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where q,L is the elasticity of the job ﬁnding rate with respect to the queue length. Intuitively, the wage is increasing in the worker’s outside option and the output from the match. This looks very similar to (), but with q,L replacing 1 − χ. The diﬀerence is that in the DMP model, wages are determined by bargaining, whereas employers determine wages (subject to job seeker’s indiﬀerence condition) in the directed search model. To close the model we assume that ﬁrms enter until proﬁts are driven to zero. That is, Π = −κ + qt(zt − wt) = 0 Substituting (17.39) into ﬂow proﬁts gives κ = qt(z − b − q,L(zt − b)) = qt(1 − q,L)(zt − b) (17.40) This pins down Lt 17.5.1 Example Suppose the matching function is Cobb-Douglas. Hi,t = ψtu1−ρ i,t V ρ i,t The job ﬁlling rate is qi,t = Hi,t Vi,t = ψt ( ui,t Vi,t 1−ρ ) = ψtL1−ρ i,t 398 The proﬁt maximization problem for the ﬁrm is max Li,t Π = −κ + ψtL1−ρ i,t (zt − b) − Li,t(U − b) The ﬁrst order condition is (1 − ρ)ψtL−ρ i,t (zt − b) = U − b As expected, each ﬁrm chooses the same queue length. Substitute the FOC into the equation for expected utility. (1 − ρ)ψtL−ρ t U = b + ft(wt − b) ⇔ (wt − b) ⇔ (zt − b) = ψtL−ρ t (1 − ρ)(zt − b) = wt − b ⇔ wt = (1 − ρ)zt + ρb (17.41) Again, this looks exactly the same as the DMP model but with ρ replacing the
bargaining weight, χ. With free entry, proﬁts are driven to 0. Thus, Substituting in the wage, we have Π = −κ + qt(zt − wt) = 0 κ = qtρ(zt − b) Solving for the equilibrium job ﬁlling rate gives qt = κ zt − b (17.42) The equilibrium queue length is Recalling that qt = ψL1−ρ t, the equilibrium queue length is 1 1−ρ Lt = ( κ ψ(zt − b) ) (17.43) If we assume that all workers start the period as unemployed, the number of vacancies is simply equal to 1 Lt or κ Thus, vacancies are increasing in match output and the eﬃciency of the matching function. They are decreasing in unemployment beneﬁts and the cost of posting vacancies. (17.44) Vt = ( ψ(zt − b) ) 1 1−ρ 399 The end of the period unemployment rate is given by u′ t = 1 − ft = 1 − ψ (ψ(zt − b) κ ρ 1−ρ ) (17.45) This looks very similar to the end of period unemployment in the DMP model. In particular, the unemployment rate is decreasing in match output and the eﬃciency of the matching function. It is increasing in vacancy posting costs and unemployment beneﬁts. 17.5.2 Eﬃciency In the central planners problem considered in the last section, we found that the central planner posts vacancies up until the point Mv(Vt, 1) = κ zt − b Is the decentralized equilibrium in the directed search model eﬃcient? Recall that the free entry condition is qt(1 − q,L) = κ zt − b Thus, the equilibrium is eﬃcient if Mv(Vt, 1) = qt(1 − L,α). To make sense of this, let’s return to the Cobb-Douglas matching function. The job ﬁlling rate is Mv(Vt, 1) = ψρV ρ−1 t qt = ψV ρ−1 t Thus, the equilibrium is eﬃcient if 1 − ρ = q,L.
That is, the elasticity of the job ﬁnding rate with respect to the queue length must equal 1 − ρ. To see holds, write qt as qt = ψtL1−ρ t Recall that an elasticity of a variable y with respect to a variable x can be written as y,x = ∂ log y ∂ log x, we have ln qt = ln ψt + (1 − ρ) ln Lt Thus,q,L = 1 − ρ. We conclude that the equilibrium in the model with directed search is eﬃcient.10 10This result can be generalized beyond a Cobb Douglas matching function. 400 In the DMP model we found that the equilibrium is eﬃcient only if ρ = χ, i.e. the bargaining weight of ﬁrms equals the elasticity of the matching function with respect to vacancies. But in the competitive search model, the equilibrium is eﬃcient for any value of ρ. What explains the diﬀerence? Since workers direct their search to the ﬁrms oﬀering the best combination of a wage and probability of ﬁnding a job, ﬁrms compete to oﬀer the best combination. With free entry driving proﬁts to zero, ﬁrms just break even and the workers get the best combination of probabilities and wages. It’s as if ﬁrms endogenize the congestion externality. 17.6 Summary • Until this point in the book all ﬂuctuations in the labor market were at the intensive margin, that is through the representative agent substituting work for leisure. In the data, there are enormous variations in the extensive margin, i.e. the number of people working. • There are millions of jobs created and destroyed each month, but in recessions, net job creation is negative. Not every worker looking for a match is successful although this success rate falls in recessions. The negative relationship between unemployment and vacancies is called the Beveridge curve. • In a one-sided matching model an individual trades oﬀ accepting a wage today versus rejecting the wage with the potential to receive a higher one tomorrow. The worker’s decision is inﬂuenced by unemployment beneﬁts and the distribution of wage oﬀers. • The
bathtub model shows the relationship between unemployment and the job ﬁnding and separation rates. A higher job ﬁnding rate lowers unemployment and a higher separation rate raises unemployment. • The matching function describes how a number of vacancies and unemployed are turned into new hires. The matching function is a reduced form way of modeling frictions in the labor market. • In the two sided matching model unemployed workers are randomly matched with ﬁrms posting vacancies. The wage is set so that the worker and ﬁrm end up splitting the surplus. In equilibrium, there is free entry of ﬁrms which drives the value of posting a vacancy to zero. This model gives us a way to understand how changes in the cost of posting vacancies, matching eﬃciency, worker productivity, and the separation rate aﬀect the labor market. 401 • In the directed search model workers direct their search to ﬁrms oﬀering the best combination of wage and job ﬁnding rate. Firms choose the wage subject to the worker’s indiﬀerence condition. In an equilibrium with free entry, ﬁrm’s choose a wage that implements the eﬃcient allocations. Key Terms • Beveridge curve • Job ﬁnding rate • Separation rate • Labor market tightness • Reservation wage • Mean preserving spread • Bargaining power • Congestion externality Questions for Review 1. What happens to net job creation during recessions? What about gross job creations? 2. List the various types of separations. How do their magnitudes change over the business cycle? 3. A claim one often hears is that more generous unemployment beneﬁts raise unemployment because unemployed workers reduce their incentive to look for a job. To what extent is this true in the McCall model? 4. In a two-sided search model, why aren’t wages set equal to their marginal products? Exercises 1. Consider a version of the McCall model with a discrete wage distribution. In particular, wages can take the value, 25, 75, and 125 with equal probability. Assume b = 40. (a) Solve for the reservation wage. 402 (b) Suppose instead the worker faces a wage distribution where wages can take the values 0, 75, 150 with equal probability. Argue that this distribution is a mean preserving spread of the distribution in part (a).
(c) Solve for the reservation wage using the distribution in part (b). Is it higher than in part (a)? 2. Suppose wages are drawn from an exponential distribution which has a density function, f (w) = λe−λw for w ≥ 0 and λ > 0 is a parameter. (a) Verify that the density function integrates to 1. (b) Solve for the distribution function. (c) Assume b = 0. Solve for the reservation wage. (d) Prove that the reservation wage is increasing in β but decreasing in λ. Explain the economic intuition for each. (e) Suppose the worker gets two wage draws from the distribution in t + 1. Solve for reservation wage and prove that it is higher than the reservation wage in part (c). Hint: P r(max{W1, W2} ≤ w) = P r(W1 ≤ w)P r(W2 ≤ w) = F (w)2. Explain the intuition. Do you think this can be generalized? 3. In the two-period McCall model, what is the reservation wage in t + 1. Is this higher or lower than the reservation wage in t? How might this generalize to reservation wages over the life cycle? 4. Suppose you can search for a job in market A or market B. The average wage is equal across markets, but market B has greater wage dispersion. Which market would you choose to search in? Justify. 5. In a recession, state governments often extend the length of time unemployed people can qualify for unemployment beneﬁts. Do you think this extension of unemployment beneﬁts would shorten or lengthen the recession? Justify your answer. 6. Suppose the matching function takes the form Ht = utVt + V φ t. 1 φ ) (uφ t (a) Assuming wages are Nash bargained, solve for the equilibrium number of vacancies. (b) Derive a condition under which the equilibrium number of vacancies is Pareto Optimal (i.e. eﬃcient in the sense of being the solution to a 403 social planner’s problem). 7. Now consider the matching function Ht = V ρ t (utet)1−ρ where et is deﬁned as search intensity. Search intensity is how hard the households are looking for work. Looking for work carries a disut
ility cost of g(et) = θeγ t where γ > 1. (a) If all households start out the period unemployed, compute the job ﬁnding and job ﬁlling rates. (b) Each household chooses how much eﬀort to put into search, taking the amount of aggregate eﬀort in the economy and the job ﬁnding rate as given. Therefore, the objective of household i is max ei U = max ei b − θeγ i,t + ei,t et f (Vt, et)(wt − b). The harder household i searches relative to the amount of the average (or aggregate) search intensity, the higher its probability of ﬁnding a job. Solve for the optimal ei,t as a function of et, ft, wt and b. (c) Observe that your optimal ei,t is only a function of economy-wide variables (nothing involving household i). From this we can infer that all households choose the same level of intensity. So ei,t = et. Solve for the economy-wide et taking ft and wt as exogenous. How does the optimal et respond to changes in ft, wt, and θ. Describe the economic intuition for all of these. (d) Derive the free-entry condition. (e) Derive the Nash bargained wage. (f) Solve for the equilibrium number of vacancies as a function of only exogenous variables and parameters. (g) How does the number of vacancies change with productivity? Does endogenous intensity amplify or dampen the response of vacancies to a change in productivity? 404 Part IV The Medium Run 405 The long run analysis carried out in Part II focuses on capital accumulation and growth. One can think about the long run as referencing frequencies of time measured in decades. In the medium run, we think about frequencies of time measured in periods of several years, not decades. Over this time horizon, investment is an important component of ﬂuctuations in output, but the capital stock can be treated as approximately ﬁxed. Prices and wages are assumed to be completely ﬂexible. Our main model for conducting medium run analysis is the neoclassical model. The building blocks of the neoclassical model are the microeconomic decision rules discussed in Part III. In this section we take these decision rules as given and focus on
a graphical analysis of the model. This part ought to be self-contained, and can be studied without having worked through Part III. Chapter 18 lays out the decision rules which characterize the equilibrium of the neoclassical model. Some intuition for these decision rules is presented, and some references are made to the microeconomic analysis from Part III. In Chapter 18 we deﬁne several curves and that will be used to analyze the model. In Chapter 19 we graphically analyze how changes in the diﬀerent exogenous variables of the model impact the endogenous variables of the model. Chapter 20 looks at the data on ﬂuctuations in the endogenous variables of the model and analyzes whether the neoclassical model can qualitatively make sense of the data, and, if so, which exogenous variable must be the main driving force in the model. The neoclassical model as presented here is sometimes called the real business cycle model, or RBC model for short. In Chapter 22 we discuss policy implications of the model. In the neoclassical model the equilibrium is eﬃcient, which means that there is no justiﬁcation for activist economic policies. In this chapter we also include a discussion of some criticisms of the neoclassical / real business cycle paradigm, particularly as it relates to economic policy. Chapter 23 considers an open economy version of the neoclassical model. 406 Chapter 18 The Neoclassical Model The principal actors in the neoclassical model are households, ﬁrms, and a government. As is common in macroeconomics, we use the representative agent assumption and posit the existence of one representative household and one representative ﬁrm. The household and ﬁrm are price-takers in the sense that they take prices as given. A strict interpretation of this assumption is that all households and ﬁrms are identical, and that there are a large number of them. A weaker interpretation permits some heterogeneity but assumes a micro-level asset market structure that ensures that households and ﬁrms behave the same way in response to changes in exogenous variables, even if they have diﬀering levels of consumption, production, etc. We assume that there are two periods – period t is the present and period t + 1 is the future. It is straightforward to extend the model to include multiple periods. The sections below discuss the decision rules of each actor and the concept of equilibrium. The microeconomic underpinning of the decision
rules are derived in Part III. After presenting the decision rules and market-clearing conditions, we develop a set of graphs that allows one to analyze the eﬀects of changes in exogenous variables on the endogenous variables of the model. 18.1 Household There is a representative household who consumes, saves, holds money, and supplies labor. The household supplies labor to the representative ﬁrm at some nominal wage Wt, and buys back goods to consume at a price of Pt dollars. The household can save some of its income through the purchase of bonds. One dollar saved in a bond pays out 1 + it dollars in the future. What is relevant for household decision-making are real prices, not nominal prices. Let wt = Wt/Pt be the real wage. The real price of consumption is simply 1 (i.e. Pt/Pt). The real interest rate is rt = it − πe t+1 is the expected growth rate of the price level between t and t + 1. We assume that expected inﬂation is exogenous. This expression for the real interest rate in terms of the nominal rate is known as the Fisher relationship. Ct is the amount of consumption the household does in period t. Yt is aggregate income. Assume that t+1, where πe 407 the household pays Tt units of real income to the government in the form of a tax each period. Nt is the labor supplied by the household. Mt is the quantity of money that the household holds. The household’s saving is St = Yt − Tt − Ct, i.e. its non-consumed income net of taxes. The decision rules for the household are a consumption function, a labor supply function, and a money demand function. One could also deﬁne a saving supply function, but this is redundant with the consumption function. These decision rules are given below: Ct = C d(Yt − Tt, Yt+1 − Tt+1, rt) Nt = N s(wt, θt) Mt = PtM d(it, Yt) (18.1) (18.2) (18.3) The consumption function is given by (18.1). C d(⋅) is a function which relates current net income, Yt − Tt; future net income, Yt+1 − Tt+1, and
the real interest rate, rt, into the current level of desired consumption. Why does consumption depend on both current and future net income? The household has a desire to smooth its consumption across time, which is driven by the assumption that the marginal utility of consumption is decreasing in consumption. If the household expects a lot of future income relative to current income, it will want to borrow to ﬁnance higher consumption in the present. Likewise, if the household has a lot of current income relative to what it expects about the future, it will want to save in the present (and hence consume less) to provide itself some resources in the future. (⋅) and ∂Cd (⋅) ∂Yt+1 Hence, we would expect consumption to be increasing in both current and future net income – i.e. the partial derivatives of the consumption function with respect to the ﬁrst two arguments, ∂Cd, ought to both be positive. While we would expect these ∂Yt partial derivatives to be positive, our discussion above also indicates that we would expect these partial derivatives to always be bound between 0 and 1. If the household gets some extra income in period t, it will want to save part of that extra income so as to ﬁnance some extra consumption in the future, which requires increasing its saving (equivalently increasing its period t consumption by less than its income). Likewise, if the household expects some more income in the future, it will want to increase its consumption in the present, but by less than the expected increase in future income – if the household increased its current consumption by more than the increase in future income through borrowing, it would have more than the extra future income to pay back in interest in period t + 1, which would mean it could not increase its consumption in period t + 1. We will refer to the partial derivative of the consumption function with respect to period t income as the marginal propensity to consume, or MPC for short. It is bound between 0 and 1: 0 < M P C < 1. While in general a partial derivative is itself a function, we will treat the MPC as a ﬁxed number. The marginal 408 propensity to save is simply equal to 1 − M P C: this denotes the fraction of an additional unit of income in period t that a household would choose to save. In terms of the analysis from the Solow model, the M P C would be 1 − s
. The consumption function depends negatively on the real interest rate, i.e. ∂Cd < 0. ∂rt Why is this? The real interest rate is the real return on saving – if you forego one unit of consumption in period t, you get back 1 + rt units of consumption in period t + 1. The higher is rt, the more expensive current consumption is in terms of future consumption. We assume that the household’s desired saving is increasing in the return on its saving, which means consumption is decreasing in rt.1 (⋅) The labor supply function is given by (18.2). It is assumed that the amount of labor supplied by the household is an increasing function of wt, i.e. ∂N s > 0; and an decreasing ∂wt function of an exogenous variable, θt, i.e. ∂N s < 0.2 The exogenous variable θt represents a ∂θt labor supply shock, which is meant to capture features which impact labor supply other than the real wage. Real world features which might be picked up by θt include unemployment beneﬁts, taxes, demographic changes, or preference changes. (⋅) (⋅) (⋅) (⋅) ; and increasing in the level of income, ∂M d ∂Yt The money demand function is given by (18.3). The amount of money that a household wants to hold, Mt, is proportional to the price level, Pt. Since money is used to purchase goods, the more expensive goods are in terms of money, the more money the household will want to hold. The demand for money is assumed to be decreasing in the nominal interest, i.e. ∂M d. Money is increasing in income because ∂it the more income a household has, the more consumption it wants to do, and therefore it needs more money. Money demand depends on the nominal interest rate because holding money means not holding bonds, which pay nominal interest of it. The higher is this interest rate, the less attractive it is to hold money – you’d rather keep it in an interest-bearing account. Money demand depends on the nominal interest rate, rather than the real interest rate, because the relevant tradeoﬀ is holding a dollar worth of money or putting a dollar in an interest-bearing account (whereas the real interest rate conve
ys information about the tradeoﬀ between giving a up a unit of consumption in the present for consumption in the future). Using the Fisher relationship between nominal and real interest rates, rt = it − πe t+1, however, the money demand function can be written in terms of the real interest rate and expected inﬂation: Mt = PtM d(rt + πe t+1, Yt) (18.4) 1Technically, the assumption here is that the substitution eﬀect of a higher rt dominates the income eﬀect, as discussed in Chapter 9. 2Technically, that labor supply is increasing in the real wage requires that the substitution eﬀect of a higher wage be stronger than the income eﬀect, as discussed in Chapter 12. 409 18.2 Firm There is a representative ﬁrm that produces output using capital and labor. We abstract from exogenous labor augmenting productivity, but there is a neutral productivity shifter which is exogenous. The production function is: Yt = AtF (Kt, Nt) (18.5) Yt denotes the output produced by the ﬁrm in period t. Kt is the capital stock, which is predetermined and hence exogenous within a period. Nt is the amount of labor used in production. At is the exogenous productivity shock which measures the eﬃciency with which inputs are turned into output. F (⋅) is an increasing and concave function in capital in labor. Mathematically, this means that FK(⋅) > 0 and FN (⋅) > 0 (i.e. the marginal products of capital and labor are both positive, so more of either input leads to more output); FKK(⋅) < 0 and FN N (⋅) < 0 (the second derivatives being negative means that there are diminishing marginal returns – having more Kt or Nt increases output but at a decreasing rate). We also assume that FN K > 0. The cross-partial derivative being positive means that the marginal product of capital is higher the more labor input there is (equivalently the marginal product of labor is higher the more capital there is). This just means that one factor of production is more productive the more of the other factor a ﬁrm has. The Cobb-Douglas production function used in Part II has these
properties. The ﬁrm hires labor on a period-by-period basis at real wage wt (equivalently at nominal wage Wt, which when divided by the price of goods is the real wage). The ﬁrm inherits the current capital stock from past investment decisions; hence Kt is exogenous within period t. It can do investment, It, so as to inﬂuence its future capital stock. The stock of capital evolves according to: Kt+1 = It + (1 − δ)Kt (18.6) Equation (18.6) says that the ﬁrm’s capital stock in the next period depends on (i) its non-depreciated current capital stock, (1 − δ)Kt, where δ is the depreciation rate, and (ii) its investment, It. The ﬁrm must borrow from a ﬁnancial intermediary to ﬁnance its investment in period t. The real interest rate it faces is rt, which is the same interest rate at which the household saves. The decision rules for the ﬁrm are a labor demand function and an investment demand function. These are given below: 410 Nt = N d(wt, At, Kt) It = I d(rt, At+1, Kt) (18.7) (18.8) (⋅) (⋅) < 0; increasing in current productivity, ∂N d ∂At The labor demand curve is given by (18.7). Labor demand is decreasing in the real wage, ∂N d > 0; and increasing in the current capital (⋅) ∂wt stock, ∂N d > 0. What is the intuition for the signs of these partial derivatives? As discussed ∂Kt in depth in Chapter 12, a proﬁt-maximizing ﬁrm wants to hire labor up until the point at which the real wage equals the marginal product of labor. If the real wage is higher, the ﬁrm needs to adjust labor input so as to equalize the marginal product with the higher wage. Since the marginal product of labor is decreasing in Nt (i.e. FN N (⋅) < 0), this necessitates reducing Nt when wt goes up. An increase in At makes the ﬁrm want to hire more
labor. The higher At raises the marginal product of labor. For a given real wage, Nt must be adjusted so as to equalize the marginal product of labor with the same real wage. Given FN N (⋅) < 0, this necessitates increasing Nt. The logic for why Nt is increasing in Kt is similar. If Kt is higher, then the marginal product of labor is bigger given our assumption that FKN > 0. To equalize a higher marginal product of labor with an unchanged real wage, Nt must increase when Kt goes up. (⋅) > 0; and decreasing in the current capital stock, ∂I d (⋅) ∂Kt The investment demand function is given by (18.8). The demand for investment is decreasing in the real interest rate, ∂I d < 0; increasing in the future level of productivity, ∂rt ∂I d < 0. To understand why investment (⋅) ∂At+1 depends on these variables in the way that it does, it is critical to understand that investment is forward-looking. The beneﬁt of investment in period t is more capital in period t + 1. The cost of investment is interest which must be paid back in the future. Hence, investment (like saving for a household) is about giving something up in exchange for something in the future. Investment depends negatively on rt because the real interest rate is the cost of borrowing by the ﬁrm. The higher is rt, the higher is the cost of current investment. Hence, the demand for investment is decreasing in rt. Investment demand depends on the future level of productivity, At+1. Investment demand does not directly depend on current productivity, At. The reason for this is that investment inﬂuences the future level of capital, and hence the future level of productivity is what is relevant when choosing current investment. Finally, investment demand is decreasing in its current capital stock. As discussed in Chapter 12, a ﬁrm has an optimal target level of Kt+1 which is independent of its current Kt. This means that the amount of current Kt it has will inﬂuence how much investment it must do to reach this target level of future capital. So, for example, if a hurricane comes and wipes out some of a 411 ﬁrm’s existing capital, it will want to do more investment
– the hurricane doesn’t aﬀect the ﬁrm’s target level of future capital, but it means that the ﬁrm needs to do more investment to reach this target capital stock. 18.3 Government There exists a government that consumes some private output (what we call “government spending”) in both period t and t+1, Gt and Gt+1. The government ﬁnances its spending with a mix of taxes, Tt and Tt+1, and by issuing debt. The amount of spending that the government does in period t, and the amount it expects to do in the future, are both exogenous to the model.3 Though we do not explicitly model any beneﬁt from government spending, we could do so by assuming that the representative household gets a utility ﬂow from government spending. As discussed in Chapter 13, we assume that something called Ricardian Equivalence holds in the model. Ricardian Equivalence states that all that matters for the equilibrium behavior in the economy are the current and future values of government spending, Gt and Gt+1. Conditional on current and expected spending, the timing and amounts of taxes, Tt and Tt+1, are irrelevant for decision-making, as is the level of debt issued by the government. The basic intuition for Ricardian equivalence is straightforward. If the government runs a deﬁcit in period t (i.e. Gt > Tt, so that its expenses are more than its revenue, which could occur because spending is high, taxes are low, or transfer payments are high), it will have to run a surplus in period t + 1 to pay oﬀ the debt carried over from period t. The household is forward-looking and cares only about the present value of its net income. The timing of tax collection has no impact on the present value of net income, given that taxes are lump sum (i.e. do not aﬀect prices relevant to the household and ﬁrm). The implication of Ricardian Equivalence is that we can act as though the government balances its budget each period, with Gt = Tt and Gt+1 = Tt+1. Agents will behave this way whether the government does in fact balance its budget or not. This greatly simpliﬁes the model, as we do
not need to worry about Tt, Tt+1, or the amount of debt issued by the government. We can re-write the household’s consumption function, (18.1), by replacing the tax terms with government spending: 3In practice, the term “government expenditure” means something diﬀerent than “government spending.” Government spending refers to expenditure on goods and services (i.e. roads, police, ﬁre departments, military, etc.). A signiﬁcant fraction of government expenditure in developed economies also includes transfer payments – things like Social Security payments, Medicare, etc.. These transfer payments are not direct expenses on new goods and services – they are transfers to households who then use the income to purchase goods and services. Therefore, in the context of our model transfer payments show up as negative taxes, not positive spending. 412 Ct = C d(Yt − Gt, Yt+1 − Gt+1, rt) (18.9) The government also decides how much money to supply, Mt. We assume that money supply is exogenous and therefore abstract from the fact that a signiﬁcant fraction of the money supply in reality is privately created. 18.4 Equilibrium Equilibrium is deﬁned as a set of prices and quantities where (i) all agents are behaving optimally, taking prices as given, and (ii) all markets simultaneously clear. Markets clearing means that total income is equal to total expenditure which equals production. In our model, this means that: Yt = Ct + It + Gt (income equals expenditure) and Yt = AtF (Kt, Nt) (income equals production). Ct = C d(Yt − Gt, Yt+1 − Gt+1, rt) Nt = N s(wt, θt) Nt = N d(wt, At, Kt) It = I d(rt, At+1, Kt) Yt = AtF (Kt, Nt) Yt = Ct + It + Gt Mt = PtM d(rt + πe t+1, Yt) rt = it − πe t+1 (18.10) (18.11) (18.12) (18.13) (18.14) (18.15) (18.16) (18.
