1 A simple two period model


 Martha Singleton
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1 A QUICK REVIEW TO INTERTEMPORAL MAXIMIZATION PROBLEMS 1 A simple two period model 1.1 The intertemporal problem The intertemporal consumption decision can be analyzed in a way very similar to an atemporal problem. The consumer has to choose between consumption of the same good at different dates. He lives for two periods and derives utility from his stream of consumption, as given by u (c 0,c 1 ). The consumer takes the gross real interest rate R as given. His budget at time 0 is given by y (expressed in terms of period 0 goods). Assume that the only source of income in period 1 is the income from savings at time 0. At date 0, the consumer chooses between the consumption level c 0 and savings s 0. Savings are in the form of consumption goods at date 0. The budget constraint at date 0 is c 0 + s 0 y. (1) At date 1, the consumer chooses the consumption level c 1. The income in that period only comes from savings in the previous period. Hence, the budget constraint is The consumer s maximization problem is thus: c 1 Rs 0. (2) max u (c 0,c 1 ) s.t. (1) (2). The difference between this problem and an atemporal problem (choice between two different goods at the same date) is that there are two budget constraints. This is artificial though, since assuming strictly increasing utility, the two budget constraints will hold as equality and can be combined as follows: so that the maximization problem is given by c R c 1 = y, (3) max u (c 0,c 1 ) s.t. (3). (P) This is similar to a standard utility maximization problem subject to a budget constraint, with the characteristic that the tradeoff is between current and future consumption and the relative price of date 1 good to 1
2 date 0 good is 1/R (it is not equal to 1 even though the two goods have the same physical characteristics). The consumer thus has the possibility of intertemporarily substituting consumption across time. A special case (yet very common) for u (.,.) is u (c 0,c 1 )=U (c 0 )+βu (c 1 ), 0 <β<1. This intertemporal utility function assumes that the consumer derives utility from consumption in each period and that intertemporal utility is a weighted sum of the utility levels in the two periods [timeadditive utility function]. β is called the discount factor. Solving (P), we get u 1 /u 2 = R (4) y c 0 c 1 /R = 0 (5) Condition (4) equates the marginal rate of substitution (relative value of current consumption to future consumption, i.e. the rate at which consumers are willing to trade one type of consumption for another) with the marginal rate of transformation (i.e. the rate of substitution between the two types of consumption available in the market). 1.2 The determinants of savings There are three important determinants of savings: the income profile, the rate of return to savings (the real interest rate) and the agent s patience toward future consumption. We enrich the model by allowing the household to have income at date 1 in addition to the income from savings (it could be labor income for example). Then, the budget constraint at date 1 is c 1 Rs 0 + y 1. The household s budget constraint at date 0 is (1) with y = y 0. The intertemporal constraint is then c 0 + c 1 R y 0 + y 1 R. (6) The level of savings is given by s 0 = y 0 c 0.Theoptimalc 0 is given by (4). Since the intertemporal budget constraint (6) binds, c 1 = y 1 + R (y 0 c 0 ). Substituting this and using time additivity of u, weget Income profile U 0 (c 0 ) U 0 = Rβ. (7) (y 1 + R (y 0 c 0 )) By the income profile, we refer to the household s income derived from human capital such as labor income, not counting income derived from savings. Hence, the income profile is (y 0,y 1 ). To illustrate how the income 2
3 profile affects savings, assume that R =1and that the household does not discount future utility (β =1). In that case, condition (7) becomes (assuming a strictly concave U (.)) c 0 = y 1 + R (y 0 c 0 ). The household achieves the same amount of consumption every period. This is due to the "consumptionsmoothing motive". In that simple case, we can explicitly solve for s 0, s 0 = y 0 y 1. 2 If the household has a flat income profile (and (R, β) =(1, 1)), the optimal level of savings is 0; if the household has an increasing income profile (y 1 >y 0 ), the optimal level of savings is negative and vice versa Real interest rate Let us now look at the rate of return to savings, i.e. the real interest rate. To isolate its role, let us assume again that β =1and and, for the moment, that the household has a smooth income profile (y 0 = y 1 = y). In that case, the optimal level of savings is 0 and the real interest rate R =1. Assume, instead, that R>1. Condition (7) becomes U 0 (c 0 )=RU 0 (y + R (y c 0 )) Unlike the previous case, the optimal consumption levels are different in the two periods. Given the smooth income profile and no time discounting, a higher real rate entices the household to save more. The intertemporal substitution effect is due to the fact that an increase in the real interest rate effectively makes consumption cheaper at date 1 than at date 0 (the real interest rate is the relative price of date 0 goods to date 1 goods). The increase in the real interest rate can also induce an income effect on savings (which does not appear in this example since, before