tax revenue raised by the government in each period. In the period t constraint, (13.1), BG is the amount of debt which the t < 0 government issues in period t. The sign convention is that BG t would correspond to a situation in which the government saves. In other words, in period t, the government can ﬁnance its expenditure, Gt, by raising taxes, Tt, or issuing debt, BG t. In period t + 1, the government has two sources of expenditure – its spending, Gt+1, and interest < 0, then this corresponds to interest revenue. payments on its outstanding debt, rtBG t. If BG t The government can again ﬁnance its expenditure by raising taxes, Tt+1, or issuing more debt, − BG BG t, where this term corresponds to the change in the quantity of outstanding debt. t+1 As in the case of the household, we assume that the government cannot die in debt, which ≤ 0. The government would not want to die with a positive stock of savings, so requires BG t+1 = 0. If we further assume BG t+1 that the government’s budget constraints hold with equality each period, we can combine (13.1) and (13.2) to get an intertemporal budget constraint for the government: ≥ 0

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