17) Expressions (18.10)-(18.17) mathematically summarize the neoclassical model. There are eight equations and eight endogenous variables. The endogenous variables are Yt, Ct, It, Nt, rt, wt, Pt, and it. The exogenous variables are At, At+1, Gt, Gt+1, θt, Mt, and πe t+1. A useful insight is that the ﬁrst six equations hold independently of any reference to nominal variables (Mt, Pt, it, or πe t+1). We will refer to these six equations as the “real block” of the model. There are also six endogenous variables in this block of equations – Yt, Ct, It, Nt, wt, and rt – four quantities and two real prices. That the real variables of the model can be determined without reference to the nominal variables is known as the classical dichotomy. We will refer to the last two equations (the money demand function and the Fisher relationship relating real to nominal variables) as the “nominal block” of the model. 413 These expressions do depend on real variables – rt and Yt – but also feature two nominal endogenous variables (Pt and it). 18.5 Graphing the Equilibrium We would like to graphically analyze equations (18.10)-(18.17). In doing so, we will split the equations up into the real and nominal block, focusing ﬁrst on the real block of equations, (18.10)-(18.15). 18.5.1 The Demand Side Focus ﬁrst on the consumption function, (18.10), the investment demand function, (18.13), and the aggregate resource constraint, (18.15). These equations summarize the demand side of the model, since the sum of demand by diﬀerent actors (the household, the ﬁrm, and the government) must equal total demand (the aggregate resource constraint). We will graphically summarize these equations with what is known as the “IS Curve.” “IS” stands for “investment=saving,” and is simply an alternative way to represent the aggregate resource constraint. To see this, add and subtract Tt from the right hand side of (18.15): Yt = Ct + Tt + It + Gt − T
t This can be re-arranged as follows: Yt − Tt − Ct + Tt − Gt = It (18.18) (18.19) The term Yt − Tt − Ct is the saving of the household, or Spr t. The term Tt − Gt is the saving t. The sum of their saving is aggregate saving, which must of the of the government, or Sg equal investment. The IS curve summarizes the combinations of (rt, Yt) for which the aggregate resource constraint holds where the household and ﬁrm choose consumption and investment optimally. Mathematically, the IS curve is given by: Yt = C d(Yt − Gt, Yt+1 − Gt+1, rt) + I d(rt, At+1, Kt) + Gt (18.20) Taking the relevant exogenous variables (Gt, Gt+1, At, At+1, and Kt) as given, and treating Yt+1 as given as well (we will return to this issue in the next chapter), this is one equation in two unknowns – rt and Yt. The IS curve simply summarizes the diﬀerent values of rt and Yt where (18.20) holds. 414 To graph the IS curve, let us deﬁne an intermediate variable, denoted Y d t. This stands for aggregate desired expenditure. Aggregate desired expenditure is the sum of desired expenditure by each agent in the economy: Y d t = C d(Yt − Gt, Yt+1 − Gt+1, rt) + I d(rt, At+1, Kt) + Gt (18.21) Aggregate desired expenditure, Y d t, is a function of aggregate income, Yt. We can plot this is a graph as follows. We assume that when current income is zero, i.e. Yt = 0, aggregate > 0. As income increases, aggregate desired desired expenditure is nevertheless positive, Y d t expenditure increase because consumption is increasing in income. Because we assume that the MPC is less than one, then a plot of aggregate desired expenditure against aggregate income is just an upward-sloping line, with a positive intercept and a slope less than one. This can be seen in Figure 18.1 below. We will refer to the plot of desired aggregate expenditure against aggregate income as the
“expenditure line.” Figure 18.1: Desired Expenditure and Income The vertical axis intercept, which is what desired expenditure would be with no current income, i.e. Yt = 0, is assumed to be positive. The level of desired expenditure which is independent of current income is sometimes called “autonomous expenditure.” Denote this: E0 = C d(−Gt, Yt+1 − Gt+1, rt) + I d(rt, At+1, Kt) + Gt (18.22) Autonomous expenditure, E0, is simply the consumption function evaluated at Yt = 0, plus desired investment plus government spending. The level of autonomous expenditure depends on several variables. First, it depends on the real interest rate, rt. If rt goes down, 415 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑 𝑌𝑌𝑡𝑡𝑑𝑑=𝐶𝐶𝑑𝑑(𝑌𝑌𝑡𝑡−𝐺𝐺𝑡𝑡,𝑌𝑌𝑡𝑡+1−𝐺𝐺𝑡𝑡+1,𝑟𝑟𝑡𝑡)+𝐼𝐼𝑑𝑑(𝑟𝑟𝑡𝑡,𝐴𝐴𝑡𝑡+1,𝐾𝐾𝑡𝑡)+𝐺𝐺𝑡𝑡 𝐸𝐸0=𝐶𝐶𝑑𝑑(−𝐺𝐺𝑡𝑡,𝑌𝑌𝑡𝑡+1−𝐺𝐺𝑡𝑡+1,𝑟𝑟𝑡𝑡)+𝐼𝐼𝑑𝑑(𝑟𝑟𝑡𝑡,𝐴𝐴𝑡𝑡+1,𝐾𝐾𝑡𝑡)
+𝐺𝐺𝑡𝑡 Slope = MPC < 1 consumption and investment will both increase for a given level of income. This has the eﬀect of increasing the vertical axis intercept and shifting the desired expenditure line up. This is shown in Figure 18.2 below. The exogenous variables which impact desired consumption and investment also will cause the expenditure line to shift. We will discuss these eﬀects below. Figure 18.2: Desired Expenditure and Income In equilibrium, expenditure must equal income, Y d t = Yt. We can graphically ﬁnd the = Yt, and equilibrium level of Yt by drawing a 45 degree line, showing all points where Y d t ﬁnding the Yt where the expenditure line crosses the 45 degree line. This is shown in Figure 18.3 below. The 45 degree line starts “below” the expenditure line, since it begins in the origin and we assume that the expenditure line has a positive vertical intercept. Since the 45 degree line has a slope of 1, while the expenditure line has a slope less than 1 (since MPC ¡ 1), graphically these lines must cross exactly once. At this point, labeled Y0,t, income is equal to expenditure. 416 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑 𝑌𝑌𝑡𝑡𝑑𝑑=𝐶𝐶𝑑𝑑�𝑌𝑌𝑡𝑡−𝐺𝐺𝑡𝑡,𝑌𝑌𝑡𝑡+1−𝐺𝐺𝑡𝑡+1,𝑟𝑟0,𝑡𝑡�+𝐼𝐼𝑑𝑑�𝑟𝑟0,𝑡𝑡,𝐴𝐴𝑡𝑡+1,𝐾𝐾𝑡𝑡�+𝐺𝐺𝑡𝑡 Slope = MPC < 1 ↓𝑟𝑟𝑡𝑡 𝑌𝑌𝑡𝑡
𝑑𝑑=𝐶𝐶𝑑𝑑�𝑌𝑌𝑡𝑡−𝐺𝐺𝑡𝑡,𝑌𝑌𝑡𝑡+1−𝐺𝐺𝑡𝑡+1,𝑟𝑟1,𝑡𝑡�+𝐼𝐼𝑑𝑑�𝑟𝑟1,𝑡𝑡,𝐴𝐴𝑡𝑡+1,𝐾𝐾𝑡𝑡�+𝐺𝐺𝑡𝑡 𝑟𝑟1,𝑡𝑡<𝑟𝑟0,𝑡𝑡 Figure 18.3: Desired Expenditure and Income: Expenditure Equals Income We can derive the IS curve graphically as follows. Draw two graphs on top of the other – the upper graph is the graph of the expenditure line, while the bottom graph has rt on the vertical axis and Yt on the horizontal axis. Thus, the horizontal axes are the same in the upper and lower graphs. This is shown in Figure 18.4. Start with some arbitrary real interest rate, r0,t, holding all other exogenous variables ﬁxed. This determines a value of autonomous spending (i.e. the vertical intercept of the expenditure line). Find the value of income where the expenditure line crosses the 45 degree line. Call this Y0,t. Hence, (r0,t, Y0,t) is an (rt, Yt) pair where income equals expenditure, taking the exogenous variables as given. Next, consider a lower value of the interest rate, call it r1,t. This leads the household and ﬁrm to desire more consumption and investment, respectively. This results in the expenditure line shifting up, shown in green in Figure 18.4. This expenditure line crosses the 45 degree line at a higher value of income, call it Y1,t. Hence, (r1,t, Y1,t) is an (rt, Yt) pair where income equals expenditure. Next, consider a higher interest rate, r2,t. This reduces desired consumption and investment for any level of Y
t, therefore shifting the expenditure line down, shown in red in Figure 18.4. This expenditure line crosses the 45 degree line at a lower level of income, call it Y2,t. Hence, (r2,t, Y2,t) is an (rt, Yt) pair where income equals expenditure. If we connect the (rt, Yt) pairs in the lower graph, we have the IS curve. 417 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑 𝑌𝑌𝑡𝑡𝑑𝑑=𝐶𝐶𝑑𝑑(𝑌𝑌𝑡𝑡−𝐺𝐺𝑡𝑡,𝑌𝑌𝑡𝑡+1−𝐺𝐺𝑡𝑡+1,𝑟𝑟𝑡𝑡)+𝐼𝐼𝑑𝑑(𝑟𝑟𝑡𝑡,𝐴𝐴𝑡𝑡+1,𝐾𝐾𝑡𝑡)+𝐺𝐺𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝑌𝑌𝑡𝑡 𝑌𝑌0,𝑡𝑡 𝑌𝑌0,𝑡𝑡 Figure 18.4: The IS Curve: Derivation The IS curve is drawn holding the values of exogenous variables ﬁxed. The exogenous variables which are relevant are Gt, Gt+1, At+1, and Kt. Changes in these exogenous variables will cause the IS curve to shift, as we will see in the next chapter. 18.5.2 The Supply Side The supply side of the economy is governed by the aggregate production function, (18.14), the labor supply curve, (18.11), and the labor demand curve, (18.12). Taking the exogenous variables At, Kt, and θt as given, equations (18.11)-(18.12) both holding determines a value of
Nt. Given a value of Nt, along with exogenous values of At and Kt, the value of Yt is determined from the production function, (18.15). We will deﬁne the Y s curve (or “output supply”) as the set of (rt, Yt) pairs where all three of these equations hold. Since rt does not enter the production function directly, and since it aﬀects neither labor demand nor supply under our assumptions, the value of Yt consistent 418 𝑌𝑌𝑡𝑡𝑑𝑑=𝐶𝐶𝑑𝑑�𝑌𝑌𝑡𝑡−𝐺𝐺𝑡𝑡,𝑌𝑌𝑡𝑡+1−𝐺𝐺𝑡𝑡+1,𝑟𝑟0,𝑡𝑡�+𝐼𝐼𝑑𝑑�𝑟𝑟0,𝑡𝑡,𝐴𝐴𝑡𝑡+1,𝐾𝐾𝑡𝑡�+𝐺𝐺𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑟𝑟𝑡𝑡 𝑟𝑟0,𝑡𝑡 𝑟𝑟2,𝑡𝑡 𝑟𝑟1,𝑡𝑡 𝑟𝑟2,𝑡𝑡>𝑟𝑟0,𝑡𝑡>𝑟𝑟1,𝑡𝑡 𝐼𝐼𝐼𝐼 𝑌𝑌𝑡𝑡𝑑𝑑=𝐶𝐶𝑑�
��𝑌𝑌𝑡𝑡−𝐺𝐺𝑡𝑡,𝑌𝑌𝑡𝑡+1−𝐺𝐺𝑡𝑡+1,𝑟𝑟2,𝑡𝑡�+𝐼𝐼𝑑𝑑�𝑟𝑟2,𝑡𝑡,𝐴𝐴𝑡𝑡+1,𝐾𝐾𝑡𝑡�+𝐺𝐺𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝐶𝐶𝑑𝑑�𝑌𝑌𝑡𝑡−𝐺𝐺𝑡𝑡,𝑌𝑌𝑡𝑡+1−𝐺𝐺𝑡𝑡+1,𝑟𝑟1,𝑡𝑡�+𝐼𝐼𝑑𝑑�𝑟𝑟1,𝑡𝑡,𝐴𝐴𝑡𝑡+1,𝐾𝐾𝑡𝑡�+𝐺𝐺𝑡𝑡 with these three equations holding is independent of rt. In other words, the Y s curve will be a vertical line in a graph with rt on the vertical axis and Yt on the horizontal axis. To derive this formally, let’s use a four part graph. This is shown in Figure 18.5. In the upper left part, we have a graph with wt on the vertical axis and Nt on the horizontal axis. In this graph we plot labor supply, (18.11), which is upward-sloping in wt, and labor demand, (18.12), which is downward-sloping in wt. The intersection of these two curves determines the wage and employment, which we denote w0,t and N0,t. Figure 18.5: The Y s Curve: Derivation Immediately below the labor market equilibrium graph, we plot the production function, with Yt against Nt, where Nt
is on the horizontal axis. This graph is in the lower left quadrant. The production function is plotted holding At and Kt ﬁxed. It starts in the origin and is upward-sloping, but at a diminishing rate, reﬂecting our assumptions about the production function. Given the value of N0,t where we are on both the labor demand and supply curves, we “bring this down” and evaluate the production function at this value, N0,t. This gives us 419 𝑤𝑤𝑡𝑡 𝑟𝑟𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑁𝑁𝑡𝑡 𝑁𝑁𝑡𝑡 𝑌𝑌𝑠𝑠 𝑟𝑟0,𝑡𝑡 𝑟𝑟2,𝑡𝑡 𝑟𝑟1,𝑡𝑡 𝑤𝑤0,𝑡𝑡 𝑌𝑌𝑡𝑡=𝑌𝑌𝑡𝑡 𝐴𝐴𝑡𝑡𝐹𝐹(𝐾𝐾𝑡𝑡,𝑁𝑁𝑡𝑡) 𝑁𝑁0,𝑡𝑡 𝑁𝑁𝑠𝑠(𝑤𝑤𝑡𝑡,𝜃𝜃𝑡𝑡) 𝑁𝑁𝑑𝑑(𝑤𝑤𝑡𝑡,𝐴𝐴𝑡𝑡,𝐾𝐾𝑡𝑡) 𝑌𝑌0,𝑡𝑡 𝑌𝑌0,𝑡𝑡 a value of output, Y0,t. In the lower right quad
rant of Figure 18.5, we simply plot a 45 degree line with Yt on both the horizontal and vertical axes. This is simply a tool to “reﬂect” the vertical axis onto the horizontal axis. So we “bring over” the value Y0,t from the production function evaluated at N0,t, and “reﬂect” this oﬀ of the 45 degree line. We then “bring this up” to the graph in the upper right quadrant, which is a graph with Yt on the horizontal axis and rt on the vertical axis. Since rt aﬀects neither the production function nor the labor market, the value of Yt is independent of rt. The Y s curve is simply a vertical line. 18.5.3 Bringing it all Together The real block of the economy is summarized by the six equations, (18.10)-(18.11). The IS curve is the set of (rt, Yt) pairs where (18.10), (18.13), and 18.15) all hold. The Y s curve is the set of (rt, Yt) pairs where (18.11), (18.12), and (18.14) all hold. All six of the equations holding requires that the economy is simultaneously on both the IS and Y s curves. Graphically, we can see this below in 18.6. 420 Figure 18.6: IS − Y s Equilibrium We can use this ﬁve part graph to determine the equilibrium values of the real wage, w0,t, employment, N0,t, output, Y0,t, and the real interest rate, r0,t. The components of output, in particular consumption and investment, are implicitly determined from the economy being on the IS curve. 421 𝑤𝑤𝑡𝑡 𝑟𝑟𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑁𝑁𝑡𝑡 𝑁𝑁𝑡𝑡 �
�𝑌𝑡𝑡𝑑𝑑 𝑌𝑌𝑠𝑠 𝐼𝐼𝐼𝐼 𝑤𝑤0,𝑡𝑡 𝑟𝑟0,𝑡𝑡 𝑌𝑌0,𝑡𝑡 𝑁𝑁0,𝑡𝑡 𝑁𝑁𝑑𝑑(𝑤𝑤𝑡𝑡,𝐴𝐴𝑡𝑡,𝐾𝐾𝑡𝑡) 𝑁𝑁𝑠𝑠(𝑤𝑤𝑡𝑡,𝜃𝜃𝑡𝑡) 𝐴𝐴𝑡𝑡𝐹𝐹(𝐾𝐾𝑡𝑡,𝑁𝑁𝑡𝑡) 𝑌𝑌𝑡𝑡=𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝐶𝐶𝑑𝑑(𝑌𝑌𝑡𝑡−𝐺𝐺𝑡𝑡,𝑌𝑌𝑡𝑡+1−𝐺𝐺𝑡𝑡+1,𝑟𝑟𝑡𝑡)+𝐼𝐼𝑑𝑑(𝑟𝑟𝑡𝑡,𝐴𝐴𝑡𝑡+1,𝐾𝐾𝑡𝑡)+𝐺𝐺𝑡𝑡 18.5.4 The Nominal Side Once we know the equilibrium values of real endogenous variables, determined graphically in Figure 18.6, we can then turn to the nominal block of the model. Money demand is
summarized by (18.16). The amount of money that a household wants to hold is proportional to the price of goods, Pt, and is a function of the nominal interest rate, which can be written using the Fisher relationship as rt + πe t+1, and the level of current income, Yt. If we graph this with Mt on the horizontal axis and Pt on the vertical axis, it is an upward-sloping line starting in the origin (intuitively, it starts in the origin because of Pt = 0, there is no reason to hold any money). This is shown in Figure 18.7. Figure 18.7: Money Demand It may strike one as odd to talk about a demand curve that is upward-sloping, as is shown in Figure 18.7. This is because Pt is the price of goods measured in units of money. The price of money, measured in units of goods, is 1. If we were to plot money demand as a function Pt of 1, as in the left panel of Figure 18.8 below, the demand curve would have its usual, Pt downward slope. Alternatively, sometimes money demand is plotted as a function of the real interest rate. This is shown in the right panel of Figure 18.8. Any of these representations are ﬁne, but we will work with the one shown in Figure 18.7, where the demand curve appears upward-sloping. 422 𝑃𝑃𝑡𝑡𝑀𝑀𝑑𝑑(𝑟𝑟𝑡𝑡+𝜋𝜋𝑡𝑡+1𝑒𝑒,𝑌𝑌𝑡𝑡) 𝑃𝑃𝑡𝑡 𝑀𝑀𝑡𝑡 Figure 18.8: Money Demand, Alternative Graphical Representations The money supply is set exogenously by the government. Denote this quantity by M0,t. In a graph with Pt on the vertical axis and Mt on the horizontal axis, the money supply curve, M s, is just a vertical line at M0,t. This is shown in Figure 18.9. Figure 18.9: Money Supply In equilibrium, money demand must equal money supply. The position of the money demand curve depends on the values of the real interest rate and output. These are determined