the increase in R, households savings are 0). If instead, households had nonzero savings, then an increase in R will affect the household s interest income from savings Rs 0, and change future income. If the household has positive savings, an increase in the interest rate increases future income relative to current income. Anticipating the rising income profile, the household reduces savings in order to smooth consumption. On the other hand, if the household has negative savings, an increase in the interest rate increases the household s interest payment at date 1. In this case, future income falls relative to current income and the household increases savings in order to smooth consumption. Therefore, income and intertemporal substitution effects work in the same direction when the household has negative savings, but in opposite directions when the household has positive savings. A measure of the strength of intertemporal substitution is the elasticity of intertemporal substitution. Because R is the relative price between goods at different dates, we can define the intertemporal elasticity as dln (c 0/c 1 ). dln (R) 3
4 Another way to compute the elasticity is dln(c 0/c 1 ) d(u 1 /u 2 ), which comes from substituting the firstorder condition F = u 1 /u 2. For the timeadditive utility function, with U (c) = c1 σ 1 1 σ, the elasticity of substitution is 1/σ (the case where σ 1 corresponds to logarithmic utility). Exercise: Suppose that U (c) = c1 σ 1 1 σ,whereσ>0. Hence, the elasticity of substitution is 1/σ. Assume β =1, R 1, y 0 > 0 and y 1 > 0. Prove the following results: (i) the optimal level of savings is s 0 = y 0R 1/σ y 1 R 1/σ + R. (ii) s 0 0 always implies ds 0/dR > 0. (iii) If R =1and y 0 >y 1,thends 0/dR > 0 if and only if σ<(y 0 + y 1 ) / (y 0 y 1 ). Solution: (i) The household s problem is to solve c 1 σ max c 0,c σ + c1 σ σ s.t. c 0 + c 1 R = y 0 + y 1 R. The first order conditions are µ u 1 c σ = 0 u 2 c = R, 1 c 0 + c 1 R = y 0 + y 1 R. After some algebra, this results in s 0 = y 0 c 0 = y 0R 1/σ y 1 R 1/σ + R. (ii) From the above, once can compute ds 0/dR. After some algebra ds 0 dr = s 0 R + R 1/σ + 1 σ R1/σ y 0 + y 1 R R + R 1/σ 2. Hence, if s 0 0, ds 0 /dr > 0. (iii) When R =1 ds 0 dr = 2s σ (y 0 + y 1 ) = (y 0 y 1 )+ 1 σ (y 0 + y 1 ) 4 4 Thus ds 0/dR > 0 if and only if (y 0 y 1 )+ 1 σ (y 0 + y 1 ) > 0, i.e. when σ< y 0 + y 1, y 0 y 1 given that y 0 >y 1. Notice that with an increasing income profile (y 0 <y 1 ), ds 0/dR > 0, for all σ. 4
5 What is the point of the exercise? We know the following: As R s 0 0 s 0 0 Subst. effect s 0 s 0 Income effect s 0 s 0 Question (ii) allowed us to check analytically that when savings are negative, an increase in the real interest rate always induces an increase in savings, as both the income and substitution effects work in the (same) direction of increasing savings. In question (iii), we assume a decreasing income profile and hence positive savings (when R =1). That is the case when the income and substitution effects work in opposite directions. We established analytically, that for the substitution effect to dominate the income effect, it has to be that σ is small enough and thus that the elasticity of substitution is high enough. The insight in this simple model will be useful when we look at a more general one in the upcoming chapter on general equilibrium, real business cycle models Impatience towards the future The degree of impatience is measured by the discount factor. When β is low, the household is more impatient. For illustration, assume that the income profile is flat, the real interest rate is equal to 1, andthatβ<1. Then condition (7) becomes U 0 (c 0 )=βu 0 (2y c 0 ), which implies that c 0 >y. In fact, the lower β, the higher c 0 is. Let us illustrate in a different manner how savings fall when the degree of impatience increases. Suppose the household chooses the same level of consumption in the two periods, i.e. that c 0 = c 1 = y. Let the household reduce savings by a slightly positive amount ε and increase consumption at date 0 by the same amount. Utility at date 0 is now approximately U (y)+εu 0 (y), while utility at date 1 is now approximately U (y) εu 0 (y). Taking into account the discounting implies that utility increases by ε (1 β) U 0 (y) > 0. 5
6 Chapter I BUSINESS CYCLES 1
7 1 A General Real Business Cycle Model 1.1 The model Theideaistovieweconomicfluctuations as the result of external shocks hitting the economy. These shocks can be of different sources. The real business cycle (RBC) literature focuses on shocks to the productive capacity of the economy as the main source of fluctuations. Hence, the reference to real shocks, as opposed to nominal or monetary shocks which had been the focus of previous literatures. In that sense, RBC models are a very definite departure from previous models. They try to see how much of the fluctuations are normal reactions to real shocks, in the absence of any market imperfections. The previous literatures tended to assume some kind of market imperfections, and explain cycles with these imperfections. The idea is to model an artificial competitive economy where aggregates determined in equilibrium (output, investment, consumption, labor) are derived from agents maximizing behavior at the microeconomic level. Instead of assuming relationships at the macroeconomic