by the intersection of the IS and Y s curves at (r0,t, Y0,t). Given these values, knowing the position of the money demand curve, the equilibrium price level can be determined at the intersection of the money demand and supply curves. This is shown in Figure 18.10 below. 423 𝑃𝑃𝑡𝑡𝑀𝑀𝑑𝑑(𝑟𝑟𝑡𝑡+𝜋𝜋𝑡𝑡+1𝑒𝑒,𝑌𝑌𝑡𝑡) 𝑀𝑀𝑡𝑡 𝑀𝑀𝑡𝑡 𝑟𝑟𝑡𝑡 1𝑃𝑃𝑡𝑡 𝑃𝑃𝑡𝑡𝑀𝑀𝑑𝑑(𝑟𝑟𝑡𝑡+𝜋𝜋𝑡𝑡+1𝑒𝑒,𝑌𝑌𝑡𝑡) 𝑃𝑃𝑡𝑡 𝑀𝑀𝑡𝑡 𝑀𝑀0,𝑡𝑡 𝑀𝑀𝑠𝑠 Figure 18.10: Equilibrium in the Money Market The nominal interest rate is determined given the real interest rate, determined by the intersection of the IS and Y s curves, and the exogenously given expected rate of inﬂation. 18.6 Summary • There are three principal actors in the Neoclassical model: the household, ﬁrms, and the government. We assume that there exists a representative household and ﬁrm both of which behave as price takers. • The household’s optimization conditions are summarized by a consumption function which relates current consumption to current and future disposable income and the real interest rate; a labor supply function which says that the quantity of hours supplied is increasing in the real wage; and a demand for real money balances. • The ﬁrm’s optimization problem is summarized by a labor demand curve and an investment demand curve. Labor demand is positively related to the level of technology and
the current capital stock and negatively related to the real wage. Investment demand depends negatively on the real interest rate and current capital stock, but positively on the level of future productivity. • The government ﬁnances itself through lump sum taxes and we assume there are suﬃcient conditions for Ricardian Equivalence to hold. Consequently, the time path of taxes is irrelevant. 424 𝑀𝑀𝑡𝑡 𝑃𝑃𝑡𝑡 𝑀𝑀0,𝑡𝑡 𝑃𝑃0,𝑡𝑡 𝑃𝑃𝑡𝑡𝑀𝑀𝑑𝑑(𝑟𝑟0,𝑡𝑡+𝜋𝜋𝑡𝑡+1𝑒𝑒,𝑌𝑌0,𝑡𝑡) 𝑀𝑀𝑠𝑠 • The IS curve is all the real interest rate / desired spending combinations such that desired spending equals total income. • The aggregate supply curve is all the real interest rate / output combinations such that households and ﬁrms are optimizing and the ﬁrm operates on their production function. • The money demand function is upward sloping in the price level since it is the inverse of the price of money. Money supply is exogenous. Key Terms • Marginal propensity to consume • Ricardian equivalence • Autonomous expenditure Questions for Review 1. In words, deﬁne the Y s curve. 2. In words, deﬁne the IS curve. 3. Evaluate the following sentence: “Demand curves should slope down. We must have made a mistake in drawing an upward-sloping demand curve for money.” Exercises 1. This exercise will ask you to work through the derivation of the IS curve under various diﬀerent scenarios. (a) Graphically derive the IS curve for a generic speciﬁcation of the con- sumption function and the investment demand function. (b) Suppose that investment demand is relatively more sensitive to the real interest rate than in (a). Relative to (a), how will this impact the shape of the IS curve? (c) Suppose that the MPC is larger than in (
a). How will this aﬀect the shape of the IS curve? 2. Suppose that labor supply were a function of the real interest rate. In particular, suppose that Nt = N s(wt, θt, rt), where ∂N s ∂rt > 0. 425 (a) Can you provide any intuition for why labor supply might positively depend on the real interest rate? (b) Suppose that labor supply is increasing in the real interest rate. Derive the Y s curve graphically. 3. [Excel Problem] Suppose that we assume speciﬁc functional forms for the consumption function and the investment demand function. These are: Ct = c1(Yt − Gt) + c2(Yt+1 − Gt+1) − c3rt It = −d1rt + d2At+1 + d3Kt (18.23) (18.24) Here, c1 through c4 and d1 through d3 are ﬁxed parameters governing the sensitivity of consumption and investment to diﬀerent factors relevant for those decisions. (a) We must have Yt = Ct + It + Gt. Use the given function forms for the consumption and investment with the resource constraint to derive an algebraic expression for the IS curve. (b) Use this to derive an expression for the slope of the IS curve (i.e. ∂Yt ). ∂rt (c) Suppose that the parameters are as follows: c1 = 0.6, c2 = 0.5, c3 = 10, d1 = 20, d2 = 1, and d3 = 0.5. Suppose that Yt+1 = 15, Gt = 10, Gt+1 = 10, At+1 = 5, and Kt = 15. Suppose that rt = 0.1. Create an Excel ﬁle to numerically solve for Yt. (d) Suppose instead that rt = 0.15. Solve for Yt in your Excel ﬁle. (e) Create a range of values of rt, ranging from 0.01 to 0.2, with a gap of 0.001 between values. Solve for Yt for each value of rt. Create a plot with rt on the vertical axis and Yt on the horizontal axis (i.e. create a plot of the IS curve).
Is it downward-sloping, as you would expect? (f) Create another version of your IS curve when At+1 = 7 instead of 5. Plot this along with the IS curve with At+1 = 5. Explain how the higher value of At+1 impacts the position of the IS curve. 426 Chapter 19 Eﬀects of Shocks in the Neoclassical Model In Chapter 18 we laid out and discussed the decision rules characterizing optimal behavior by the household and ﬁrm in the neoclassical model. We also derived a graphical apparatus to characterize the equilibrium. In this chapter, we use this graphical apparatus to analyze the eﬀects of changes in exogenous variables on the endogenous variables of the model. 19.1 Equilibrium The neoclassical model is characterized by the following equations all simultaneously holding: Ct = C d(Yt − Gt, Yt+1 − Gt+1, rt) Nt = N s(wt, θt) Nt = N d(wt, At, Kt) It = I d(rt, At+1, Kt) Yt = AtF (Kt, Nt) Yt = Ct + It + Gt Mt = PtM d(rt + πe t+1, Yt) rt = it − πe t+1 (19.1) (19.2) (19.3) (19.4) (19.5) (19.6) (19.7) (19.8) Equations (19.1)-(19.6) comprise the “real block” of the model, while equations (19.7)(19.8) comprise the “nominal block” of the model. The IS curve summarizes (19.1), (19.4), and (19.6), while the Y s curve summarizes (19.2), (19.3), and (19.5). Graphically: 427 Figure 19.1: IS − Y s Equilibrium The equilibrium real interest rate and level of output, r0,t and Y0,t, are determined at the intersection of the IS and Y s curves. Once these are known, the position of the money demand curve, which is given by (19.7), is known, and the equilibrium price level can be determined by the intersection of this demand curve with the exogenous quantity
of money supplied. This is shown in Figure 19.2. 428 𝑤𝑤𝑡𝑡 𝑟𝑟𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑁𝑁𝑡𝑡 𝑁𝑁𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑 𝑌𝑌𝑠𝑠 𝐼𝐼𝐼𝐼 𝑤𝑤0,𝑡𝑡 𝑟𝑟0,𝑡𝑡 𝑌𝑌0,𝑡𝑡 𝑁𝑁0,𝑡𝑡 𝑁𝑁𝑑𝑑(𝑤𝑤𝑡𝑡,𝐴𝐴𝑡𝑡,𝐾𝐾𝑡𝑡) 𝑁𝑁𝑠𝑠(𝑤𝑤𝑡𝑡,𝜃𝜃𝑡𝑡) 𝐴𝐴𝑡𝑡𝐹𝐹(𝐾𝐾𝑡𝑡,𝑁𝑁𝑡𝑡) 𝑌𝑌𝑡𝑡=𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝐶𝐶𝑑𝑑(𝑌𝑌𝑡𝑡−𝐺𝐺𝑡𝑡,𝑌𝑌𝑡𝑡+1−𝐺�
��𝑡𝑡+1,𝑟𝑟𝑡𝑡)+𝐼𝐼𝑑𝑑(𝑟𝑟𝑡𝑡,𝐴𝐴𝑡𝑡+1,𝐾𝐾𝑡𝑡)+𝐺𝐺𝑡𝑡 Figure 19.2: Equilibrium in the Money Market 19.2 The Eﬀects of Changes in Exogenous Variables on the En- dogenous Variables The exogenous variables of the model include the current and future levels of productivity, At and At+1; the current and future levels of government spending, Gt and Gt+1; the current capital stock, Kt; the value of the labor supply shifter, θt; the quantity of money supplied, Mt; and the rate of expected inﬂation, πe t+1. We will refer to changes in an exogenous variable as “shocks.” Our objective is to understand how the endogenous variables of the model react to diﬀerent shocks. Some of these shocks will be analyzed in the text that follows, while the remainder are left as exercises. Focusing on the equations underlying the curves, we can split the shocks into three diﬀerent categories. At and θt are supply shocks in that they appear only in the equations underlying the Y s curve; Gt, Gt+1, and At+1 are demand shocks in that they only appear in the equations underlying the IS curve; Mt and πe t+1 are nominal shocks that do not appear in the equations underlying the Y s or IS curves. Kt is both a demand shock (it inﬂuences the amount of desired investment, and hence the IS curve) as well as a supply shock (it inﬂuences the amount of output that can be produced given labor). We will not focus on ﬂuctuations in Kt here. While Kt can exogenously decrease (say, due to a hurricane that wipes out some of a country’s capital), it cannot exogenously increase (capital must itself be produced). Thus, ﬂuctuations in Kt are not a candidate source for business cycle ﬂuctuations (deﬁned as increases and
decreases in output relative to trend). 429 𝑀𝑀𝑡𝑡 𝑃𝑃𝑡𝑡 𝑀𝑀0,𝑡𝑡 𝑃𝑃0,𝑡𝑡 𝑃𝑃𝑡𝑡𝑀𝑀𝑑𝑑(𝑟𝑟0,𝑡𝑡+𝜋𝜋𝑡𝑡+1𝑒𝑒,𝑌𝑌0,𝑡𝑡) 𝑀𝑀𝑠𝑠 There is one potentially thorny issue that bears mentioning here. Current consumption demand depends on expectations of future income, Yt+1. Future income is an endogenous variable. The complication arises because changes in all of the exogenous variables will induce changes in current investment, which would aﬀect the future stock of capital, and hence future output. We will ignore these eﬀects. As noted at the onset of this part of the book, when thinking about the medium run we think about the capital stock as eﬀectively being ﬁxed. While investment will ﬂuctuate in response to shocks, the ﬂuctuations in investment relative to the size of the capital stock will be small, and we can therefore safely ignore the eﬀects of changes in current investment on future capital over a short enough period of time (say a few years). Concretely, our assumption means that we will treat Yt+1 as invariant to changes in period t exogenous variables – i.e. we will treat Yt+1 as ﬁxed when At, Gt, θt, or Mt change. We will not treat Yt+1 as ﬁxed when expected future exogenous variables change – i.e. we will permit changes in At+1 or Gt+1, anticipated in period t, to aﬀect expectations of Yt+1. Change in these variables will impact Yt+1 in exactly the same way that changes in the period t versions of these exogenous variables would aﬀect Yt. As we shall see, changes in πe t+1 will not have any e
ﬀect on real variables, and so we can treat Yt+1 as ﬁxed with respect to πe t+1 as well, even though this variable is dated t + 1. In the subsections below, we work through the eﬀects on the endogenous variables of shocks to each of the exogenous variables. In doing so, we assume that the economy is initially in an equilibrium characterized by a 0 subscript (i.e. the initial equilibrium level of output is Y0,t). The new equilibrium, taking into account a change in an exogenous variable, will be denoted by a 1 subscript (i.e. the new equilibrium level of output will be Y1,t). We will consider exogenous increases in a subset of exogenous variables; the exercises would be similar, with reversed signs, for decreases. 19.2.1 Productivity Shock: Increase in At: Consider ﬁrst an exogenous increase in At, from A0,t to A1,t, where A1,t > A0,t. This is a supply side shock, so let’s focus on the curves underlying the supply side of the model. An increase in At shifts the labor demand curve to the right. This results in a higher level of Nt and a higher wt, which we denote w1,t and N1,t. The higher At also shifts the production function up – for a given Nt, the ﬁrm produces more Yt when At is higher for given levels o Nt and Kt. If you combine the higher Nt from the labor market with the production function that has shifted up, you get a higher level of Yt, call it Y1,t. Output on the supply side rises for two reasons – the exogenous increase in At, and the endogenous increase in Nt. Since the value of Yt from the supply side is independent of the level of rt under our assumptions, the 430 vertical Y s curve shifts to the right. These eﬀects are shown with the blue lines in Figure 19.3 below. Figure 19.3: Increase in At There is no shift of the IS curve. The rightward shift of the Y s curve, combined with no shift in the IS curve, means that rt must fall, to r1,t. The lower rt causes the expenditure line to shift up in such a way that income equals
expenditure at the new higher level of Yt. 431 𝑤𝑤𝑡𝑡 𝑟𝑟𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑁𝑁𝑡𝑡 𝑁𝑁𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑 𝑌𝑌𝑠𝑠 𝐼𝐼𝐼𝐼 𝑤𝑤0,𝑡𝑡 𝑟𝑟0,𝑡𝑡 𝑌𝑌0,𝑡𝑡 𝑁𝑁0,𝑡𝑡 𝑁𝑁𝑑𝑑(𝑤𝑤𝑡𝑡,𝐴𝐴0,𝑡𝑡,𝐾𝐾𝑡𝑡) 𝑁𝑁𝑠𝑠(𝑤𝑤𝑡𝑡,𝜃𝜃𝑡𝑡) 𝐴𝐴0,𝑡𝑡𝐹𝐹(𝐾𝐾𝑡𝑡,𝑁𝑁𝑡𝑡) 𝑌𝑌𝑡𝑡=𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝐶𝐶𝑑𝑑�𝑌𝑌𝑡𝑡−𝐺𝐺𝑡𝑡,𝑌𝑌𝑡𝑡+1−�
�𝐺𝑡𝑡+1,𝑟𝑟0,𝑡𝑡�+𝐼𝐼𝑑𝑑�𝑟𝑟0,𝑡𝑡,𝐴𝐴𝑡𝑡+1,𝐾𝐾𝑡𝑡�+𝐺𝐺𝑡𝑡 𝐴𝐴1,𝑡𝑡𝐹𝐹(𝐾𝐾𝑡𝑡,𝑁𝑁𝑡𝑡) 𝑌𝑌𝑠𝑠′ 𝑁𝑁𝑑𝑑(𝑤𝑤𝑡𝑡,𝐴𝐴1,𝑡𝑡,𝐾𝐾𝑡𝑡) 𝑟𝑟1,𝑡𝑡 𝑌𝑌1,𝑡𝑡 𝑁𝑁1,𝑡𝑡 𝑤𝑤1,𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝐶𝐶𝑑𝑑�𝑌𝑌𝑡𝑡−𝐺𝐺𝑡𝑡,𝑌𝑌𝑡𝑡+1−𝐺𝐺𝑡𝑡+1,𝑟𝑟1,𝑡𝑡�+𝐼𝐼𝑑𝑑�𝑟𝑟1,𝑡𝑡,𝐴𝐴𝑡𝑡+1,𝐾𝐾𝑡𝑡�+𝐺𝐺𝑡𝑡 This is an “indirect” eﬀect of the lower real interest rate, and is hence shown in green in the diagram. Eﬀectively, when At goes up, ﬁrms produce more output. Since the higher level of output must translate into higher
expenditure in equilibrium, the real interest rate must fall, which induces the household to consume more and the ﬁrm to investment more. Hence, Ct and It both rise. We can think about general equilibrium in the model as being characterized by rt falling as the economy “moves down” along the IS curve until the point where the economy is on both IS and Y s curves. Now, let us examine the eﬀects on nominal endogenous variables. Since πe t+1 is taken to be exogenous, a lower real interest rate translates into a lower nominal interest rate. The lower interest rate leads to an increase in money demand, as does the higher level of income. Hence, the money demand curve shifts out to the right, which is shown in Figure 19.4. With no change in money supply, the price level must fall so that the money market is in equilibrium. Hence, a higher At causes Pt and it to both fall. Figure 19.4: Increase in At: The Money Market 19.2.2 Expected Future Productivity Shock: Increase in At+1 Suppose that agents in the economy expect the future level of productivity, At+1, to increase. In the recent academic literature, a shock such as this has come to be called a “news shock.” More generally, we could think about expectations of higher future productivity as representing a wave of optimism or “animal spirits” as Keynes originally coined the term. A change in At+1 aﬀects the demand side of the model. The supply side in period t only depends on the current level of productivity. There is both a direct and an indirect eﬀect on 432 𝑀𝑀𝑡𝑡 𝑃𝑃𝑡𝑡 𝑀𝑀0,𝑡𝑡 𝑃𝑃0,𝑡𝑡 𝑀𝑀𝑑𝑑(𝑟𝑟0,𝑡𝑡+𝜋𝜋𝑡𝑡+1𝑒𝑒,𝑌𝑌0,𝑡𝑡) 𝑀𝑀𝑠𝑠 𝑀𝑀𝑑𝑑(𝑟𝑟1,�
��𝑡+𝜋𝜋𝑡𝑡+1𝑒𝑒,𝑌𝑌1,𝑡𝑡) 𝑃𝑃1,𝑡𝑡 current demand. First, higher At+1 makes the ﬁrm want to do more investment. Second, an increase in At+1 is like an increase in current productivity from the perspective of period t + 1. Hence, Yt+1 will rise, as we saw above when analyzing the eﬀects of an increase in current At. This will make the household want to consume more in the present as well. Both the increase in desired investment and consumption raise autonomous desired expenditure in period t. This shifts the expenditure line up (shown in blue below) and causes the IS curve to shift out to the right. This is shown in Figure 19.5 below. The higher At+1 raises expectations about future income from Y0,t+1 to Y1,t+1: 433 Figure 19.5: Increase in At+1 There is no shift in the Y s curve. Hence, in equilibrium, Yt is unchanged. The real interest rate must rise. This is demonstrated with the green arrow in the expenditure line graph, where the increase in rt is suﬃcient to make the desired expenditure line shift back to where it began. Nothing happens in the labor market. It is ambiguous as to what happens to Ct and It. Since Gt is exogenous and Yt is unchanged, we know that Ct + It (what one might 434 𝑤𝑤𝑡𝑡 𝑟𝑟𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑁𝑁𝑡𝑡 𝑁𝑁𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑 𝑌𝑌𝑠𝑠 𝐼𝐼𝐼𝐼 𝑤𝑤0,
𝑡𝑡 𝑟𝑟0,𝑡𝑡 𝑌𝑌0,𝑡𝑡 𝑁𝑁0,𝑡𝑡 𝑁𝑁𝑑𝑑(𝑤𝑤𝑡𝑡,𝐴𝐴𝑡𝑡,𝐾𝐾𝑡𝑡) 𝑁𝑁𝑠𝑠(𝑤𝑤𝑡𝑡,𝜃𝜃𝑡𝑡) 𝐴𝐴𝑡𝑡𝐹𝐹(𝐾𝐾𝑡𝑡,𝑁𝑁𝑡𝑡) 𝑌𝑌𝑡𝑡=𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝐶𝐶𝑑𝑑�𝑌𝑌𝑡𝑡−𝐺𝐺𝑡𝑡,𝑌𝑌0,𝑡𝑡+1−𝐺𝐺𝑡𝑡+1,𝑟𝑟0,𝑡𝑡�+𝐼𝐼𝑑𝑑�𝑟𝑟0,𝑡𝑡,𝐴𝐴0,𝑡𝑡+1,𝐾𝐾𝑡𝑡�+𝐺𝐺𝑡𝑡 𝑟𝑟1,𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝐶𝐶𝑑𝑑�𝑌𝑌𝑡𝑡−𝐺𝐺𝑡𝑡,𝑌𝑌1,𝑡
𝑡+1−𝐺𝐺𝑡𝑡+1,𝑟𝑟1,𝑡𝑡�+𝐼𝐼𝑑𝑑�𝑟𝑟1,𝑡𝑡,𝐴𝐴1,𝑡𝑡+1,𝐾𝐾𝑡𝑡�+𝐺𝐺𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝐶𝐶𝑑𝑑�𝑌𝑌𝑡𝑡−𝐺𝐺𝑡𝑡,𝑌𝑌1,𝑡𝑡+1−𝐺𝐺𝑡𝑡+1,𝑟𝑟0,𝑡𝑡�+𝐼𝐼𝑑𝑑�𝑟𝑟0,𝑡𝑡,𝐴𝐴1,𝑡𝑡+1,𝐾𝐾𝑡𝑡�+𝐺𝐺𝑡𝑡 𝐼𝐼𝐼𝐼′ call private expenditure) must be unchanged. rt being higher works to make both Ct and It lower, counteracting the positive eﬀect of the higher At+1. Which eﬀect dominates for which variable is unclear, so we cannot say with certainty what happens to Ct or It. We do know, however, that if Ct rises, It must fall (and vice-versa). Let us turn next to the money market. Since rt rises and πe t+1 is taken to be exogenous, then it must rise as well. Higher rt works to pivot the money demand curve in. Since Yt is unaﬀected, we know that money demand therefore pivots in. Along a stable money supply curve, this means that the price level, Pt, must rise. This is shown in Figure 19.6. Figure 19.6: Increase in At+1: The Money Market 19.2.3 Government Spending Shock: Increase in Gt: Suppose that there is
an exogenous increase in Gt. As noted above, we are here assuming that Ricardian Equivalence holds, so it is irrelevant how this spending increase is ﬁnanced. The household behaves as though the government fully ﬁnances the increase in spending with an increase in current taxes. Gt is a demand-side shock, and will aﬀect the position of the IS curve. How will it do so? Gt shows up twice in the expressions underlying the IS curve – once directly as an independent component of expenditure, and once indirectly inside the consumption function. The direct eﬀect is positive, whereas the indirect eﬀect is negative. So how is desired expenditure impacted? It turns out that desired expenditure increases for every level of income. This is shown formally in the Mathematical Diversion below. The intuition for it is straightforward. Since the MPC is less than 1, the negative indirect eﬀect of higher Gt (the reduction in 435 𝑀𝑀𝑡𝑡 𝑃𝑃𝑡𝑡 𝑀𝑀0,𝑡𝑡 𝑃𝑃0,𝑡𝑡 𝑀𝑀𝑑𝑑(𝑟𝑟0,𝑡𝑡+𝜋𝜋𝑡𝑡+1𝑒𝑒,𝑌𝑌0,𝑡𝑡) 𝑀𝑀𝑠𝑠 𝑃𝑃1,𝑡𝑡 𝑀𝑀𝑑𝑑(𝑟𝑟1,𝑡𝑡+𝜋𝜋𝑡𝑡+1𝑒𝑒,𝑌𝑌0,𝑡𝑡) consumption) is smaller than the direct eﬀect (the increase in one of the components of expenditure). Hence, total autonomous expenditure increases, which shifts the IS curve to the right. Mathematical Diversion Autonomous expenditure, deﬁned in Chapter 18 in equation (18.22), is total desired expenditure when current income is zero. Formally: E0 = C d(−Gt, Y
t+1 − Gt+1, rt) + I d(rt, At+1, Kt) + Gt (19.9) The partial derivative of E0 with respect to Gt is: ∂E0 ∂Gt = − ∂C d(⋅) ∂Yt + 1 = 1 − M P C (19.10) The ﬁrst term on the right hand side of (19.10) is the negative of the partial derivative of the consumption function with respect to its ﬁrst argument, which we denote as ∂Cd (the argument is Yt − Gt). This is simply the MPC, which we ∂Yt take to be a constant less than 1. Hence, an increase in Gt raises autonomous expenditure (the vertical intercept of the expenditure line) by 1 − M P C, which is positive given that the MPC is less than 1. The increase in Gt therefore raises autonomous expenditure. This means that the expenditure line shifts up for a given rt, resulting in the IS curve shifting out to the right. These eﬀects are shown in blue in Figure 19.7. A rightward shift of the IS curve with no eﬀect on the Y s curve means that nothing happens to Yt, while rt increases. The increase in rt reduces autonomous expenditure (both investment and consumption), in such a way that the expenditure line shifts back down to where it began. A higher rt means that on net It is lower. A higher rt, in conjunction with higher Gt, also means that Ct is lower. There are no eﬀects on labor market variables. 436 Figure 19.7: Increase in Gt Having determined the eﬀects of an increase in Gt on the real variables of the model, we turn next to the nominal variables. A higher rt, in conjunction with no change in Yt, means that the money demand curve pivots in. Along a stable money supply curve, this results in an increase in Pt. Given that we take πe t+1 to be exogenous, the nominal interest rate simply moves in the same direction as the real interest rate. 437 𝑤𝑤𝑡𝑡 𝑟𝑟𝑡𝑡 𝑌𝑌𝑡�
�� 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑁𝑁𝑡𝑡 𝑁𝑁𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑 𝑌𝑌𝑠𝑠 𝐼𝐼𝐼𝐼 𝑤𝑤0,𝑡𝑡 𝑟𝑟0,𝑡𝑡 𝑌𝑌0,𝑡𝑡 𝑁𝑁0,𝑡𝑡 𝑁𝑁𝑑𝑑(𝑤𝑤𝑡𝑡,𝐴𝐴𝑡𝑡,𝐾𝐾𝑡𝑡) 𝑁𝑁𝑠𝑠(𝑤𝑤𝑡𝑡,𝜃𝜃𝑡𝑡) 𝐴𝐴𝑡𝑡𝐹𝐹(𝐾𝐾𝑡𝑡,𝑁𝑁𝑡𝑡) 𝑌𝑌𝑡𝑡=𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝐶𝐶𝑑𝑑�𝑌𝑌𝑡𝑡−𝐺𝐺0,𝑡𝑡,𝑌𝑌𝑡𝑡+1−𝐺𝐺𝑡𝑡+1,𝑟𝑟0,𝑡𝑡�+𝐼𝐼𝑑𝑑�𝑟𝑟0,�
��𝑡,𝐴𝐴𝑡𝑡+1,𝐾𝐾𝑡𝑡�+𝐺𝐺0,𝑡𝑡 𝑟𝑟1,𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝐶𝐶𝑑𝑑�𝑌𝑌𝑡𝑡−𝐺𝐺1,𝑡𝑡,𝑌𝑌𝑡𝑡+1−𝐺𝐺𝑡𝑡+1,𝑟𝑟1,𝑡𝑡�+𝐼𝐼𝑑𝑑�𝑟𝑟1,𝑡𝑡,𝐴𝐴𝑡𝑡+1,𝐾𝐾𝑡𝑡�+𝐺𝐺1,𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝐶𝐶𝑑𝑑�𝑌𝑌𝑡𝑡−𝐺𝐺1,𝑡𝑡,𝑌𝑌𝑡𝑡+1−𝐺𝐺𝑡𝑡+1,𝑟𝑟0,𝑡𝑡�+𝐼𝐼𝑑𝑑�𝑟𝑟0,𝑡𝑡,𝐴𝐴𝑡𝑡+1,𝐾𝐾𝑡𝑡�+𝐺𝐺1,𝑡𝑡 𝐼𝐼𝐼𝐼′ Figure 19.8: Increase in Gt: The Money Market One often hears about the government spending multiplier – how much output changes for a one unit change in government spending. This can be cast in terms of derivatives, or dYt. dGt In the neoclassical model under our assumptions concerning labor supply, the government spending multiplier is zero – output does not change.
1 This is a result of our assumption that the supply of output is invariant to Gt – there is no mechanism in this model through which higher Gt could entice the ﬁrm to produce more output. On the demand side, a multiplier of zero obtains because the real interest rate rises, which reduces both It and Ct suﬃciently so that total expenditure remains unchanged. Put a little bit diﬀerently, private expenditure is completely “crowded out” by the increase in public expenditure. Crowding out is a term used in economics to refer to the fact that increases in government spending may result in decreases in private expenditure due to equilibrium eﬀects on the real interest rate. In the case of the neoclassical model, crowding out is said to be complete – the reduction in private spending completely oﬀsets the increase in public spending, leaving total expenditure unchanged. One can derive an expression for the “ﬁxed interest rate multiplier,” or the change in Yt for a change in Gt, if the real interest rate were held ﬁxed. We can think about this as representing how output would change if the Y s curve were horizontal instead of vertical. For the neoclassical model with Ricardian Equivalence, the ﬁxed interest rate multiplier turns 1The relevant assumption giving rise to this result is that labor supply only depends on the real wage and the exogenous variable θt. Under alternative assumptions about preferences, it could be the case that labor supply is increasing in rt, and hence the Y s curve is upward-sloping. Under this assumption, which is laid out in Appendix C, the government spending multiplier will be positive but will nevertheless be less than one. 438 𝑀𝑀𝑡𝑡 𝑃𝑃𝑡𝑡 𝑀𝑀0,𝑡𝑡 𝑃𝑃0,𝑡𝑡 𝑀𝑀𝑑𝑑(𝑟𝑟0,𝑡𝑡+𝜋𝜋𝑡𝑡+1𝑒𝑒,𝑌𝑌0,𝑡𝑡) 𝑀𝑀𝑠𝑠 �
��𝑃1,𝑡𝑡 𝑀𝑀𝑑𝑑(𝑟𝑟1,𝑡𝑡+𝜋𝜋𝑡𝑡+1𝑒𝑒,𝑌𝑌0,𝑡𝑡) out to be 1, as is shown formally in the mathematical diversion below. If there were no Ricardian Equivalence, and the increase in government spending were ﬁnanced with debt > 1, which is what is as opposed to taxes, the ﬁxed interest rate multiplier would be often presented in textbook treatments. This expression for the multiplier only holds if (i) there is no Ricardian Equivalence, (ii) the increase in spending is ﬁnanced via debt, and (iii) the real interest rate is ﬁxed. 1 1−M P C Mathematical Diversion The IS equation can be written mathematically as: Yt = C d(Yt − Gt, Yt+1 − Gt+1, rt) + I d(rt, At+1, Kt) + Gt (19.11) Here, this is an implicit function – Yt appears on both the right and left hand sides. Another term for the total derivative is the “implicit derivative,” which is a way to derive an expression for a derivative of an implicit function. Take the total derivative of (19.11), holding all exogenous variables but Gt ﬁxed: dYt = ∂C d(⋅) ∂Yt (dYt − dGt) + ∂C d(⋅) ∂rt drt + ∂I d(⋅) ∂rt drt + dGt (19.12) Now, suppose that rt is held ﬁxed, so drt = 0. Denoting ∂Cd ∂Yt can be re-written: (⋅) as MPC, (19.12) dYt = M P C(dYt − dGt) + dGt (19.13) This can be re-arranged to yield: dYt dGt = 1 (19.14) In other words, the �
�ﬁxed interest rate” multiplier is 1. In words, what this says is that the IS curve shifts out horizontally to the right one-for-one with an increase in Gt – if rt were held ﬁxed, Yt would increase by Gt. What is the intuition for this result? It is easiest to think about this by thinking about a period being broken into many “rounds” with many diﬀerent households. The following example hopefully makes this clear. In “round 1,” the government increases spending by dGt and increases the taxes of a household by the same amount. This increases total expenditure by (1 − M P C)dGt – the 1 is the direct eﬀect of the expenditure, while the −M P C is the indirect eﬀect from the household 439 on whom the tax is levied reducing its consumption by the MPC times the change in its take-home income. Since the MPC is less than 1, (1 − M P C) > 0, so total expenditure rises in round 1. But that additional expenditure is additional income for a diﬀerent household. In “round 2,” with (1 − M P C)dGt extra in income, that household will increase its consumption by M P C(1 − M P C)dGt – i.e. it will consume MPC of the additional income. Hence, in “round 2,” there is an additional increase in expenditure of M P C(1 − M P C)dGt. But that extra expenditure is income for some other household. In “round 3,” that household will increases its consumption by M P C × M P C(1 − M P C)dGt, or the MPC times the extra income generated from the previous round. This process continues until there is no additional expenditure. Formally, we can summarized the eﬀect on expenditure in each round as: Round 1 = (1 − M P C)dGt Round 2 = M P C(1 − M P C)dGt Round 3 = M P C 2(1 − M P C)dGt Round 4 = M P C 3(1 − M P C)dGt ⋮ Round j = M P C j−1(1 − M P C)
dGt The total change in income/expenditure is the sum of changes from each “round,” or: dYt = (1 − M P C)dGt [19.15) Using the formula for an inﬁnite sum derived in Appendix A, the term inside brackets is equal to 1−M P C. The MPC’s cancel, and one gets dYt = dGt. 1 Suppose that we instead assumed that consumption was not forward-looking and that Ricardian Equivalence did not hold. In particular, suppose that the consumption function is given by: Ct = C d(Yt − Tt, rt) (19.16) 440 In (19.16), consumption depends only on current net income and the real interest rate. Since consumption is not forward-looking, Ricardian Equivalence does not necessarily hold, and we cannot act as though Tt = Gt. With this consumption function, the mathematical expression for the IS curve is given by: Yt = C d(Yt − Tt, rt) + I d(rt, At+1, Kt) + Gt (19.17) Totally diﬀerentiate (19.17): dYt = ∂C d(⋅) ∂Yt (dYt − dTt) + ∂C d(⋅) ∂rt drt + ∂I d(⋅) ∂rt drt + dGt (19.18) Now, re-label the partial derivative of the consumption function with respect to its ﬁrst argument as M P C, and suppose that the real interest rate is held ﬁxed. (19.18) can be written: dYt = M P CdYt − M P CdTt + dGt (19.19) Now, suppose that the government spending increase is “tax ﬁnanced,” so that dTt = dGt (i.e. taxes increase by the same amount as the increase in spending). Then (19.19) reduces to the same expression in the Ricardian Equivalence case, (19.14). But suppose that the increase in spending is “deﬁcit ﬁnanced,” so that dTt = 0.