level, aggregates outcomes are derived from optimal behavior of individual agents. Agents: Large number of identical households (of measure 1). Each household is small enough that it cannot influence market outcomes through its behavior (competitive economy) and lives forever. Large number of identical small firms. Both types of agents are price takers, i.e. they believe they can buy or sell any quantity of the good or labor but not affect price. Households are expected utility maximizers, while firms are profit maximizers. The population size is assumed constant. Technology: Each period, households rent capital and provide labor to firms, at a wage rate w t and a rental rate r t, respectively. The production function is Y t = e zt f(h t,k t ) (1) where Y t, H t,andk t are output, labor and capital in period t, respectively. z t is a random shock to the economy s productive capacity. Assume that the stochastic process is characterized as follows: z t+1 = ρz t + ε t+1 (2) where 0 <ρ<1 and ε t N(0,σ 2 ε). Each period, a new shock hits the economy. That shock is known before any decision is to be taken. f is defined on R 2 + with values in R +, and it is increasing, concave, and continuously differentiable in H and K. It is also homogenous of degree 1 (f(ah, ak) =af(h, K), i.e. if the inputs of capital and labor are doubled, output is also doubled). In addition, f(h, 0) = f(0,k)=0. Also, f 1 (H, K) + as H 0 and f 2 (H, K) + as K 0. Endowments: 2
8 Households are initially endowed with k 0 and start every period with a new total time endowment equal to 1, to be allocated between labor and leisure. Utility functions: For the households, the utility derived from an infinite sequence of consumptions and labor choices, {c t } + t=0 and {h t } + t=0 is h i + X u {c t } + t=0, {h t} + t=0 = β t U (c t, 1 h t ) t=0 where the discount factor β satisfies 0 <β<1. The one period utility function U is defined from R + to R and is assumed to be increasing, strictly concave and continuously differentiable in both arguments. Moreover, lim 1 (c, 1 h) c 0 = lim U 2 (c, 1 h) =+ h 1 lim 1 (c, 1 h) c + = lim U 2 (c, 1 h) =0 h 0 These restrictions, as well as the restrictions on the production function f ensure that there exists an interior solution 1. Resource constraint: Each period capital depreciates at a rate δ. The resource constraint is c t + i t = c t + k t+1 (1 δ) k t = w t h t + r t k t where i t is the household s investment in period t. This constraint states that households need to allocate their period income (from renting capital and working) between consumption and investment. The problem has already been set up as a competitive equilibrium problem. It could also be solved as a social planner problem, given that, in this particular case, they are equivalent. However, for the sake of generality, we will solve the problem as an equilibrium problem. The method retained can also be used in cases where the welfare theorems do not apply (distortions, externalities...). The concept used is called Recursive Competitive Equilibrium. We will solve the problem using dynamic programming and start by defining the state variables, as well as the control variables. Individual state variable: k t (capital owned by an individual household) Aggregate state variables: z t, K t (total capital in the economy) Control variables: c t, h t, i t We will start by describing the maximization problems faced by the firms and the households. Firm s decision problem: 1 These conditions are referred to as the Inada conditions. 3
9 Every period, the firm chooses capital and labor to maximize profits. The firm, as opposed to the worker, faces a one period problem. Because it does not have any claim to either labor or capital, it does not have to consider how its choices this period will affect its decision next period. Hence, the firm s problem is Differentiating with respect to h t and k t,weobtain t, Max h t,k t Π t = e z t f (h t,k t ) w t h t r t k t e zt f 1 (h t,k t )=w t (3) e zt f 2 (h t,k t )=r t (4) These two equations define the individual demands, given factor prices. Remark that, since the production functions is assumed to exhibit constant return to scale, firms make zero profits in equilibrium 2. The aggregate variables H t and K t satisfy (3) and (4) if the number of firmsisnormalizedto1. Hence, (3) and (4) imply that w t = bw (z t,h t,k t ) (5) r t = br (z t,h t,k t ) (6) Equations (5)(6) represent the market clearing conditions, pinning down prices. Household s decision problem: Households are expected utility maximizers. Hence, they solve 3 " + # X MaxE β t u (c t, 1 h t ) t=o s.t. c t + k t+1 (1 δ) k t = w t h t + r t k t, t given stochastic processes for w t, r t and given k 0 The households need to make expectations on the future behavior of variables relevant to their intertemporal decision making. We will assume that households expect wage and capital rental rates to be functions of z t, K t and H t, as indicated by the solution to the firm s problem, and hence that they know (5) and (6) (remember h t is a control variable, but not H t ). Assuming that the other households also behave optimally, the individuals also know, or make expectations on H t = H (z t,k t ) (7) K t+1 = K (z t,k t ) (8) 2 By the application of Euler s theorem for homogenous functions of degree 1, f(h, k) =hf 1 (h, k)+kf 2 (h, k). 3 Because firms make zero profit in equilibrium, we do not need to add dividends from ownership of firms in the household s budget constraint. 4