Then, (19.19) reduces to: dYt dGt = 1 1 − M P C (19.20) Since the MPC is less than 1, this expression is greater than 1. In other words, without Ricardian Equivalence, a deﬁcit-ﬁnanced increase in government spending raises output by a multiple of the initial increase in spending. Note that this expression only holds if rt is ﬁxed. Were we to incorporate a consumption function like (19.16) into the model, the government spending multiplier in equilibrium would still be 0 – rt would rise to completely crowd out private expenditure given our assumptions about the supply side of the economy. Compared to the Ricardian equivalence case, rt would have to rise more, but output would still not change in response to an increase in Gt. 441 19.2.4 An Increase in the Money Supply: Increase in Mt Now, consider an exogenous increase in Mt. Mt does not appear anywhere in the “real block” of the model (the ﬁrst six equations). Hence, neither the IS nor the Y s curves shift. There is no eﬀect of the change in Mt on any real variable. We therefore say that “money is neutral,” by which we mean that a change in the money supply has no eﬀect on any real variables. The only eﬀect of an increase in Mt will be on the price level. We can see this in a money market graph, shown below in Figure 19.9. The vertical money supply curve shifts to the right. The money demand curve does not shift. The only eﬀect is an increase in Pt. The nominal interest rate is unchanged, since πe t+1 is taken as given and rt is unaﬀected. Figure 19.9: Increase in Mt In this model, money is completely neutral – changes in Mt have no eﬀect on any real variables. Though monetary neutrality is not how most people think about the real world (i.e. most people seem to think that what the Fed does matters for the real economy), the intuition for monetary neutrality is pretty clear once one thinks about it. Changing the quantity of Mt, in a sense, just changes the measurement of the units of account. Whether I call one can of soda 2 dollars or 4 dollars shouldn�
�t impact how much soda I buy – when I purchase something like soda, I am functionally trading my time (which generates income in the form of the wages) for a good. Money is just an intermediary used in exchange, and how I value money shouldn’t impact how much exchange I conduct. To get monetary non-neutrality, we need some form of “stickiness” in prices (how much I pay for the soda) or wages (how 442 𝑀𝑀𝑡𝑡 𝑃𝑃𝑡𝑡 𝑀𝑀0,𝑡𝑡 𝑃𝑃0,𝑡𝑡 𝑃𝑃𝑡𝑡𝑀𝑀𝑑𝑑(𝑟𝑟0,𝑡𝑡+𝜋𝜋𝑡𝑡+1𝑒𝑒,𝑌𝑌0,𝑡𝑡) 𝑀𝑀𝑠𝑠 𝑀𝑀𝑠𝑠′ 𝑀𝑀1,𝑡𝑡 𝑃𝑃1,𝑡𝑡 much income I earn from my time spent working, which inﬂuences how much soda I can purchase). If prices and/or wages are unable to instantaneously adjust to the change in Mt, changes in Mt could impact real variables like how much soda I consume. When we study Keynesian models later in Part V, we will do just this. 19.2.5 Expected Future Inﬂation: Increase in πe t+1 Finally, suppose that there is an exogenous increase in expected inﬂation, πe t+1. Like a change in Mt, this has no eﬀect on any of the real variables in the model – neither the IS nor the Y s curves shift, and there is no change in rt or Yt. For a ﬁxed real interest rate, an increase in πe t+1 raises the nominal interest rate, it, from the Fisher relationship. This higher nominal interest rate depresses the demand for money, causing the money demand curve to pivot in. Along a stable
money supply curve, this results in an increase in Pt. This is shown below in Figure 19.10. Figure 19.10: Increase in πe t+1 From Figure 19.10, we see that there is an element of “self-fulﬁllment” in terms of an increase in expected future inﬂation. Put diﬀerently, expecting more future inﬂation results in more current inﬂation (i.e. an increase in Pt). An increase in expected future inﬂation could be triggered by the central bank promising to expand the money supply in the future. From our analysis, this would have the eﬀect of raising the price level in the present. This is, in a nutshell, what much of the non-standard monetary policy of the last several years has sought to accomplish. 443 𝑀𝑀𝑡𝑡 𝑃𝑃𝑡𝑡 𝑀𝑀0,𝑡𝑡 𝑃𝑃0,𝑡𝑡 𝑃𝑃𝑡𝑡𝑀𝑀𝑑𝑑(𝑟𝑟0,𝑡𝑡+𝜋𝜋0,𝑡𝑡+1𝑒𝑒,𝑌𝑌0,𝑡𝑡) 𝑀𝑀𝑠𝑠 𝑃𝑃𝑡𝑡𝑀𝑀𝑑𝑑(𝑟𝑟0,𝑡𝑡+𝜋𝜋1,𝑡𝑡+1𝑒𝑒,𝑌𝑌0,𝑡𝑡) 𝑃𝑃1,𝑡𝑡 19.2.6 Summary of Qualitative Eﬀects Table 26.1 summarizes the qualitative eﬀects of increases in the diﬀerent exogenous variables on the eight endogenous variables of the model. A + sign indicates that the endogenous variable in question increases when the relevant exogenous variable increases, a − sign indicates that the endogenous variable decreases, a? indicates that the
eﬀect is ambiguous, and a 0 indicates that the endogenous variable is unaﬀected. Note that the eﬀects of changes in θt and Gt+1 are left as exercises. Table 19.1: Qualitative Eﬀects of Exogenous Shocks on Endogenous Variables Variable Yt ↑ At + Exogenous Shock ↑ Mt ↑ Gt 0 0 ↑ At+1 0 ↑ πe 0 t+1 Ct It Nt wt rt it Pt + + + + - - -?? + + 19.3 Summary • We can use the IS and Y s curves to graphically analyze how the diﬀerent endogenous variables of the neoclassical model react to changes in the exogenous variables. In doing so, we treat the future capital stock as eﬀectively ﬁxed, which means that Yt+1 does not react to changes in period t exogenous variables which potentially impact period t investment. • The neoclassical model oﬀers a supply-driven theory of economic ﬂuctuations. Because the Y s curve is vertical, only supply shocks (changes in At or θt) can result in movements in output and labor market variables. Demand shocks (changes in At+1, Gt, or Gt+1) only aﬀect the composition of output between consumption and investment, not the level of output. 444 • The real interest rate is a key price in the model which adjusts to shocks to force aggregate expenditure to equal aggregate production. Diﬀerent assumptions on labor supply (see the relevant discussion in Chapter 12 or Appendix C) could generate an upward-sloping Y s curve, but plausible parameterizations would generate a nearly vertical Y s curve, wherein supply shocks would account for the vast majority of output and labor market ﬂuctuations. • The model features monetary neutrality and the classical dichotomy holds. Monetary neutrality means that changes in exogenous nominal variables do not aﬀect the equilibrium values of real variables. The classical dichotomy means that real variables are determined in equilibrium independently of nominal variables – one need not know the values of the exogenous nominal variables to determine the equilibrium values of the endogenous real variables. • The converse is not true – changes in real exogenous variables will aﬀect nominal endogenous variables. Positive supply shocks (increase in At or a decrease in �
�t) result in a lower price level; positive demand shocks (increases in At+1 in and Gt, or a decreases in ft or in Gt+1) raise the price level. Key Terms • Classical dichotomy • Monetary neutrality • Fixed interest rate government spending multiplier • Crowding out Questions for Review 1. Can you provide any intuition for the neutrality of money in the neoclassical model? Do you think monetary neutrality is a good benchmark when thinking about the real world? 2. Deﬁne what is meant by the “classical dichotomy.” If the classical dichotomy holds, can we ignore nominal variables when thinking about the real eﬀects of changes in real exogenous variables? 3. Explain why shocks to the IS curve have no eﬀect on output in the neoclassical model. Exercises 445 1. Consider the basic neoclassical model. Suppose that there is an increase in At. Draw out two versions of the model, one in which labor supply is relative elastic (i.e. sensitive to the real wage), and one in which labor supply is relatively inelastic (i.e. relatively insensitive to the real wage). Comment on how the magnitudes of the changes in Yt, rt, wt, and Nt depend on how sensitive labor supply is to the real wage. 2. Consider the basic Neoclassical model. Suppose that there is an increase in θt. (a) Graphically analyze this change and describe how each endogenous variable changes. (b) Now, draw out two versions of the model, one in which labor demand is relatively elastic (i.e. sensitive to the real wage), and one in which labor supply is relatively inelastic (i.e. relatively insensitive to the real wage). Comment on how the magnitudes of the changes in Yt, rt, wt, and Nt depend on how sensitive labor supply is to the real wage. 3. Consider the basic neoclassical model. Suppose that there is an increase in θt. Draw out two versions of the model, one in which labor demand is relatively elastic (i.e. sensitive to the real wage), and one in which labor supply is relatively inelastic (i.e. relatively insensitive to the real wage). Comment on how the magnitudes of the changes in Yt, rt, wt, and Nt depend on how sensitive labor supply
is to the real wage. 4. Consider the basic neoclassical model. Suppose that there is a reduction in At. In which direction will Pt move? Will it change more or less if money demand is less sensitive to Yt? 5. Consider the basic Neoclassical model. Graphically analyze the eﬀects of: (a) An increase in Gt+1. (b) An increase in At+1. (c) A permanent increase in productivity (i.e. At and At+1 increase by the same amount). In each case. In each case, clearly describe how each endogenous variable changes. 6. Consider two diﬀerent versions of the basic neoclassical model. In one, the marginal propensity to consume (MPC) is relatively large, in the other the MPC is relatively small. 446 (a) Show how a higher or lower value of the MPC aﬀects the slope of the IS curve. (b) Suppose that there is an increase in ft. Show graphically how this impacts equilibrium rt in the two cases considered in this problem – one in which the MPC is relatively large, and one in which the MPC is relatively small. 7. [Excel Problem] Suppose that we have a neoclassical model. This problem will give speciﬁc functional forms for the equations underlying the model. Begin with the supply side. Suppose that labor demand supply are given by: Nt = a1wt − a2θt Nt = −b1wt + b2At + b3Kt (19.21) (19.22) (19.21) is the labor supply curve and (19.22) is labor demand. a1, a2, and b1 − b3 are positive parameters. (a) Use (19.21)-(19.22) to solve for expressions for Nt and wt as a function of parameters and exogenous variables. (b) Suppose that a1 = 1, a2 = 0.4, b1 = 2, b2 = 0.5, and b3 = 0.3. Suppose further that θt = 3, At = 1, and Kt = 20. Create an Excel ﬁle to solve for numerical values of Nt and wt using your answer from the previous part. (c) Suppose that the production function is Yt = AtK
α. Suppose that α = 1/3. Use your answer from the previous parts, along with the given values of exogenous variables and parameters, to solve for Yt. t N 1−α t Now let us turn to the demand side. Suppose that the consumption and investment demand functions are: Ct = c1(Yt − Gt) + c2(Yt+1 − Gt+1) − c3rt It = −d1rt + d2At+1 + d3Kt (19.23) (19.24) The aggregate resource constraint is Yt = Ct + It + Gt. (d) Use the aggregate resource constraint, plus (19.23)-(19.24), to derive an expression for rt as a function of Yt and other variables (for the purposes of this exercises, treat Yt+1 as exogenous). In other words, derive an expression for the IS curve. 447 (e) Suppose that c1 = 0.5, c2 = 0.4, c3 = 1, d1 = 20, d2 = 0.5, and d3 = 0.1. Suppose further that At+1 = 1, Yt+1 = 1.2, Gt = 0.2, and Gt+1 = 0.2. Given your answer for the value of Yt above, your expression for the IS curve, and these parameter values to solve for numeric values of rt, Ct, and It. (f) Suppose that At increases from 1 to 1.2. Solve for new numeric values of Yt, Nt, wt, rt, Ct, and It. Do these move in the same direction predicted by our graphical analysis? (g) Set At back to 1. Now suppose that Gt increases from 0.2 to 0.3. Solve for numerical values of the endogenous variables in your Excel ﬁle. Do these variables change in the way predicted by our graphical analysis? 448 Chapter 20 Taking the Neoclassical Model to the Data In Chapter 19, we analyzed how changes in diﬀerent exogenous variables would impact the endogenous variables of the neoclassical model. This Chapter, we seek to investigate whether or not the basic neoclassical model can produce movements in endogenous variables that look like what we observe in the data. To the extent to which the model can do this, which exogenous driving force must be
the main driver of the business cycle? Is there any model-free evidence to support this mechanism? These are the questions we take up in this Chapter. 20.1 Measuring the Business Cycle When economists talk about the “business cycle” they are referring to ﬂuctuations in real GDP (or other aggregate quantities) about some measure of trend. As documented in Part II, the deﬁning characteristic of real GDP is that it trends up. When moving away from the long run, we want to focus on movements in real GDP and other aggregate variables about the long run trend. As such, it is necessary to ﬁrst remove a trend from the observed data. Formally, suppose that a series can be decomposed into a “trend” component, which we demarcate with a superscript τ, and a cyclical component, which we denote with a superscript c. Suppose that the series in question is log real GDP. The decomposition of real GDP into its trend and cyclical component is given by (20.1) below: ln Yt = ln Y τ t + ln Y c t (20.1) ln Y c t = ln Yt − ln Y τ Given a time series, ln Yt, our objective is to ﬁrst come up with a time series of the trend component, ln Y τ t. Once we have this, the cyclical component is simply deﬁned as the residual, t. In principle, there are many ways in which one might remove a i.e. trend from a trending time series. The most obvious way to do this is to ﬁt a straight line through the series. The resulting straight line would be the “trend” component while the deviations of the actual series from trend would be the cyclical component. Another way to come up with a measure of the trend component would be to take a moving average. In 449 = average (Yt−4, Yt−3, Yt−2, Yt−1, Yt, Yt+1, Yt+2, Yt+3, Yt+4). particular, one could deﬁne the trend component at a particular point in time as the average realization of the actual series in a “window” around that point in time. For example, if the data are quarterly, a
two-sided one year moving average measure of the trend would be ln Y τ t As is common in academic work, we will measure the trend using the Hodrick-Prescott (HP) Filter. The HP ﬁlter picks out the trend component to minimize the volatility of the cyclical component, subject to a penalty for the trend component itself moving around too much. The HP ﬁlter is very similar to a two-sided moving average ﬁlter. In Figure 20.1 below, we plot the time series of the cyclical component of real GDP after removing the HP trend from the series. The shaded gray regions are recessions as deﬁned by the National Bureau of Economic Research (NBER). For more on recession dates, see here. The cyclical component of output rises and falls. It tends to fall and be low during periods identiﬁed by the NBER as recessions. Figure 20.1: Cyclical Component of Real GDP In modern macroeconomic research, one typically studies the “business cycle” by looking at second moments (i.e. standard deviations and correlations) of aggregate time series. Second moments of all series are frequently compared to output. One typically looks at standard deviations of diﬀerent series relative to output as measures of relative volatilities of series. For example, in the data, investment is signiﬁcantly more volatile than output, which is in turn more volatile than consumption. Correlations between diﬀerent series and output are taken to be measures of cyclicality. If a series is positively correlated with output, we say that series is procyclical. This means that when output is above trend, that series tends to also be above its trend (and vice-versa). If the series is negatively correlated with output, we 450 -.05-.04-.03-.02-.01.00.01.02.03.0455606570758085909500051015HP Filtered Component of Real GDP say it is countercylical. If it is roughly uncorrelated with output, we say that it is acyclical. Because the model with which we have been working is qualitative in nature, it is not possible to focus on relative volatilities of series. Instead, we will focus on cyclicalities of diﬀerent series, by which we simply mean the correlation coeﬃcient between
the cyclical component of a series with the cyclical component of output. The ﬁrst inner column of Table 20.1 below shows correlations between the cyclical components of diﬀerent aggregate times series with output. The six variables on which we focus are aggregate consumption, investment, labor input, the real wage, the real interest rate, and the price level. These correspond to the key endogenous variables (other than output) in our model. Consumption corresponds to total consumption expenditures and investment to gross private ﬁxed investment. These series, along with the real GDP series, are available from the BEA. The total labor input series is total hours worked in the non-farm business sector, available here. The real wage series is real compensation in the non-farm business sector, available here. The real interest rate is constructed using the Fisher relationship. We use the Federal Funds Rate as the nominal interest rate, and use next-period realized inﬂation as the measure of expected inﬂation to compute the real interest rate. The price level is the GDP price deﬂator, also available from the BEA. Table 20.1: Correlations Among Variables in the Data and in the Model Variable Corr w/ Yt in Data Corr conditional on At Corr conditional on θt Ct It Nt wt rt Pt 0.88 0.91 0.87 0.20 0.10 -0.46 + + + - + + + + - We see that consumption, investment, and labor hours are strongly positively correlated with output – these correlations are all above 0.85. This means that when output is high (low) relative to trend, these other series are on average also high (low) relative to trend. The real wage is procyclical, with a positive correlation with GDP of 0.20. This correlation is substantially lower than the cyclicalities of consumption, investment, and hours. The real interest rate is essentially acyclical – its correlation with output is about 0.10. Depending on how the real interest rate is measured (i.e. which nominal interest rate to use, or how to measure expected inﬂation), this correlation could be closer to zero or mildly negative. Regardless of construction, the real interest rate is never strongly cyclical in one direction or another. The price level is countercyclical. 451 There are good reasons to think that the observed cyclicality of the
real wage understates the true cyclicality of real wages in the real world. This is due to what is known as the “composition bias.” The aggregate wage series used to measure the aggregate real wage is essentially a measure of the average wage paid to workers. If the real world featured one type of worker (like our simple model does), this wouldn’t be a problem. But in the real world workers are paid substantially diﬀerent wages. It is an empirical fact that employment ﬂuctuations over the business cycle tend to be relatively concentrated among lower wage workers. If job loss during a recession tends to be concentrated among lower wage workers, even if every worker’s individual wage is unchanged the average wage will tend to rise due to the composition of the workforce shifting from low to high wage workers. This will tend to make the real wage look high when output is low (i.e. countercylical). The reverse would be true in an expansion. Solon, Barsky, and Parker (1994) study the importance of this so-called composition bias for the cyclicality of the aggregate real wage and ﬁnd that is quantitatively important. In particular, the correlation of a composition-corrected real wage series with aggregate output is likely substantially larger than the 0.20 shown in the table above. 20.2 Can the Neoclassical Model Match Business Cycle Facts? Having now established some basic facts concerning business cycle correlations in the data, we now want to take our analysis a step further. We ask the following question: can the basic neoclassical model qualitatively match the correlations documented in Table 20.1? If so, which exogenous variable could be responsible for these co-movements? Since the neoclassical model features a vertical Y s curve, the only exogenous variables which can generate a business cycle (i.e. changes in output) are At (productivity) and θt (labor supply). Changes in variables which impact the IS curve (Gt, Gt+1, and At+1) only aﬀect the composition of output, not output itself, and are therefore not candidates to explain ﬂuctuations in output within the context of the neoclassical model. When we move to the short run in Part V, we will extend our analysis into a framework in which changes in these variables can impact output, but in the neoclassical model they cannot. The second and
third inner columns of Table 20.1 present the qualitative correlations among diﬀerent variables with output in the neoclassical model conditional on changes in At and θt. A + sign indicates that the variable in the relevant row co-moves positively with output (i.e. increases when output increases, and decreases when output decreases). A − sign indicates that the variable in question co-moves negatively with output. An increase (decrease) in At causes Yt to increase (decrease), along with increases (decreases) in Ct, It, Nt and wt. This means that, conditional on a change in At, these variables 452 co-move positively with output, hence the + signs in the relevant parts of the table. In the model, an increase (decrease) in At causes rt to decline (increase) and Pt to decline (increase), so these variables co-move negatively with output, hence the − signs. Focus next on the co-movements implied by changes in θt. An increase (decrease) in θt causes output to decline (increase). Along with output, consumption, investment, and labor input all decrease (increase), hence these series co-move positively with output. Diﬀerently than conditional on changes in At, changes in θt cause the real wage to co-move negatively with output, hence the − sign in the table. The real interest rate increases when θt increases, and so co-moves negatively with output. So too does the price level. Compared to the data, changes in At or θt can generate (at least qualitatively) the correct co-movements with output for consumption, investment, hours, and the price level. For both At and θt, the implied correlation between the real interest rate and output is oﬀ relative to the data. Changes in At generate positive co-movement between the real wage and output, consistent with what is observed in the data. Diﬀerently than the data, changes in θt generate negative co-movement between the real wage and output. To the extent to which the so-called composition bias is important, the implied countercyclicality of the real wage conditional on shocks to θt is problematic. We can conclude that the neoclassical model can best match observed business cycle correlations when it is primarily driven by changes
in At.1 There is a still a problem in the sense that the model implies that increases (decreases) in At ought to trigger a decrease (increase) in the real interest rate, implying negative co-movement between the interest rate and output, whereas in the data the real interest rate is approximately acyclical. This is fairly easy to reconcile within the context of the model. In our previous analysis, we have focused on a change in At, holding At+1 ﬁxed. In reality, changes in productivity are likely to be quite persistent in the sense that an increase in At likely means that At+1 will increase as well. In Figure 20.2, we consider the eﬀects of a simultaneous increase in At and At+1 in the neoclassical model. The increase in At shifts the Y s curve out, which on its own would result in an increase in Yt and a reduction in rt. The increase in At+1 shifts the IS curve out, which on its own would have no impact on Yt but would result in rt increasing. In other words, At and At+1 have competing eﬀects on rt. Depending on how much At+1 increases relative to At, as well as how sensitive investment is to At+1, the real interest rate could on net fall (as it does when just At increases), rise (as it does when just At+1 increases), or do nothing at all (as we have shown here). Note that in a hypothetical situation in which both At and At+1 increase, leaving the real interest rate 1Note that this is not meant to suggest that changes in At are the only source of business cycle ﬂuctuations in the model. At and θt (along with exogenous variables which aﬀect the position of the IS curve) could all be changing simultaneously. We simply mean that At must be the predominant source of exogenous changes for the model to best ﬁt the data, at least on the dimensions which we are studying. 453 unaﬀected, the changes in Yt, Nt and wt would be identical to the case where just At changes in isolation. Even with no decline in rt, both Ct and It would increase – Ct because of the higher Yt and anticipation of higher Yt+1 due to the anticipated increase in At+1, and It due to the anticipation
of higher At+1. In other words, with a persistent change in productivity, the neoclassical model can qualitatively generate the co-movements we observe in the data – output, consumption, investment, labor hours, and the real wage all moving together, with the real interest rate roughly unchanged and the price level moving opposite output. 454 Figure 20.2: Increase in At and At+1 20.3 Is there Evidence that At Moves Around in the Data? We have established that the neoclassical model can generate movements in output and other endogenous variables which qualitatively resemble what we observe in the data when the model is predominantly driven by shocks to productivity – i.e. exogenous changes in 455 𝑤𝑤𝑡𝑡 𝑟𝑟𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑁𝑁𝑡𝑡 𝑁𝑁𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑 𝑌𝑌𝑠𝑠 𝐼𝐼𝐼𝐼 𝑤𝑤0,𝑡𝑡 𝑟𝑟0,𝑡𝑡=𝑟𝑟1,𝑡𝑡 𝑌𝑌0,𝑡𝑡 𝑁𝑁0,𝑡𝑡 𝑁𝑁𝑑𝑑(𝑤𝑤𝑡𝑡,𝐴𝐴0,𝑡𝑡,𝐾𝐾𝑡𝑡) 𝑁𝑁𝑠𝑠(𝑤𝑤𝑡𝑡,𝜃𝜃𝑡𝑡) 𝐴𝐴0,𝑡𝑡𝐹𝐹(𝐾𝐾𝑡𝑡