10 z t+1 = z (z t,ε t ) (9) Using (7), the households expect the future wage and capital rental rates to be functions of the aggregate state variables only w t = w (z t,k t ) (10) r t = r (z t,k t ) (11) Once these expectations are taken, the households are able to solve their maximization problem for consumption, investment and labor supplied. Expectations are rational in the sense that K t+1 = K (z t,k t ), z t+1 = z (z t,ε t ), w t = w (z t,k t ), r t = r (z t,k t ) are known by the agents. But be careful, what is known by the agents is the rule of motion for K t+1 and z t+1, as well as factor prices as a function of aggregate state variables, not the exact future sequences. The agents make exact predictions about these on average, but are not assumed to be correct every period! Due to the recursive nature of the problem, the individual household s decision problem can be written as 4 : where v (z t,k t,k t )= Max h t,k t+1 {u (w t h t + r t k t +(1 δ) k t k t+1, 1 h t )+βe ε [v (z t+1,k t+1,k t+1 ) z t ]} (12) The solution to this problem is w t = w (z t,k t ) r t = r (z t,k t ) K t+1 = K (z t,k t ) z t+1 = z (z t,ε t ) z 0,k 0,K 0 are given h t = h (z t,k t,k t ) k t+1 = k (z t,k t,k t ) DEFINITION: A recursive competitive equilibrium is a list of value function V (z t,k t,k t ), individual decision rules h t (z t,k t,k t ), k t+1 (z t,k t,k t ) for the representative household, aggregate laws of motion H t (z t,k t ), K t+1 (z t,k t ), factor price functions w t (z t,k t ) and r t (z t,k t ) such that : (i) the household s problem is satisfied, i.e. h t (z t,k t,k t ),andk t+1 (z t,k t,k t ) solve (12) (ii) the firm s problem is satisfied and markets clear, i.e. e zt f 1 (H t,k t )=w t (z t,k t ) and e zt f 2 (H t,k t )=r t (z t,k t ) 4 Because the budget constraint will always be binding, there are really only two decision variables: how much labor to supply and how much to invest. Once investment is chosen, consumption is automatically given. Also, choosing investment is equivalent to choosing how much capital to bring to the next period (k t+1 ). 5
11 (iii) consistency of individual and aggregate decisions 5, i.e. h t (z t,k t,k t )=H t (z t,k t ) and k t+1 (z t,k t,k t )=K t+1 (z t,k t ) (iv) aggregate resource constraint is satisfied, i.e. C(z t,k t )+I(z t,k t )=Y (z t,k t ) The concept used to construct the equilibrium is illustrated in figure Expectations? optimization Equilibrium Behavior ¾? Outcome mechanism/institution Figure 1: Recursive competitive equilibrium concept 1.2 Calibration The concept presented above abstracts away from considerations of growth in the economy. Implicitly, we only looked at fluctuations around a steady state growth path. However, this is not a problem because one can always start from an economy that is allowed to grow along a steady state path and transform it into a stationary economy and solve for the Recursive Competitive Equilibrium. Hence, the techniques outlined can be used to study both how an economy grows over time and how it fluctuates around its growth trend. We present below how the problem can be stationarized (in later applications, we will start directly from the stationarized version of the economy). Restriction on the production function The steady state growth path is defined as the path where all rates of growth are constant, except for labor, which is bounded above (by H). Assume that there is a permanent component to technological change X t 5 This is because all households are identical. The typical household must be typical in equilibrium. However, it cannot be imposed on the decision maker. Prices move to make it desirable to the household. 6
12 in addition to the temporary one z t. We restrict ourselves to the case where the permanent change is labor augmenting 6,thatis,X t affects the efficiency of labor f (H t,k t,x t )=f (X t H t,k t ) Along the steady state growth path, H t = H. Let us call γ the growth rates. Hence, γ X = X t+1 /X t, γ K = K t+1 /K t, γ C = C t+1 /C t, γ I = I t+1 /I t, γ Y = Y t+1 /Y t. Writing the resource constraint for any dates t and t +1givesusthat and hence Therefore, for all t C t + I t = Y t γ c C t + γ I I t = γ Y Y t γ c C t + γ I I t = γ Y C t + γ Y I t (γ C γ Y ) C t +(γ I γ Y ) I t =0 For that to hold for all t, it has to be the case that C t and I t grow at the same rate. If this were not the case, the above equation cannot hold for all t, unless the two coefficients are equal to zero, which was ruled out by assumption. Hence it must be the case that γ C = γ I, which in turn implies that (γ C γ Y )(C t + I t )=0. Thus γ C = γ Y. We know that I t = K t+1 (1 δ)k t. Therefore γ K =1 δ + I t K t and thus I t /K t is constant for all t, and as a result γ K = γ I. Finally, Y t = f(x t H t,k t )=f(x t H,K t ). Because of the constant returns to scale assumption on f γ Y = Y t+1 = f Xt+1 H,K t+1 Y t f = f γx X t H,γ K K t X t H,K t f X t H,K t The only way for this expression to be constant for all t, isthatγ X = γ K. Hence 7 γ X = γ I = γ Y = γ C = γ K This matches the empirical observations that (i) real output grows at a more or less constant rate, (ii) the stock of real capital grows at a more or less constant rate greater than the rate of growth of the labor input, the growth rates of real output and the stock of capital tend to be about the same 8. 6 For a steady state growth to be feasible, we need that the permanent technological change be labor augmenting. 7 Hence a steady state growth path requires that the growth rates be constant. This is enough to conclude that they are not only constant, but also equal. 8 Of course, there has been a marked break in the trend of productivity growth around 1973, a phenomenon for which a satisfactory explanation still has to be found. Nevertheless, the facts mentioned still hold. The growth rates may have changed, but the relationships between them are still valid. 7