,𝑁𝑁𝑡𝑡) 𝑌𝑌𝑡𝑡=𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝐶𝐶𝑑𝑑�𝑌𝑌𝑡𝑡−𝐺𝐺𝑡𝑡,𝑌𝑌𝑡𝑡+1−𝐺𝐺𝑡𝑡+1,𝑟𝑟0,𝑡𝑡�+𝐼𝐼𝑑𝑑�𝑟𝑟0,𝑡𝑡,𝐴𝐴𝑡𝑡+1,𝐾𝐾𝑡𝑡�+𝐺𝐺𝑡𝑡 𝐴𝐴1,𝑡𝑡𝐹𝐹(𝐾𝐾𝑡𝑡,𝑁𝑁𝑡𝑡) 𝑌𝑌𝑠𝑠′ 𝑁𝑁𝑑𝑑(𝑤𝑤𝑡𝑡,𝐴𝐴1,𝑡𝑡,𝐾𝐾𝑡𝑡) 𝑌𝑌1,𝑡𝑡 𝑁𝑁1,𝑡𝑡 𝑤𝑤1,𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝐶𝐶𝑑𝑑�𝑌𝑌𝑡𝑡−𝐺𝐺𝑡𝑡,𝑌𝑌𝑡𝑡+1−𝐺𝐺𝑡𝑡+1,𝑟𝑟0,𝑡𝑡
�+𝐼𝐼𝑑𝑑�𝑟𝑟0,𝑡𝑡,𝐴𝐴0,𝑡𝑡+1,𝑓𝑓𝑡𝑡,𝐾𝐾𝑡𝑡�+𝐺𝐺𝑡𝑡=𝐶𝐶𝑑𝑑�𝑌𝑌𝑡𝑡−𝐺𝐺𝑡𝑡,𝑌𝑌𝑡𝑡+1−𝐺𝐺𝑡𝑡+1,𝑟𝑟1,𝑡𝑡�+𝐼𝐼𝑑𝑑�𝑟𝑟1,𝑡𝑡,𝐴𝐴0,𝑡𝑡+1,𝐾𝐾𝑡𝑡�+𝐺𝐺𝑡𝑡 𝐼𝐼𝐼𝐼′ At. Is there any evidence that At in fact moves around much in the data, and to the extent to which it does, are those movements consistent with what the model would imply output should be doing? One can come up with an empirical measure of At without reference to all of the model if one is willing to make an assumption about the aggregate production function. As in the Solow model, assume that the production function is Cobb-Douglas: Take natural logs of (20.2) and re-arrange: Yt = AtK α t N 1−α t ln At = ln Yt − α ln Kt − (1 − α) ln Nt (20.2) (20.3) If one observes time series on Yt, Kt, and Nt (which, in principle, are available from the national economic accounts), and if one is willing to take a stand on a value of α, one can back out a measure of ln At as the part of output that cannot be explained given observable capital and labor inputs. If factor markets are competitive, 1 − α should correspond to labor’s total share of income. In other words, if the real wage equals the marginal product
of labor, then for the Cobb-Douglas production function we ought to have: wt = (1 − α)AtK α t N −α t (20.4) (20.4) is nothing more than the condition wt = AtFN (Kt, Nt). One can multiply and divide the right hand side of (20.4) to get: Re-arranging terms in (20.5), one gets: wt = (1 − α) Yt Nt 1 − α = wtNt Yt (20.5) (20.6) (20.6) says that 1 − α ought to equal total payments to labor (wtNt) divided by total income (Yt). This is sometimes called “labor’s share” of income. In the data, labor’s share of income is approximately constant at around 2/3 from the end of World War 2 through about 2000. This implies a value of α = 1/3. Since 2000, labor’s share has been steadily declining, and is about 0.6 at present. Although this recent decline in labor’s share is quite interesting, we will ignore it and treat α as a constant equal to 1/3. Given this, as well as measurements on Yt, Kt, and Nt, we can use (20.3) to back out an empirical measure of At. This empirical measure of At is sometimes called “total factor productivity” (or TFP) since it is that part 456 of output which cannot be explained by the factors capital and labor. The empirical measure of At is sometimes also called a “Solow residual” after Bob Solow of Solow model fame. Figure 20.3 plots the cyclical components of real GDP (black line) along with the cyclical component of TFP (blue line). The shaded gray bars are recessions as dated by the NBER. One observes from the ﬁgure that TFP and GDP seem to co-move strongly. TFP tends to rise and fall at the same time output rises or falls. TFP declines and is low relative to trend in all identiﬁed recessions. The correlation between the cyclical component of the TFP series and the cyclical component of output is very high, at 0.78. It is also the case that the cyclical component of TFP is
quite persistent in the sense of being positively autocorrelated. This is consistent with productivity shocks being persistent (i.e. increases in At portend increases in At+1 in a way consistent with the analysis immediately above). Figure 20.3: Cyclical Components of Real GDP and TFP The visual evidence apparent in Figure 20.3 is often taken to be evidence in support of the neoclassical model. For the model to generate the qualitatively right co-movements among aggregate variables, it needs to be driven by persistent changes in productivity (by persistent we meanAt and At+1 increase or decrease together). We see this in the data – At moves around quite a bit, and is quite persistent in the sense of being highly autocorrelated. Furthermore, the increases and decrease in At we observe over time line up with the observed increases and decreases in output. In particular, recessions seem to be times when productivity is low, and expansions times when productivity is high. This seems to provide evidence in favor of the model. We should mention at this point that there are potentially important measurement issues with regard to TFP, some of which cast doubt on this apparent empirical support for the 457 -.05-.04-.03-.02-.01.00.01.02.03.0455606570758085909500051015GDPTFP neoclassical model. Will return to criticisms of TFP measurement in Chapter 22. 20.4 Summary • All data series consist of a trend component and a cyclical component. The cyclical component is how the series moves over the business cycle. The cyclical component is not invariant to how the trend component is computed. Most macroeconomists use an HP ﬁlter. • A series is procyclical if it is positively correlated with output. A series is countercyclical if it is negatively correlated with output. A series is acylical if it is uncorrelated with output. In the data, consumption, investment, real wages, and hours are procyclical. The price is countercyclical and the real interest rate is approximately acyclical. • No exogenous variable in isolation can induce the same correlations in the Neoclassical model as we see in the data. However, the Neoclassical model is consistent with all these comovements if business cycles are driven by persistent changes in productivity. • We can construct an empirical measure of productivity by subtracting output from its share-weighted inputs
. As in the Solow model, we call this diﬀerence ”Total Factor Productivity.” TFP is strongly procyclical and persistent. To the extant empirical TFP is a good measure of productivity, the Neoclassical model performs quite well in matching the data. Key Terms • Linear trend • Moving average • HP ﬁlter • Cyclicality • Composition bias • Total Factor Productivity Questions for Review 1. Rank the following series from most to least volatile: output, consumption, investment. 458 2. Describe the cyclicality of consumption, investment, hours, the real wage, and real interest rate. 3. Why might the true correlation of real wages with output be understated in the data? 4. Is there one exogenous variable in the Neoclassical model that can explain all the correlations in the data? If so, which one? If not, can any two shocks simultaneously explain the correlations? 5. How is the productivity series constructed in the data? Does it move positively or negatively with output? Exercises 1. [Excel Problem] Go to the Federal Reserve Bank of St. Louis FRED website. Download data on real GDP, real personal consumption expenditures, real gross private domestic investment, the GDP price deﬂator, total hours worked per capita in the non-farm business sector, and real average hourly compensation in the non-farm business sector. All series should be at a quarterly frequency. Download these data from 1947q1 through the most recent available date. Take the natural log of each series. (a) Isolate the cyclical component of each series by ﬁrst constructing a moving average trend measure of each series. In particular, deﬁne the trend component of a series is the two year, two-sided moving average of a series. This means that you will lose two years (eight quarters) worth of observations at the beginning and end of the sample. Concretely, your measure of trend real GDP in 1949q1 will be the average value of actual real GDP from 1947q1 (eight observations prior to 1949q1) through 1951q1 (eight observations subsequent to 1949q1. Compute this for every observation and for each series. Then deﬁne the cyclical component of a series as the actual value of the series minus its trend value. Produce a time series plot of the cyclical component of real GDP. Do observed declines in real
GDP align well with the NBER dates of recessions (which can be found here)? (b) Compute the standard deviations of the cyclical component of each series. Rank the series in terms of their volatilities. (c) Compute the correlations of the cyclical component of each series with the cyclical component of output. Do the signs of the correlations 459 roughly match up with what is presented in Table 20.1? (d) If the production function is Cobb-Douglas, then the real wage (which equals the marginal product of labor) ought to be proportional to the average product of labor (since with a Cobb-Douglas production function the marginal product and average product of each factor are proportional to one another). In particular: wt = (1 − α) Yt Nt (20.7) Yt is average labor productivity. Download data on this series from Nt the St. Louis Fed, which is called real output per hour of all persons. Compute the trend component of the log of this series like you did for the others, and then compute the cyclical component by subtracting the trend component from the actual series. Compute the correlation between this series and the empirical measure of wt (real average hourly compensation in the non-farm business sector). The theory predicts that this correlation ought to be 1. Is it? Is it positive? (e) Take your series on the log wage and log labor productivity (the levels of the series, not the trend or cyclical components) and compute ln wt − ). If the theory is correct, this series ought to be proportional to ln ( Yt Nt 1 − α, which is labor’s share of income (it won’t correspond to an actual numeric value of 1 − α since the units of the wage and productivity series are indexes). What does this plot look like? What can you conclude has been happening to 1 − α over time? 460 Chapter 21 Money, Inﬂation, and Interest Rates How is the quantity of money measured? What determines the average level of inﬂation in the medium run? What about expected inﬂation (which we have taken to be exogenous)? And what about the level of the nominal interest rate? Although money is neutral with respect to real variables in the neoclassical model, does this hold up in the data? In this Chapter, we use the building blocks of the neoclassical model to explore
these questions. 21.1 Measuring the Quantity of Money In Chapter 14, we deﬁned money as an asset which serves the functions of a medium of exchange, a store of value, and a unit of account. Most modern economies operate under a ﬁat money system, wherein the thing used as money has no intrinsic value and only has value because a government (by ﬁat) issues that thing and agents accept it in exchange for goods and services. In the United State, the dollar is the unit of money. In Europe, it is the Euro, and in Japan the Yen. How does one measure the quantity of money in an economy? This may seem like a silly question – wouldn’t one just count up the number of dollars (or euro, or Yen)? It turns out that this is not such an easy question to answer. Most of the dollars out there do not exist in any tangible form. While there is currency (physical representations of dollars), much of the money supply is electronic and therefore does not exist in any tangible way. Because these electronic entries serve as a store of value, a unit of account, and a medium of exchange, they are money as well. Indeed, many diﬀerent assets can be denominated in dollars and used in exchange, so measuring the money supply is not in fact so clear. One can think about the quantity of money as the dollar value of assets which serve the three functions deﬁned by money. Currency is one particular kind of asset. An “asset” is deﬁned as “property owned by a person or community, regarded as having value available to meet debts, commitments, or legacies” (this deﬁnition comes from a Google search of the word “asset”). Currency (a physical representation of money – i.e. a dollar bill or a quarter) is an asset. Another kind of asset is a demand deposit, which refers to the funds people hold in checking accounts (it is called a “demand deposit” because people can demand the funds 461 in their account be paid out in currency at any time). Checks, which are simply claims on demand deposits, are used all the time to transfer resources from buyers and sellers (a debit card is simply a paperless form of a check). Other forms of assets could serve the functions of money. For example, money market mutual funds are ﬁn
ancial instruments against which checks can often be written. Some savings accounts allow checks to be written against them, and in any event is relatively seamless to transfer money from a savings to a checking account. Because there are many assets (all denominated in units of money) which can be used in transactions in addition to currency, there are many diﬀerent ways to deﬁne the quantity of money. The most basic deﬁnition of the quantity of money is the currency in circulation. In 2016 in the United States, there were roughly 1.4 trillion dollars of currency in circulation. If you add in the total value of demand deposits (and other similar instruments) to the quantity of currency in circulation, the money supply would be about 3.2 trillion dollars. This means that there is close to 2 trillion more dollars in demand deposits than there is in currency. The next most basic deﬁnition of the money supply is called M1, and includes all currency in circulation plus demand deposits. We can continue going further, including other assets into a deﬁnition of the quantity of money. M2 is deﬁned as M1, plus money market mutual funds and savings deposits. Generally speaking, we can think about diﬀerent assets according to their liquidity, by which we mean the ease with which these assets can be used in exchange. By construction, currency is completely liquid as it is “legal tender for all debts public and private.” Demand deposits are not quite currency, but because funds can be converted to currency on demand, they are nearly as liquid as currency and can be used directly for most types of transactions. Hence, relative to currency, M1 includes currency plus a slightly less liquid asset (demand deposits). M2 includes M1, plus some other assets that are not quite as liquid as demand deposits (money market mutual funds and savings accounts). M3 is another measure of the money supply. In addition to M2 (which in turn includes M1, which itself includes currency), M3 includes institution money market funds (money market funds not held by individual investors) and short term repurchase agreements. Wikipedia has a decent entry on deﬁnitions of the money supply and how they are employed around the world. Figure 21.1 plots the time series of currency, M1, M2, and M3 for the United States over the period 1975-2005. The series are plotted in logs. One can
visually see that M1 is substantially bigger than currency in circulation – for most periods, M1 is about 1 log point higher than currency, which means M1 is about 100 percent bigger than currency, or double, which is consistent with the numbers presented above. M2, in turn, is about 1 log point (or more) bigger than M1 in most periods, so M2 is about 100 percent bigger than M1, or about double the size of M1. M3, in contrast, is not much larger than M2. Most economists use M1 462 or M2 as their preferred measure of the quantity of money.1 For most of this book, when referring to the quantity of money in the United States, we will be referring to M2. Figure 21.1: Diﬀerent Measures of the Money Supply 21.1.1 How is the Money Supply Set? Who sets the money supply? How is it set? While these questions seem rather trivial, in reality they are pretty complicated. While the government is a monopoly supplier of currency, other assets which serve as money are privately created. In your principles class you might have studied fractional reserve banking and the money creation process. We will not bore you with those details here, giving only a highly condensed version. If one is interested in more details, Wikipedia has a good entry on money creation. It is also discussed in more detail later in Chapter 32. Modern economies have central banks, like the US Federal Reserve. In addition to regulating banks and serving as a “lender of last resort” in periods of high demand for liquidity (see the discussion in Chapter 33), the central bank can inﬂuence (though not completely control) the supply of money. The central bank can directly set the quantity of currency in circulation. Call this CUt. The central bank can also set the quantity of reserves 1Indeed, the Federal Reserve discontinued measuring M3 in 2006. 463 45678910767880828486889092949698000204CurrencyM1M2M3 in the banking system. Reserves are balances held by banks which have not been lent out to others. These reserves are either kept as “vault cash” (i.e. currency sitting in the bank as opposed to in circulation), or on account with the central bank, where a bank’s reserve balance with the central bank is isomorphic to an individual’s checking account with a bank.
In a 100 percent reserve banking system, bank loans must be backed by reserves of an equal amount. So if a bank has 500 dollars in deposits, it must hold 500 dollars in reserves – i.e. it has to keep the entire value of the deposits as vault cash or on account with the central bank. Modern economies feature what are called “fractional reserve” banking systems. Banks make money by not holding reserves. The entire business of banking involves accepting deposits and lending the funds out for other uses (as is discussed in more detail in Chapter 31). For this reason, a bank would never choose to hold the value of its deposits in reserves. Rather, a bank would want to keep only a small amount of reserves on hand to be able to meet withdrawal demands, lending the rest out to households and businesses. Central banks often require banks to hold a certain amount of deposits in reserves. Economists refer to this amount as the require reserve ratio. Reserves not required to be held by a central bank, or so-called “excess reserves,” can be lent out to households and ﬁrms. In lending these reserve out, a bank creates deposits. While banks can lend out excess reserves, in the process creating deposits, they are not required to do so. The central bank can inﬂuence the amount of deposits by adjusting reserves in the banking system. If the central bank takes actions which generate excess reserves (either by lowering the required reserve ratio or purchasing other assets, such as government debt, held by banks in exchange for reserves), then this can result in an expansion of deposits, and hence in the money supply. These issues are discussed in much more detail in Chapter 32, but we provide a short and stylized description here. Denote the total quantity (both excess and reserved) reserve in the banking system as REt. Deﬁne the term M Bt, for monetary base, as the sum of currency in circulation plus reserves: Deﬁne the monetary base, M Bt, as the sum of currency plus reserves: M Bt = CUt + REt (21.1) A central bank can directly set M Bt by either creating bank reserves through asset purchases or by printing more currency. But the money supply, as noted above, includes more than just currency – it also includes demand deposits, and potentially other forms of ﬁnancial assets depending on which measure of money one prefers. While a central bank can
directly set the monetary base, it can only indirectly set the money supply. This is because, as noted above, commercial banks can themselves create money by issuing loans, thereby 464 creating deposits. Inﬂuencing the quantity of reserves in the banking system will impact the quantity of loans made by banks, but only to the extent to which banks choose not to hold more reserves than required by law. Figure 21.2 plots the time series of the natural logs of the monetary base (blue line) and the money supply (as measured by M2) for the United States. Visually, we can see that M2 is substantially higher than the monetary base. For the most part, the monetary base and the money supply move together. One does observe some anomalous behavior post-2008, when the monetary base increased substantially without much noticeable eﬀect on the money supply. We will return more to this in Chapter 37. Figure 21.2: M2 and the Monetary Base We can think about the money supply as equaling a multiple of the monetary base. In particular: Mt = mmtM Bt (21.2) Here, Mt is the money supply, and mmt is what is called the money multiplier. In the simplest possible model in which banks hold no excess reserves and households hold no currency (see Chapter 32), the money multiplier is one divided by the required reserve ratio. So, if the central bank requires commercial banks to hold 20 percent of total deposits in the 465 345678910606570758085909500051015M2Monetary BaseM2 and Monetary Base form of reserves, the money multiplier would be 5 – the money supply would be ﬁve times larger than the monetary base. This expression for the money multiplier assumes that banks do not hold excess reserves and that individuals do not withdraw deposits for cash. Figure 21.3 plots the implied money multiplier for the US over time (using M2 as the measure of the quantity of money). Figure 21.3: M2 Divided by the Monetary Base One can observe that the money multiplier is not constant. It consistently rose from 1960 through the 1980s. The implied money multiplier was very nearly constant from 1990 through the middle of the 2000s. The money multiplier then fell drastically post-2008 and has not recovered. The real world phenomenon driving this behavior is that commercial banks have been holding excess reserves – they have not been lending out the maximum amount of reserves. As noted above, the central bank can
directly control the monetary base, M Bt. It can only inﬂuence mmt through its control of the required reserve ratio, but otherwise mmt is out of the control of the central bank. It is therefore not particularly accurate to think of the central bank as having control over the supply of money. However, we will hereafter ignore this fact. We will therefore think of Mt as being an exogenous variable set by a central bank. But in reality, one must keep in mind that the central bank can really only directly control 466 24681012606570758085909500051015Monetary Supply divided by Monetary Base the monetary base, M Bt, and hence indirectly the money supply, Mt. For more on the money creation process and the money multiplier, the interested reader is referred to Chapter 32. 21.2 Money, the Price Level, and Inﬂation We are treating the supply of money, Mt, as being set exogenously by a central bank (subject to the caveats above). The demand for money is determined by actors in the economy. The price level, the inﬂation rate, the rate of expected inﬂation, and the nominal interest rate are in turn all determined by supply equaling demand in the market for money. Recall our generic speciﬁcation for the demand for money from the neoclassical model: Mt Pt = M d(it, Yt) (21.3) We have assumed that the demand for money is proportional to the price level, decreasing in the nominal interest rate (which can be written in terms of the real interest rate via rt = it − πe t+1, where we have taken expected inﬂation to be exogenous), and increasing in total output, Yt. Let’s assume a particular functional form for this money demand speciﬁcation, given by: Mt Pt = ψti−b1 t Yt (21.4) In (21.4), b1 is assumed to be a constant parameter. Hence, we are assuming that the demand for real balances is decreasing in the nominal interest rate and proportional to total output. We have introduced a new term, ψt, which we take to be exogenous. We can think about ψt as measuring preferences for holding money – the bigger is ψt, the more money people would like to hold. We will return to this variable more below
. In terms of a micro-founded money demand speciﬁcation, we can think about ψt as being a parameter which scales the utility a household receives from holding money. This money demand function can be written in terms of the real interest rate via: Mt Pt = ψt(rt + πe t+1 )−b1Yt (21.5) In the neoclassical model, the classical dichotomy holds, and Yt and rt are determined independently of Mt or other nominal variables. We are treating (for now) the expected inﬂation rate as exogenous. This mean that the right hand side of (21.5) is determined completely independently from the left hand side. As such, what this tells us that changes in Mt will result in proportional changes in Pt. In other words, conditional on rt, Yt, and πe t+1, what determines the price level is the quantity of money, Mt. 467 What about the rate of change in the price level (i.e. the inﬂation rate)? Let’s take natural logs of (21.4): ln Mt − ln Pt = ln ψt − b1 ln it + ln Yt (21.6) This equation must hold at every point in time. Subtract oﬀ the same expression dated in period t − 1 from (21.6) and re-arrange terms a bit to get: ln Mt − ln Mt−1 = ln Pt − ln Pt−1 + ln ψt − ln ψt−1 − b1 (ln it − ln it−1) + (ln Yt − ln Yt−1) (21.7) Recall that the ﬁrst diﬀerence across time of the natural log of a variable is approximately equal to the growth rate of that variable. If we are willing to assume that the nominal interest rate and the new exogenous variable ψt are roughly constant across time, we can write (21.7) as: gM t = πt + gY t (21.8) In other words, (21.8) says that the growth rate of the money supply equals the sum of the inﬂation rate (the growth rate of the price level) and the growth rate of output. This expression can
be re-arranged to yield: πt = gM t − gY t (21.9) (21.9) says that the inﬂation rate equals the excess growth rate of the money supply over output (i.e. the diﬀerence between the growth rates of the money supply and output). So what determines the inﬂation rate? According to (21.9) and the assumptions going into it, inﬂation is caused by excessive money growth relative to output growth. Over a suﬃciently long period of time, output grows at an approximately constant rate (recall the stylized facts from Part II). Taken literally, then, (21.9) implies that money growth ought to translate one-for-one into the inﬂation rate. This would be consistent with the famous quote by Nobel prize winner Milton Friedman, who once said that “Inﬂation is everywhere and always a monetary phenomenon.” Does this implication hold up in the data? Figure 21.4 is a scatter plot of the (annualized) inﬂation rate (as measured by the GDP price deﬂator) and the (annualized) growth rate of the M2 money stock. Each circle represents a combination of inﬂation and money growth observed at a point in time. The straight line is a best-ﬁtting regression line. One can observe that the two series move together, but the relationship is relatively weak. The correlation between the two series is 0.22 – positive, but 468 not overwhelmingly so. Figure 21.4: Scatter Plot: Money Growth and Inﬂation Figure 21.4 measures the inﬂation rate and the growth rate of the money via quarter-overquarter changes in the M2 stock of money and the GDP price deﬂator (then expressed at annualized rates). Nominal interest rates are clearly not constant quarter-to-quarter, nor is there reason to think that ψt would necessarily be constant. Further, it could be that, in the short run, changes in the money supply impact real output (as we will see in Part V). For these reasons, looking at the correlation between money growth and the inﬂation rate at a quarterly frequency may be asking too much of the theory. Figure 21.5 plots the time series of “smoot
hed” money growth and inﬂation against time. These series are smoothed to remove some higher frequency (i.e. quarter-to-quarter) variation. The smoothed series ought to instead pick up lower frequency variation (i.e. changes in the series over the course of several years). Our smoothing technique is to look at the HP ﬁlter trend component of each series. The HP ﬁlter trend is essentially a two-sided moving average. In other words, the trend (or smoothed) value at a point in time is the average value of observations in a window around that point. The details of this smoothing procedure are unimportant. We can observe in the ﬁgure that the smoothed components of money growth and inﬂation do seem to move together. In particular, the correlation between these series is 0.66, which is substantially higher than the correlation between quarter-over-quarter growth rates of the two series of 0.22 mentioned above. 469 -2024681012-404812162024Money GrowthInflationScatter Plot of Money Growth and Inflation Figure 21.5: Smoothed Money Growth and Inﬂation While Figure 21.5 seems to indicate that money growth and inﬂation seem to move together over longer periods of time, there is an interesting diﬀerence in the Figure pre- and post-1990. In particular, from 1960-1990, the plots of smoothed money growth and inﬂation are very similar. Indeed, the correlation between the two series over this sample is 0.79, which is substantially higher than over the full sample (0.66). After 1990, the series do not seem to move together nearly as much. While the smoothed inﬂation rate fell throughout the 1990s, money growth actually picked up. Further, while money growth has been increasing since 2005, the smoothed inﬂation rate has been falling. If one computes the correlation between smoothed money growth and inﬂation since 1990, it actually comes out to be negative (-0.51). What gives? Let’s re-write equation (21.4) by deﬁning a term V −1 t = ψti−b1 t. We will call this term Vt the “velocity” of money. The money demand speci�
��cation can then be written: Re-arranging terms: Mt Pt = V −1 t Yt MtVt = PtYt (21.10) (21.11) Expression (21.11) is often times called the “quantity equation.” In words, it says that money times velocity equals the price level times real output (which is nominal GDP). There is a natural economic interpretation of the velocity term in (21.11). Since PtYt is nominal 470 123456789100123456789606570758085909500051015Smoothed Money GrowthSmoothed InflationMoney GrowthInflation GDP, if money must be used for all transactions, then PtYt equals the average number of Mt times that each unit of money is used (i.e. what is called velocity). The quantity equation, (21.11), can be deﬁned independently of any economic theory. Given observed values of nominal GDP and the stock of money, one can then use this equation to determine Vt. Figure 21.6 below plots velocity as implicitly deﬁned by (21.11) for the US since 1960: Figure 21.6: Velocity Figure 21.6 is quite interesting, particularly in light of Figure 21.5. From 1960-1990, we can see that velocity is approximately constant. Constant velocity in conjunction with the quantity equation is a central tenet of a school of thought called monetarism (see here for more). We can see from Figure 21.6 that the assumption of constant velocity clearly breaks down around 1990. Velocity increases during the ﬁrst part of the 1990s and has been steadily declining ever since. As noted above, (21.11) can be deﬁned independently of any economic theory – one can use it to infer Vt from the data, given data on PtYt and Mt (which is what we do in Figure 21.6). But (21.11) can also be motivated from economic theory given a speciﬁcation of money demand. In particular, using the money demand speciﬁcation with which we have been working, Vt can be written: Vt = ψ−1 t ib1 t (21.12) From the perspective of our theory, the velocity of money could not be constant for two reasons – changes in ψt and changes in it. Increases in it increase the velocity of
money, while 471 1.41.51.61.71.81.92.02.12.22.3606570758085909500051015Velocity of Money increases in ψt reduce it. Figure 21.7 below plots the time series of the (annualized) eﬀective Federal Funds Rate over the period 1960-2016. Figure 21.7: Nominal Federal Funds Rate Visually, it appears as though the nominal interest rate and velocity are positively correlated, consistent with our theory. That said, it is diﬃcult to square the near constancy of measured velocity from 1960-1990 with the highly volatile Federal Funds rate over that same period. Over the entire sample, the correlation between the Funds rate and velocity is 0.20, which is positively but not particularly strong. Since 1990, however, the correlation between the two series is much larger, at 0.74. All this said, it is clear that changes in it alone cannot explain all of the observed behavior in velocity. From the perspective of our theory, the decline in velocity since 1990 must also be due to increases in ψt, which as noted above reﬂects a household’s desire to hold money. In other words, since 1990, there has been an increasing demand for money, with this increasing demand for money particularly stark since the onset of the Great Recession (about 2008 or so). What real-world phenomena can explain this? Part of this is changes in transactions technology. Holding money used to be more costly in the sense that it was diﬃcult to transfer cash into interest-bearing assets. Now this is much easier due to online banking, etc.. Part of the increase in the demand for money since the onset of the Great Recession is likely driven by uncertainty about the future and ﬁnancial turmoil. 472 048121620606570758085909500051015Federal Funds Rate 21.3 Inﬂation and Nominal Interest Rates The previous section established that, to the extent to which velocity is constant (which is aﬀected by nominal interest rates and the desire to hold money), in the medium run inﬂation is caused by excessive money growth over output growth. In this section, we explore the question of what determines the level of nominal interest rates in the medium run. In our most basic model, the consumption Euler equation for an optimizing household can be written: u′
(Ct) = βu′(Ct+1)(1 + rt) (21.13) Let’s assume a speciﬁc functional form for ﬂow utility, the natural log. This means that (21.13) can be written: Ct+1 Ct If we take natural logs of this, we get: = β(1 + rt) (21.14) ln Ct+1 − ln Ct = ln β + rt In (21.15), we have used the approximation that ln(1 + rt) ≈ rt. The log ﬁrst diﬀerence of consumption across time is approximately the growth rate of consumption, which over suﬃciently long periods of time is the same as the growth rate of output. Call this gY t+1. Then we can expression rt as: (21.15) rt = gY t+1 − ln β (21.16) Since β < 1, ln β < 0, so if the economy has a positive growth rate rt > 0. (21.16) tells us that, over suﬃciently long time horizons, the real interest rate depends on the growth rate of output (it is higher the faster output grows) and how impatient households are (the smaller β is, the higher will be the real interest rate). The real interest rate in the medium run is independent of any nominal factors. Recall that the Fisher relationship says that rt = it − πe t+1. Plug this into (21.16) to get: it = πe t+1 + gY t+1 − ln β (21.17) Although we have taken expected inﬂation, πe t+1, to be an exogenous variable, over long periods of time we might expect expected inﬂation to equal realized inﬂation (at least in an average sense). This just means that household expectations of inﬂation are correct on 473 average, not each period. If we replace expected inﬂation with realized inﬂation, (21.17) can be written: it = πt + gY t+1 − ln β (21.18) To the extent to which output growth is fairly constant across time (which is one of the stylized growth
facts), and that β is roughly constant over time, (21.18) implies that the level of the nominal interest rate ought to be determined by the inﬂation rate (which is in turn determined by money growth relative to output growth). In US data for the period 1960-2016, the correlation between the Federal Funds rate and the inﬂation rate (as measured by percentage changes in the GDP price deﬂator) is 0.70, which is consistent with (21.18). Figure 21.8 plots smoothed time series of inﬂation and interest rates for the US over this period. To get the smoothed series, we use the HP trend component of each series, similarly to what we did for money growth and inﬂation in Figure 21.5. Figure 21.8: Smoothed Inﬂation and Interest Rates Visually, we can see that these series move together quite strongly. The correlation coeﬃcient between the smoothed interest rate and inﬂation rate series is 0.76, which is a bit higher than the correlation between the actual series without any smoothing (0.70). From this, we can deduce that the primary determinant of the level of nominal interest rates over a suﬃciently long period of time is the inﬂation rate (which is in turn determined by money growth, among other factors). An interesting current debate among academics (and policymakers) concerns the connec- 474 -202468101214012345678606570758085909500051015Smoothed Fed Funds RateSmoothed Inflation RateFed Funds RateInflation tion between inﬂation rates and interest rates. As we will see in Part V, standard Keynesian analysis predicts that monetary expansions result in lower interest rates and higher inﬂation (perhaps with some lag). This is the conventional stabilization view among most people – lowering interest rates increases demand, which puts upward pressure on inﬂation. An alternative viewpoint, deemed “Neo-Fisherianism” by some, reaches the reverse conclusion. It holds that raising inﬂation rates requires raising interest rates. The Neo-Fisherian viewpoint is based on the logic laid out in this chapter – if the real interest rate is independent of monetary factors, interest rates and inﬂation ought to move together. This is certainly what
one sees in the data, particularly over longer time horizons. In the very short run, when, as we will see, monetary policy can aﬀect real variables (including the real interest rate), the Neo-Fisherian result may not hold, and lower interest rates may result in higher inﬂation. In some respect, the debate between Neo-Fisherians and other economists centers on time horizons – in the medium run, the Neo-Fisherian view ought to hold (and does in the data), while in the short run monetary non-neutrality may result in it not holding. 21.4 The Money Supply and Real Variables The basic neoclassical model makes the stark prediction that money is neutral with respect to real variables – changes in the quantity of money do not impact real GDP or other variables. Does this hold up in the data? Figure 21.9 plots the cyclical components (obtained from removing an HP trend) of the M2 money supply and real GDP. Visually, it appears as though the money supply and output are positively correlated. For the full sample, the series are in fact positively correlated, albeit relatively weakly. In particular, the correlation between the cyclical components of M2 and GDP is about 0.20. 475 Figure 21.9: Cylical Components of Real GDP and the Money Supply Does the positive correlation (however mild) between the money supply and real GDP indicate that changes in the money supply cause changes in real GDP? Not necessarily. Remember that correlation does not imply causation. It could be that the central bank chooses to increase the money supply whenever real GDP increases, for example. This could result in a positive correlation between the series, but would not imply that changes in the money supply cause real GDP to change. A slightly better, though still imperfect, way to assess whether changes in the money supply cause changes in real GDP is to instead look at dynamic correlations. By dynamic correlations, we mean looking at how the money supply observed in date t correlates with real GDP in date t + j, where j > 0. Table 21.1 presents correlation of the cyclical component of the M2 money supply with the cyclical component of real GDP lead several periods. The frequency of observation is a quarter. 476 -.05-.04-.03-.02-.01.00.01.02.03.04606570758085909500051015Cyclical Component of GDPCyclical Component
of M2 Table 21.1: Dynamic Correlations between M2 and Output Variable Correlation with ln Mt ln Yt ln Yt+1 ln Yt+2 ln Yt+3 ln Yt+4 ln Yt+5 ln Yt+6 ln Yt+7 ln Yt+8 0.22 0.32 0.37 0.37 0.33 0.26 0.19 0.10 0.03 We observe from the table that the period t money supply is positively correlated with the cyclical component of real GDP led several periods. Interestingly, these correlations are larger (about 0.35) when output is led several quarters (up to a year) than the contemporaneous correlation of 0.22. This is suggestive, but only suggestive, that changes in the money supply do impact real GDP. It is only suggestive because it could be that the Fed anticipates that output will be above trend in a year, and increases the money supply in the present in response. While this is a possibility, it seems somewhat unlikely. The fact that these correlations are larger when output is led several periods than the contemporaneous correlation seems to suggest that changes in the money supply do have some causal eﬀect on real GDP. There are more sophisticated statistical techniques to try and determine whether changes in the money supply cause changes in real GDP and other real aggregate variables. Most of these studies do ﬁnd that changes in the money supply do impact real GDP in a positive manner, though the eﬀects are generally modest. See Christiano, Eichenbaum, and Evans (1999) for more. Nevertheless, it is interesting to note that the money supply ceases to be strongly correlated with output after about two years (eight quarters). To the extent to which money is nonneutral empirically, it is only so for a couple of years at most. After a period of several years, changes in the money supply do not seem to impact real variables, and monetary neutrality seems to be an empirically valid proposition. This fact forms the basis of our dividing things into the medium run, where the neoclassical model holds and money is neutral, and the short run, which we will study in Part V, where price or wage rigidity can allow increases in the money supply to result in a temporary increase in real GDP and changes in other real variables. 477 21.5 Summary • Money is diﬃcult to
measure because many diﬀerent kinds of assets can and do serve as money. Three conventional deﬁnitions of the money supply include M1, M2, and M3. M1 is the sum of all currency in circulation and demand deposits. M2 includes M1 plus some assets that are not as liquid as M1 such as money market mutual funds. M3 includes M2 plus institution money market funds and short term repurchase agreements. Economists usually prefer using M1 or M2 as their preferred measure of money supply. • Central banks can set the monetary base which consists of reserves plus currency in circulation, but can only partially inﬂuence the money supply. The money supply is some multiple of the monetary base. This multiple is a function of the reserve requirements set by central bank, banks’ willingness to lend out excess reserves, and houehold preferences for holding currency as opposed to demand deposits. • Under a conventional money demand function and assuming a constant nominal interest rate, inﬂation is the diﬀerence between the growth in the money supply and the growth in output. Over the long run, output has grown at a roughly constant rate which implies inﬂation rises one-for-one with growth in the money supply. • The relationship between growth in the money supply and inﬂation is positive but relatively weak at quarterly frequencies. However, the trend component of these series is much more highly correlated. • The quantity equation is an identity. It says that the money supply times the velocity of money equals nominal GDP. The velocity of money is not measured directly, but rather inferred so as to make sure the quantity equation holds. In terms of economics, velocity can be interpreted as the number of times the average unit of money is used. Velocity was relatively constant from 1960-1990, but has been quite volatile since 1990. • Over the long run, nominal interest rates should move one for one with the inﬂation rate. In the data there is indeed a strong relationship between these two variables. • The Neoclassical model predicts that the determination of real variables is independent of nominal variables. In the data, the cyclical component of M2 is positively correlated with cyclical component of output. While this is suggestive evidence against the classical dichotomy, correlation does not imply causation. However, increases in the money supply are also correlated with future increases in output which is stronger evidence against the classical dichotomy. 478
Key Terms • Currency • Asset • M1 • M2 • M3 • Reserves • Fractional reserve banking • Monetary base • Money multiplier • Velocity of money • Neo-Fisherian • Dynamic correlation Questions for Review 1. Describe some of the diﬃculties in measuring the money supply. To what extant do alternative measures of the money supply move together? 2. Do central banks control the money supply? 3. To what extent is inﬂation a monetary phenomenon? 4. Evaluate the Neoclassical model’s prediction about the velocity of money. 5. Evaluate the Neoclassical model’s prediction about the correlation between the nominal interest rate and inﬂation. 6. What evidence is there that changes in the money supply aﬀect output? Exercises 1. [Excel Problem] Download quarterly data on real GDP and M1 from the St. Louis Fed FRED website for the period 1960 through the second quarter of 2015. Our objective here is to examine how the money supply and output are correlated, with an eye towards testing the prediction of the neoclassical model of monetary neutrality. 479 (a) Before looking at correlations we need to come up with a way of detrending the series – both the money supply and real output trend up, and correlations are not well-deﬁned for trending series. We will focus on natural logs of the data. We will use a moving average ﬁlter. In particular, we will deﬁne the “trend” value of each series as a two-sided three year (12 quarter) moving average of the natural log of the data. This involves losing three years of data at both the beginning and end of the sample. Our data sample begins in 1960q1 and ends in 2015q2. Your trend value for a series in 1963q1 will equal the average of the series from 1960q1 to 1966q1 (12 observations before the period in question, and 12 observations after). Your trend value in 1963q2 will equal the average of the series from 1960q2 to 1966q2. Your trend value of a series in 2012q2 will equal the average of the series from periods 2009q2 through 2015q2. And so on. The ﬁrst observation in your trend series should be 1963q1 and the last should be 2012q2. (b) After you have
constructed your trend series for both log M1 and log real GDP, deﬁne the detrended series as the diﬀerence between the log of the actual series and its trend value. You will then have a time series of detrended values of log M1 and log real GDP running from 1963q1 to 2012q2. Plot the detrended values of log M1 and log real GDP against time and show them here. What do you see happening to real GDP around the time of the Great Recession (loosely, 2008 and 2009)? What about detrended M1? (c) Compute the correlation coeﬃcient between detrended M1 and detrended output. Does this correlation suggest that money is non-neutral? Why might it not be suggestive of that? Explain. (d) Now, to get a better sense of causality, let’s look correlations between M1 and output at diﬀerent leads. First, compute the correlation between output and M1 led four quarters (i.e. compute the correlation between detrended output from 1963q1 to 2011q2 with the detrended M1 from 1964q1 to 2012q2). Next, compute the correlation between M1 and output led four quarters (i.e. compute the correlation between detrended M1 from 1963q1 to 2011q2 with detrended output from 1964q1 to 480 2012q2). Are these correlations suggestive that money is non-neutral? Explain. 481 Chapter 22 Policy Implications and Criticisms of the Neoclassical Model In Chapter 15, we showed that a hypothetical benevolent social planner would choose the same allocations of consumption, labor supply, and investment as emerge in a decentralized equilibrium. What we have been doing in Part IV is simply a graphical analysis of the micro-founded equilibrium conditions derived in Part III. The implication of this analysis is that the equilibrium of the neoclassical model is eﬃcient in the sense of being exactly what a hypothetical benevolent social planner would choose. In other words, it is not possible for aggregate economic policy to improve upon the equilibrium allocations of the neoclassical model. This means that there is no role in the neoclassical model for activist economic policies designed to “smooth” out business cycle ﬂuctuations. If the economy goes into a recession because At declines, for example, the recession is eﬃcient
– it is not optimal for policy to try to combat it, taking the reduction in At as given. The neoclassical model with which we have been working also goes by the name “Real Business Cycle” (RBC) model. Economists Fynn Kydland and Ed Prescott won a Nobel Prize for developing this model. One can read more about RBC theory here. The model is called “Real” because it features monetary neutrality and emphasizes productivity shocks as the primary source of economic ﬂuctuations. It has the surprising and important policy implication that there is no role for activist economic policies. This was (and is) a controversial proposal. In this chapter, we (brieﬂy) discuss several criticisms which have been levied at the neoclassical / RBC model, criticisms which may undermine this strong policy proscription. 22.1 Criticisms In the subsections below we (brieﬂy) lay out several diﬀerent criticisms of the neoclassical model. Some of these question how well the neoclassical model can ﬁt the data (which we discussed in Chapter 20), some question assumptions in the model, and others point out things which are missing from the model. 482 22.1.1 Measurement of TFP A defender of the policy proscriptions which follow from the neoclassical model might say something along the lines of “Well, you might not like the implications of the model, but the model ﬁts the data well. Therefore it is a good model and we ought to take seriously its policy implications.” Several people have questioned just how well the neoclassical model ﬁts the data, beginning with Larry Summers in Summers (1986). The neoclassical model needs ﬂuctuations in At to be the main driving force behind the data in order to qualitatively ﬁt the data well. In Chapter 20, we showed that one could construct a measure of aggregate productivity given observations on Yt, Kt, and Nt. The resulting empirical measure, which is often called total factor productivity or just TFP, moves around a lot and is highly correlated with output – in periods where Yt is low, TFP tends to be low, in a way consistent with decreases in At causing declines in output. One of the main areas of criticism of the neoclassical model is that the measure of TFP is only as good as the empirical measures