13 Restriction on the utility function The dynamic programing problem can be solved for a variety of utility functions. We would like to use the data to restrict the class of utility functions to be considered. We will use the observation that the longrun aggregate labor supply is constant to pick a particular class of utility function. A general treatment can be found in King, Plosser & Rebelo (JME 1988). The question asked is: what kind of utility function do we need to be consistent with balanced growth and a constant labor supply in the long run? Suppose you solve the following maximization problem: V (k t ) = Max {u (c t, 1 h t )+βv (k t+1 )} k t+1,h t s.t. c t + k t+1 (1 δ) k t = f (X t h t,k t ) You are interested in finding restrictions on the utility functions that are consistent with a steady state growth path, where labor is constant at h and other variables grow at the same rate γ as the technological progress X t.theefficiency conditions are: u 1 c, 1 h = β f 2 +1 δ u 1 γc,1 h (13) γ t f 1 u 1 c, 1 h = u 2 c, 1 h (14) These are the standard efficiency conditions. The marginal products of capital of labor (f 1 ) and capital (f 2 ) are constant along the steady state growth path, due to the constant return to scale assumption on the production function. X t was normalized to γ t.bydifferentiating (13) with respect to c and writing the expression c u 11(c,1 h) at c u 1(c,1 h) t = c and c t+1 = γc, one can verify that c u 11(c,1 h) must remain constant along the u 1(c,1 h) SSGP as consumption increases (and equal to σ). Solving the differential equation, one gets that: u (c, 1 h) = c 1 σ a (1 h)+b (1 h) if σ 6= 1 u (c, 1 h) = Ln (c) d (1 h)+e (1 h) if σ =1 Similarly, writing (14) at c t = c and c t+1 = γc and using the functional form just found for u, one can verify that one needs the restrictions that b 0 (1 h) =0if σ 6= 1and d 0 (1 h) =0if σ =1. Hence: u (c, 1 h) = c 1 σ a (1 h) if σ 6= 1 u (c, 1 h) = Ln (c)+e (1 h) if σ =1 For a CRRA utility function, the income effect (h ) of an increase in wage cancels out the substitution effect (h ), which matches the empirical result that the longrun aggregate supply of labor is constant, even though real wages increased over time. With this class of utility function, we can now compute the intertemporal elasticity of substitution ( IES ), as well as the intratemporal elasticity of substitution ( ES ). Suppose you are solving the following standard 8
14 equilibrium problem. The household s maximization problem is (state variable: k t, control variables: h t, k t+1 ): + X Max t=0 β t u (c t, 1 h t ) s.t. w t h t + r t k t = c t + k t+1 (1 δ) k t In dynamic programming terms, this can be rewritten as This gives us the following first order conditions: V (k t )= Max k t+1,h t {u (c t, 1 h t )+βv (k t+1 )} U c w = U l (15) Uc t = βuc t+1 (1 + r t+1 δ) (16) Hence, the marginal rate of substitution between consumption this period and leisure this period is U l /U c = w, and the marginal rate of substitution between consumption this period and consumption next period is Uc/βU t c t+1 =1+r t+1 δ. Wecannowdefine the two elasticities as: IES = ES = dln (c t+1 /c t ) dln MRS ct,c t+1 dln (c t /l t ) dln (MRS ct,l t ) ³ σ With a CRRA utility function, U c = c σ g(l) and U l = c1 σ 1 σ g0 (l). Hence, MRS ct,c t+1 = 1 ct+1 g(lt) β c t g(l t+1 ) and MRS ct,l t = 1 c t l t g 0 (l t ) 1 σ l t g(l.thus, t) IES = 1 σ ES = 1 Making the problem stationary Now, for the variable A t,define a s t = At X t,except for L t and H t, which are not normalized (we know that all variables grow at the same rate along the steady state growth path). Since the utility function is CRRA, we can rewrite the maximization problem as + X Max t=0 β t Ct 1 σ 1 σ g (L t) s.t. Y t = C t + K t+1 (1 δ) K t 9
15 and further The Bellman equations are + X Max t=0 β t (c s tx 0 γ t x) 1 σ g (lt s ) 1 σ s.t. e zt f (h s t,k s t ) = c s t + γ x k s t+1 (1 δ) k s t V (kt s,z t ) = Max kt+1 s,hs t where β = βγ 1 σ x ( (c s t) 1 σ 1 σ g (ls t )+β E V ) kt+1,z s t+1 The problem thus redefined is stationary and can be solved using dynamic programing Choosing the parameters of the model We can now start the calibration. Let us start by defining the capital and labor shares of output. Take a production function y = Zf(h, k) Then by differentiation, we obtain and thus y = Zf(h, k)+zf 1 (h, k) h + Zf 2 (h, k) k y y = Z Z + Zf 1(h, k)h h y h + Zf 2(h, k)k y Zf 1 (h, k)h and Zf 2 (h, k)k are the labor and capital shares of output (income from production going to the owner of labor and capital). Notice that because of the perfect competition assumption, Zf 1 is the wage received by the worker and Zf 2 is the rental income to the owner of capital. By the constant returns to scale and perfect competition assumptions, the two shares add up to 1. Since everything else is observable, one can calculate time series for z, using (17). It is found, looking at data, that capital and labor shares of output have been approximately constant over time 9, even while their relative prices have changed. This suggests a CobbDouglas production function f (h t,k t )=h θ t k 1 θ t To be consistent with observed values of labor and capital shares of output, θ 2/3. We know that the utility function must be CRRA. The only parameter we have to choose is 1/σ (intertemporal elasticity of substitution). Most empirical studies point towards a value of σ between 1 and 2. As the artificial 9 This is also true across countries. k k (17) 10