of Kt and Nt. Over suﬃciently long time horizons, most economists feel that we have pretty good measures of capital and labor, but what about month-to-month or quarter-to-quarter? Many economists have pointed out that observed inputs might not correspond to the true inputs relevant for production. For example, suppose a ﬁrm has ten tractors. One quarter, the ﬁrm operates each tractor for 18 hours a day. The next quarter, the ﬁrm operates the tractors only 9 hours a day. To an outside observer, the ﬁrm’s capital input will be the same in both quarters (ten tractors), but the eﬀective capital input is quite diﬀerent in each quarter, because in the ﬁrst quarter the tractors are more intensively utilized than in the second quarter. To the extent to which eﬀective capital and labor inputs are mismeasured, what is measured as TFP may not correspond to the concept of At in the model. To be concrete, suppose that the aggregate production function is given by: Yt = At (utKt)α N 1−α t (22.1) Here, ut is capital utilization (in terms of the example given above, one might think of this as representing the number of hours each unit of capital is used). With this production function, what one measures in the data as TFP will be: ln T F Pt = ln At + α ln ut (22.2) In other words, if the utilization of capital moves around, measured TFP will not correspond one-to-one to the exogenous variable At in the model. One can see why this 483 might matter. Suppose that there is an increase in the demand for a ﬁrm’s product. The ﬁrm chooses to work its capital harder, increasing ut. This results in higher output. One will then observe TFP being high at the same time output is high, and might falsely attribute it to At being, though in this example At is not high – output is high because demand is high. How important might this problem be in practice? While most economists think that the utilization ought to be fairly stable over long time horizons, in the short run it might move around quite a bit. Basu, Fernald, and Kimball (2006) argue that this problem is
important. They come up with a way to measure utilization and “correct” a traditional measure of TFP for it. They ﬁnd that the corrected TFP measure is close to uncorrelated with output, which suggests that utilization moves around quite a bit. John Fernald of the Federal Reserve Bank of San Francisco maintains an updated, quarterly measure of the corrected TFP series. Figure 22.1 below plots the cyclical component of output along with the cyclical component of the adjusted TFP series Figure 22.1: Cyclical Components of GDP and Utilization-Adjusted TFP It is instructive to compare this ﬁgure with Figure 20.3. Whereas the conventional TFP series is highly correlated with output, it is clear here that the corrected TFP series is not. In particular, the correlation coeﬃcient between the corrected TFP series and output is -0.13. In Figure 22.1, one can see many periods which are near recessions but in which corrected TFP is high and/or rising. To the extent to which the corrected TFP series accurately measures the model concept of At (it may not, for a variety of reasons), this is a problem for the neoclassical theory of business cycles. For the model to match co-movements in the data, it needs to be driven by changes in At. If changes in At do not line up with observed changes in Yt, then the model is missing some important ingredient, and one should be weary about 484 -.05-.04-.03-.02-.01.00.01.02.03.0455606570758085909500051015GDPAdjusted TFP taking its policy implications too seriously. 22.1.2 What are these Productivity Shocks? One might dismiss the corrected version of TFP as being wrong on some dimension, or in attributing too much of the variation in observed TFP to utilization. Nevertheless, there remains a nagging question: what exactly are these productivity shocks causing output to move around? One can phrase this question in a slightly diﬀerent way. If there is a big, negative productivity shock which causes output to decline, why can’t we read about that in the newspaper? Another question is: what does it mean for productivity to decline? To the extent to which one thinks about productivity as measuring things like knowledge, how can it decline? Do we forget things we once knew
? These questions do not have simple answers, and have long left many uncomfortable with real business cycle theory. 22.1.3 Other Quantitative Considerations In Chapter 20, we focused only on the ability of the model to qualitatively capture co-movements of diﬀerent aggregate variables with output. In more sophisticated versions of the model which are taken to a computer, one can also look at how volatile diﬀerent series are (i.e. what their standard deviations are) and compare that with what we see in the data. While the neoclassical model successfully predicts that output and labor input are strongly positively correlated, it has diﬃculty in matching the relative volatility of hours. In the data, total hours worked is about as volatile as total output (i.e. they have roughly the same standard deviations). The basic model has great diﬃculty in matching this – in quantitative simulations of the model, total hours usually ends up about half as volatile as output. Put another way, the model seems to be missing some feature which drives the large swings in aggregate labor input we observe in the data. 22.1.4 An Idealized Description of the Labor Market The labor market in the neoclassical model is particularly simplistic. There is one kind of labor input, and this labor input is supplied by a representative household in a competitive spot market. There is no attachment between workers and ﬁrms, there is nothing like on the job training or human capital acquisition, and there is no unemployment as it is deﬁned in the national accounts (indeed, taken literally our model predicts that all movements in labor input are along the intensive margin, i.e. hours of work instead of whether or not to work). It 485 is possible to write down versions of the model with a more sophisticated description of the labor market, but it is diﬃcult to adequately model the richness of real-world labor markets. 22.1.5 Monetary Neutrality The basic neoclassical model features the classical dichotomy and the neutrality of money. Nominal shocks have no real eﬀects, and there is no role for monetary policy to try to react to changing economic conditions. Evidence presented in Chapter 21 casts doubt on the assumption that money is completely neutral, at least over short horizons. In particular, we showed that the cyclical component of the aggregate money supply is positively correlated with the cyclical component of output led over
several quarters. While not dispositive, this is at least strongly suggestive that changes in the money supply have real eﬀects. A large body of research supports that money is indeed non-neutral, although most of this research suggests that the real eﬀects of money are not particularly large and not particularly long-lasting. The notion of monetary neutrality also seems to run counter to our every day experience. People seem to think that what central banks do matters in ways beyond aﬀecting the price level and inﬂation rate. 22.1.6 The Role of Other Demand Shocks The basic neoclassical model, as we have written it, has output being completely supply determined. This means that only changes in At or θt can impact output. There is no role for other demand side disturbances (i.e. shocks to the IS curve), such as changes in Gt, Gt+1, or At+1. The model can be amended in such a way that these shocks can impact output by permitting the real interest rate to impact labor supply (as was discussed brieﬂy in Chapter 12 and as is developed in more detail in Appendix C). With such a modiﬁcation, the eﬀects of demand shocks on output are nevertheless small, and the model has diﬃculty generate positive co-movement between consumption and labor input conditional on demand-side shocks. Both casual experience and academic research suggests that demand shocks might be important drivers of output, at least in the short run. For example, a large body of research tries to estimate the government spending multiplier. Most of this research ﬁnds that the multiplier is positive (i.e. increases in Gt cause Yt to increase), though the literature is divided on whether the multiplier is greater than or less than one. Other work looks at how news or optimism about the future (e.g. anticipated changes in At+1) might impact output. 486 22.1.7 Perfect Financial Markets The basic neoclassical model does not have much to say about ﬁnancial intermediation. In the setup we have pursued, the household saves through a ﬁnancial intermediary (i.e. a bank), and this intermediary funnels these savings to the representative ﬁrm for investment in productive capital. The interest rate on savings and investment are the same, and the solution to the model
would be equivalent if the ﬁrm instead ﬁnanced itself with equity instead of debt. In reality ﬁnancial markets seem to be imperfect, and interest rates relevant for investment are often quite diﬀerent than what a household can safely earn on its savings. Figure 22.2 below plots an empirical measure credit spreads, deﬁned as the spread between the Baa rated corporate bond rate and the interest rate on a Treasury note of 10 year maturity. We can see that the credit spread tends to rise in periods identiﬁed as recessions (as demarcated with gray shaded bars), and seem to co-move negatively with output. Indeed, in the data since the early 1950s, the correlation between the cyclical (HP ﬁltered) component of real GDP and the Baa credit spread is almost -0.4. One might interpret the credit spread as a measure of the health of ﬁnancial intermediation, and the countercyclicality of the observed credit spread in the data seemingly suggests that ﬁnancial intermediation works poorly during recessions. This certainly aligns with conventional wisdom concerning the recent Great Recession. Figure 22.2: Cyclical Components of GDP and Baa Credit Spread The basic neoclassical model does not allow us to meaningfully address ﬁnancial market imperfections or the role of credit spread shocks. We will take this up later in the book in Part VI. 487 0123456789-.05-.04-.03-.02-.01.00.01.02.03.0455606570758085909500051015Baa Credit SpreadCyclical Real GDPBaa spreadCyclical Real GDPCorrelation = -0.38 22.1.8 An Absence of Heterogeneity The basic neoclassical model with which we have been working features a representative household and ﬁrm – there is no interesting heterogeneity. In the real world, there is lots of heterogeneity – some households earn substantially more income than others, for example. By abstracting from heterogeneity, the neoclassical model may substantially understate the welfare costs of recessions, and might therefore give misleading policy implications. In a typical recession in the data, output falls by a couple of percentage points relative to trend. If everyone’s income in the economy fell by a couple of percentage points, no one would like this but it wouldn’
t be that big of a deal. In the real world, recessions tend to impact individuals diﬀerently. Some people see their income drop a lot (say, because they lose a job), whereas others see virtually no change in their income. If there is imperfect insurance across households, then the utility of the individuals hardest hit will decline by a lot, whereas the utility of those who are not aﬀected will be unchanged or will decline only little. A benevolent planner may desire to redistribute resources from the unaﬀected households to the aﬀected households (e.g. those households who lose their jobs). To the extent to which this redistribution is diﬃcult/impossible, the planner may prefer to ﬁght recessions with stimulative policies of one sort or the other. Because the neoclassical model abstracts from heterogeneity, it cannot successfully speak to these issues, and its policy implications may therefore be misguided. 22.2 A Defense of the Neoclassical Model The basic neoclassical model is fully based on microeconomic decision-making. It takes dynamics and forward-looking behavior seriously. It is therefore immune from many of the criticisms levied by economists during the 1970s against the macroeconomic models of the middle of the 20th century. The neoclassical model can potentially ﬁt the data well in a qualitative sense if it is predominantly driven by changes in productivity. It has the stark policy implication that there is no need for aggregate economic policy to try to smooth out business cycle ﬂuctuations. As we have documented here, the neoclassical model, for all its desirable features and potential empirical successes, is not immune from criticism. Our own view is that these criticisms have much merit, and that the neoclassical model is probably not a good framework for thinking about economic ﬂuctuations in the short run. Why then, have we spent so much space of this book working through the neoclassical model? It is because the neoclassical model is a good benchmark model for thinking about ﬂuctuations, and it provides a good description of the data over longer time horizons, what we have deemed the “medium run” 488 (periods of a couple to several years). When thinking about building a better model for short run ﬂuctuations, one needs to clearly articulate the deviation from the neoclassical benchmark. In practice, this is how modern macro
economics is done. A phrase commonly used is that “it takes a model to beat a model.” The neoclassical model serves as the “backbone” for virtually all short run macroeconomic models. Models designed to understand short run ﬂuctuations introduce one or more “twists” to the neoclassical model. These “twists” are usually only operative for up to a couple of years. Keynesian models, which are the most popular alternative to the neoclassical model, assume that, in the short run, prices and/or wages are imperfectly ﬂexible. As we will see in more depth in Part V, this short run “stickiness” will change the behavior of the model and alter its policy implications in an important way. Most economists agree that prices and/or wages are subject to some level of “stickiness” in the short run. Where they diﬀer is in how important this stickiness is and how long it lasts – in other words, part of the disagreement is over how long the short run is. Neoclassical economists (sometimes called “freshwater” economists) tend to think that nominal stickiness is not that important and does not last that long. They prefer to use the neoclassical model (or some close variant thereof) to think about short run ﬂuctuations. Keynesian economists (or sometimes “saltwater”) think that nominal stickiness is important and might last a very long time. While most Keynesian economists would agree that the neoclassical model is a good benchmark for understanding medium run movements in output and other quantities, they feel that nominal stickiness means that the economy can deviate from this neoclassical benchmark by a signiﬁcant amount and for a signiﬁcant length of time. As such, they prefer to use Keynesian models to understand short run ﬂuctuations. We will study these models in Part V. 22.3 Summary • The Neoclassical model, also known as the Real Business Cycle model, makes the stark proposition that business cycles are optimal in the sense that a government cannot make people better oﬀ by following some activist policy. In fact, activist policy can only make people worse oﬀ. This is a controversial idea. • One criticism is that measured TFP poorly captures productivity. If input utilization varies
over the business cycle, measured TFP will be mis-speciﬁed. Measures of TFP that correct for input utilization show that TFP and output have a much lower, and possibly even negative, correlation. 489 • Also, no one knows what TFP really is. To the extent it measures something like technology or knowledge, what does it mean for TFP to decline? • The Neoclassical model is also criticized because it predicts monetary neutrality. This runs counter to the evidence discussed in 21. • Academic research shows that demand shocks are an important determinant of short-run output ﬂuctuations. However, the Neoclassical model predicts that output is invariant to demand shocks. • Finally, the Neoclassical model has no heterogeneity. This is a problem because the burden of recessions is not shared equally. Some people do not lose anything at all while others lose their jobs. By abstracting from this heterogeneity there is no role for redistribution or ﬁscal policy that may substitute for redistribution. • These criticisms have merit and taken together imply that the Neoclassical model may not be the best model for business cycles. However, it is a useful benchmark and does a good job describing the economy over the medium run. Key Terms • Variable utilization • Corrected TFP series • Freshwater economist • Saltwater economist Questions for Review 1. Evaluate the following statement: Because there is no role for activist policy in the Neoclassical model, declines in productivity are welfare improving. 2. Why might measured TFP be an incorrect measure of true productivity? 3. What is concerning about excluding meaningful heterogeneity in the Neo- classical model? 490 Chapter 23 Open Economy Version of the Neoclassical Model In this chapter we consider an open economy version of the neoclassical model. This introduces a new expenditure category, net exports, which we will denote N Xt. Net exports is the diﬀerence between exports (stuﬀ produced in an economy and sold elsewhere) and imports (stuﬀ produced elsewhere but purchased in an economy of interest). As we discussed in Chapter 1, the reason that imports gets subtracted oﬀ is because the other expenditure categories (consumption, investment, and government spending) do not discriminate on where a good was produced. Hence, a household buying a foreign good increases consumption, but does not aﬀect total domestic spending, so subtracting oﬀ imports is
necessary for the positive entry in consumption to not show up in total aggregate expenditure. For simplicity, we will think of a world with two countries – the “home” country (the country whose economy we are studying) and the foreign economy, which we take to represent the rest of the world. Net exports depends on the real exchange rate, which governs the terms of trade between domestic and foreign goods. In real terms, this exchange rate measures how many “home” goods one foreign good will purchase (in contrast, the nominal exchange rate measures how many units of “home” currency one unit of foreign currency will purchase). Because of international mobility of capital, the real exchange rate will depend on the real interest rate diﬀerential between the home and foreign economies. This means that net exports will in turn depend on the real interest rate diﬀerential, where we take the foreign real interest rate as given. In eﬀect, the opening of the economy will just add another term to the expenditure identity (which manifests graphically in terms of the IS curve) which depends negatively on the real interest rate. 23.1 Exports, Imports, and Exchange Rates In this section, we introduce a foreign sector into our neoclassical model of an economy. This introduces a new expenditure category, net exports, which we will denote N Xt. Net exports is the diﬀerence between exports (goods and services produced in the home country and sold to foreigners) and imports (goods and services produced abroad and purchased by domestic residents). Net exports in turn depends on the real exchange rate, which is the 491 relative price of home and foreign produced goods. In what follows, we will think of the country whose economy we are modeling as the “home” country (where relevant, denoted with a h superscript) and will simply model all other foreign countries as one conglomerate foreign country (where relevant, denoted with a F superscript). We will sometimes also refer to the foreign sector as the “rest of the world.” t, the ﬁrm on investment, I h Total desired expenditure on home production is the sum of desired expenditure by the household, C h t. There is an additional term, Xt, which stands for exports. Exports represent expenditure by the rest of the world on home-produced goods and services. Total desired expenditure on home-produced goods and services is the sum of these four components,
as given in (23.1). t, and the government, Gh + Gh t + Xt (23.1) The household can consume goods either produced at home or abroad and similarly for the ﬁrm doing investment and government expenditure. That is, total consumption, investment, and government expenditure are the sums of home and foreign components: Ct = C h t Gt = Gh t It = I h t + C F t + GF t + I F t If we plug these in to (23.1) and re-arrange terms, we get: Y d t = Ct + It + Gt + Xt − (C F t + I F t + GF t ) (23.2) (23.3) (23.4) (23.5) We will refer to the term C F t in (23.5) as imports – this term denotes total desired expenditure by home residents on foreign produce goods and services. Labeling this term IMt, (23.5) can be written: + I F t + GF t Y d t = Ct + It + Gt + Xt − IMt Or, deﬁning N Xt = Xt − IMt: Y d t = Ct + It + Gt + N Xt (23.6) (23.7) We assume that total desired consumption and investment are the given by the same functions we have previously used: 492 Ct = C d(Yt − Gt, Yt+1 − Gt+1, rt) It = I d(rt, At+1, Kt) (23.8) (23.9) Consumption is an increasing function of current and future perceived net income (it is perceived because we continue to assume that Ricardian Equivalence holds, so that the household behaves as though the government balances its budget each period) and a decreasing function of the real interest rate. Investment is a decreasing function of the real interest rate, an increasing function of expected future productivity, and a decreasing function of the existing capital stock, Kt. We continue to assume that government spending is exogenous with respect to the model. What determines desired net exports? Mechanically, net exports depends on how much foreign stuﬀ home residents want to purchase less how much home stuﬀ foreigners want to purchase. In principle, this diﬀerence depends on many factors. One critical factor is the real
exchange rate, which measures the relative price of home produced goods to foreign produced goods. We will denote the real exchange rate by t. This is simply a relative price between home and foreign produced goods, and the units are home goods foreign goods. So if the real exchange rate is 1, one unit of a foreign good will purchase one home produced good. Exchange rates can be tricky in that the relative price of goods can be deﬁned in the opposite way (i.e. foreign goods to domestic goods). We will always think of the real exchange rate as being denoted home goods relative to foreign goods. The building block of the real exchange rate is the nominal exchange rate, which we will denote by et. The nominal exchange rate measures how many units of the home currency one unit of foreign currency can purchase. As an example, if the units of the domestic currency are dollars, and the units of the foreign currency are euros, then the nominal exchange rate is dollars per euro. If the nominal exchange rate is 2, it says that one euro will purchase 2 dollars. If the exchange rate were deﬁned in the other way, it would be 1/2, and would say that one dollar will purchase half of a euro. The real and nominal exchange rates are connected via the following identity: t = et P F t Pt (23.10) Here, P F t is the nominal price of foreign goods and Pt is the nominal price of home goods. The logic emboddied in (23.10) is as follows. t measures how many home goods can be purchased with one foreign good. One foreign good requires P F t units of foreign currency. This P F t units of the home currency (since the units t units of foreign currency purchases etP F 493 of et are home currency divided by foreign currency, etP F t t units of home currency will purchase etP F currency). etP F t Pt is denominated in units of home units of home goods. We assume that desired net exports depend positively on the real exchange rate. Why is this? If t increases, then foreign goods will purchase relatively more home goods (and vice-versa). Put diﬀerently, home goods are relatively cheap for foreigners, and foreign goods are relatively expensive for home residents. This will tend to make exports rise (the home country will sell more of its relatively cheaper goods abroad) and imports will fall (home residents will buy relatively fewer foreign goods, since these are now
more expensive). We say that an increase in t so deﬁned represents a real depreciation of home goods (home goods are relatively cheaper for foreigners). Thus, we assume that net exports are increasing in t. We will not model other sources of ﬂuctuations in net exports (which could include changes in home or foreign income, etc.), but will instead use an exogenous variable to denote all other sources of change in desired net exports. We will denote this exogenous variable as Qt. We will normalize it such that an increase in Qt results in an increase in desired net exports (and vice-versa for a decrease in Qt). One source of changes in Qt could be tariﬀs and other barriers to trade or trade unions and agreements that lower barriers to trade. Now, what determines the real exchange rate, t? We will assume that the real exchange rate depends on the diﬀerential between the home and foreign real interest rates, rt − rF t, where rF t denotes the foreign real interest rate (which we take to be exogenous in the model). Why is this? If rt > rF t, one earns a higher real return on saving in the home country than in the foreign country. This ought to drive up the demand for home goods relative to foreign goods, which would result in a reduction in t, what we would call a home appreciation (and vice versa). Hence, the real exchange rate itself ought to be decreasing function of the real interest diﬀerential between the home country and the rest of the world. In particular, we will assume: t = h(rt − rF t ) (23.11) Here, h(⋅) is some unknown but decreasing function, i.e. h′(⋅) < 0. This speciﬁcation omits other factors which might inﬂuence the real exchange but focuses on one of the most important that is relevant to the rest of our model. Since net exports are assumed to be increasing in t, but t is decreasing in the real interest rate diﬀerential, we can conclude that net exports are decreasing in the real interest rate diﬀerential between the home and foreign country. Formally: N Xt = N X d(rt − rF t − ), Qt + (23.12) 494 The + and − signs indicate the net exports is decreasing in the real interest

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