16 economy is not very sensitive to the exact value of σ, avalueofσ =1is generally chosen. Hence, the form generally retained for the utility function is u (c, 1 h) =(1 α) Ln (c)+αln (1 h) By solving for the first order conditions, we can obtain the deterministic steady state. maximization problem is " + # X MaxE β t [(1 α) Ln (c t )+αln (1 h t )] t=0 s.t. c t + k t+1 (1 δ) k t = f (h t,k t ), t The household s The first order conditions are u 1 (c t, 1 h t ) = βv 0 (k t+1 ) f 1 (h t,k t ) u 1 (c t, 1 h t ) = u 2 (c t, 1 h t ) and from the envelope theorem V 0 (k t )=u 1 (c t, 1 h t )[f 2 (h t,k t )+1 δ] Hence u 1 (c t, 1 h t )=βu 1 (c t+1, 1 h t+1 )[f 2 (h t+1,k t+1 )+1 δ] Given our assumptions on the production function, this implies that α 1 h = θ 1 α c µ 1 θ k 1 α h c Ã = β 1 α c (1 θ) µ θ h +1 δ! k or α 1 h = θ y 1 α h c 1 = β ³(1 θ) y +1 δ k (18) (19) We also that know that in steady state i = δk The depreciation rate is chosen to match the average investment to capital ratio in the economy. Using (19) and the average value for the output to capital ratio, β can be determined. Microeconomic evidence points toward a value of h.3. Finally, given the average value for y/c, α can be found using (18). As you can see, 11
17 calibrated values comes either from the requirement that the steady state values match the corresponding average values in the economy or from independently conducted microeconomic studies. To complete the calibration of the model, we need to determine values for ρ and σ ε (see (2)). Using (1), we see that z t+1 z t =(LnY t+1 LnY t ) θ (LnH t+1 LnH t ) (1 θ)(lnk t+1 LnK t ) (20) From there, the series of {z t } observed in the economy can be calculated. The residuals calculated are quite persistent, hence a very high value for ρ is generally retained (typically ρ =.95). Knowing ρ, the standard deviation of the error terms (σ ε )canbedetermined. 1.3 Defining and measuring the business cycle Business cycles are generally considered as a deviation from a trend. The question is then to define the trend. Once the trend is known, fluctuations can be easily calculated. Several method can be used (linear trend, piecewise linear trend). Most authors use a technique known as the HodrickPrescott filter. Consider the series of real output {y t } for example. It can be decomposed as the sum of a growth component y g t and a cyclical component y c t. The problem is to choose a trend to minimize the cyclical component, while still retaining a smooth trend. In other words, the problem is equivalent to Min s.t. TX (yt c ) 2 t=0 TX t=0 y g t+1 yg t y g t yt 1 g 2 not too big Take λ 10 a parameter reflecting the relative variance of the growth component to the cyclical component. Then the problem is to choose {y g t } to minimize the loss function Min {y g t } TX TX (y t y g t ) 2 + λ t=0 t=0 y g t+1 yg t y g t y g 2 t 1 The point of the exercise is to trade off the extent to which the growth component tracks the actual series against the smoothness of the trend. Please notice that for λ =0, y g t = y t and for λ +, thegrowth component is a purely linear time trend. For quarterly data, a value of λ =1, 600 is generally retained 11.Using this method, we can get a smooth time varying trend. The rational behind that choice is that it eliminates fluctuations at frequencies lower than eight years (business cycles are generally considered as fluctuations around the growth path occurring at frequencies of three to five years). 10 Penalty coefficient. 11 λ =400for annual data. 12
18 The standard detrending procedure is the following. Let X t be the series of interest. Take LnX t. HPfilter LnX t. Take the standard deviation of filtered LnX t. This is the percentage standard deviation of X t. The relative percentage standard deviations are usually relative to the percentage standard deviation of GDP 12. Once the U.S. time series have been detrended, business cycle facts can be presented. Several measures are of interest. First, we can look at the amplitude of fluctuations in the data. Second, we can also measure the correlation of aggregate variables with real GDP. That allows us to verify if a particular variable is proor countercyclical with respect to y t. Third, we can look at crosscorrelation over time to see if one variable tends to lead or lag another variable. Below is a table summarizing the U.S. business cycle data from 1954 to Variable x t Std Dev Corr (x t 1,y t ) Corr (x t,y t ) Corr (x t+1,y t ) GNP 1.72% Consumption: NonDurables & Services.86% Investment 5.34% NonFarm Hours 1.59% Productivity 13.90% The magnitude of fluctuations in output and hours are similar. This confirms the general consensus that the effects of business cycles are most clearly felt in the labor market. Consumption is the smoothest of the series 14. Investment fluctuates the most. Productivity is slightly procyclical, but varies less than output. This can now be compared with the same measures as simulated with the model. The model has been simulated 100 times, each simulation lasting for 150 period long (or the length of the observation period) 15. The simulated data were HPfiltered to give the same representation as the U.S. data. The results of the simulated economy are provided below: 12 Take a series X t =(1+α t ) T t. T t represents the trend component, while α t represent the percentage deviation from trend. Hence, after taking logs, LnX t = Ln (1 + α t)+lnt t.ifoneweretohpfilter LnX t, one would get a trend term and a deviation term. What the procedure explained above amounts to, is to isolate the deviation term. By HPfiltering LnX t, one tries to pick up Ln (1 + α t ) α t. 13 GNP/Hours 14 This should not be surprising given the concavity of the utility function. 15 For each simulation, the first observations have been discarded, in order to get rid of dependence on initial values. However, the simulation still produced 150 observations. 13
19 Variable x t Std Dev Corr (x t 1,y t ) Corr (x t,y t ) Corr (x t+1,y t ) GNP 1.35% Consumption: NonDurables & Services.33% Investment 5.95% NonFarm Hours.77% Productivity 16.61% Performance of simulated model: One question can be answered using these simulations: assuming the economy is a perfectly competitive one, how much of the variations in output can be explained by optimal adjustments to purely real shocks to the productive capacity of the economy? This question is an interesting one, because before the advent of RBC theory, all models assumed that the fluctuations were due to nominal (monetary) shocks, and fluctuations were how an economy with market imperfections reacted to these nominal shocks. In that sense, RBC is a radical departure from previous literature. In the artificial economy, output fluctuates less than in the U.S. economy, but still a large share of the fluctuations can be accounted for, without assuming any kind of market imperfections. The investment time series is very volatile, both in the U.S. and in the artificial economy. All times series are procyclical in the U.S. and this is reflected in the model economy. This, however, is not surprising, given that there is only one source of uncertainty in the economy, z t. The model does not perform as well, when looking at hours of work or productivity, which suggests that some elements of the labor market are missing. Another point that the model is missing is the fact that the consumption series is not volatile enough in the model economy. Interpretation of the model: There are several channels through which a shock is propagated in the economy. The shocks considered affect the productive capacity of the economy, and they are propagated by the manner in which optimizing agents (at the micro level) react and alter their economic decisions (investment and consumption). Because their utility functions are concave (they are risk averse), households smooth out their consumption throughout their lifetime, so that a change in output will manifest itself partly through a change in consumption and partly through a change in investment. As households try to avoid wide fluctuations in consumption, it is not surprising to find in the model and in the data, that investment is more volatile than consumption. Of course, variations in investment now, affect future output. Hence shocks are transmitted through time. Finally, households substitute leisure across periods, in response to a rise/decrease in wages in this period (due to a rise/decrease in z t, and thus in labor productivity). It is interesting to point out the role of capital accumulation. Suppose, for the sake of argument, that output is only a function of labor (i.e. abstract away from capital). Then the household s problem becomes static. Assume that a positive productivity shock hits the economy. Then, output y t and wage w t increase proportionately and a change in z t has the same secular effect as a trend. Then, the income and substitution effects cancel out and labor h t is constant (but consumption c t increases in proportion to w t ). With capital, 16 GNP/Hours 14
20 however, investment would increase and consumption would not increase as much. Thus, it is also efficient to lower consumption and raise labor hours relative to the noinvestment case. Another point that is often stressed in the RBC literature is how the persistence of productivity shocks affects the model. Suppose ρ =0. Assume a onetime (temporary) shock to the economy. The marginal productivity of labor increases this period, and the representative household faces an unusually high opportunity cost of taking leisure this period. While there are offsetting income and substitution effects, the model s preferences were chosen so that a permanent increase in the real wage generates exactly offsetting income and substitution effects, so that labor is left unchanged following such an increase. An implication is that labor has to rise in response to a temporary productivity increase. With a temporary shock, there is a much smaller income effect and there is great incentive to substitute intertemporally, since the current wage is high relative to expected future wages. On net, the positive labor response amplifies the productivity shock. The agents must decide what to do with this additional income. One possibility is to consume it all in one period. This would be inefficient, given that the marginal utility of consumption is decreasing, thus inducing a preference for smooth consumption paths. It is optimal to increase consumption both today and tomorrow. When there is serial correlation in the productivity shocks (ρ >0),the same mechanism is at work,but the effects are drawn out over time. Conclusion: In conclusion, the artificial economy performs relatively well, but can be improved along certain lines, particularly the labor market. Also, the model can be generalized by adding new sources of uncertainty, such as government spending shocks or monetary shocks, or any other type of shock. We will look at how the model can be improved by adding new shocks, how it performs when including monetary shocks, and how it can be modified to capture essential aspects of the labor market. Although we will not study the case, RBC theory can be used to study economies with heterogeneous agents 17. You should retain from this chapter that RBC theory is really a rigorous methodology and is flexible enough to study a lot of problems. 1.4 An application of RBC theory We present here a paper entitled Variance properties of Solow s productivity residual and their cyclical implications (Finn, JEDC 1995). It is intended first as an application of the methodology just learned, but also to show the concept of RBC can be extended to very different situations. We will see other examples of that later, when we study the labor market. The question is what constitutes a technology shock?. Theoretically, it is any real shock that influences the productive capacity of an economy, since it enters as a multiplicative factor in the production function. When we compute the Solow residuals, we can obtain time series for the {z t }, but it does not tell us what these z t stand for. Weather shocks may be considered as (negative) technology shocks (it decreases the output produced given, a certain amount of labor and capital inputs). What else may constitute a temporary 17 Agents characterized by age or skill, or subject to idiosyncratic shocks. 15
21 technology shock? The paper looks at how energy price shocks 18, which are not part of the standard model, affect the economy. The impetus for the paper is the observation that the correlation between the growth rates of the Solow residual and oil prices is 0.55 and the correlation between the growth rates of the Solow residual and total government spending is It seems that the Solow residuals, as computed, may include more than pure technology shocks and are influenced by such events as energy price shocks and to a lesser degree, government spending shocks. The paper investigates if energy shocks influence economic outcomes, in a way that cannot be captured by a production function, whose only inputs are labor and capital. The channel through which energy prices influence the productive capacity of the economy is capital utilization. The capital rented by firms can now be used more or less intensively. Energy costs (and depreciation) depend on how intensively the machinery is being used. For example, one may expect that if energy prices increase, capital utilization (hours of service per period or speed of utilization per hour) will decrease and hence output will decrease. The model makes the standard assumptions that the economy is perfectly competitive, with a representative firm, a representative household and a government. The production function exhibits constant returns to scale, and labor augmenting technological change. The utility function has constant relative risk aversion and unitary intertemporal elasticity of substitution. The defining characteristics of this model is variable energy costs, as well as variable depreciation costs. Energy prices are exogenous (open economy). There is endogenous capital utilization, that is it is left to the agent to optimally choose how intensively to use their capital. The economy is hit by stochastic shocks to technology, energy prices and government spending. Firm s problem: The production function is CobbDouglas in labor input (l t ) and capital services. Capital services is equal to the physical capital rented (k t ) times the rate of capital utilization (h t ). Capital utilization is defined as hours of service per period or speed of utilization per hour. Firms pay for what they use in production: l t and (k t h t ). Hence, firms maximize profits, taking wage rates (w t ) and capital services rental rates (r t )asgiven: Max l t,(k t h t ) y t w t l t r t (k t h t ) where y t = f (z t l t,k t h t )=(z t l t ) θ (k t h t ) 1 θ Household s problem: The household maximizes its expected discounted lifetime utility, subject to its budget constraint. The novelty is that energy costs and depreciation are functions of how intensively the capital is being used. In particular, both energy costs and depreciation are increasing and convex functions of capital utilization. Higher rate of utilization increases wear and tear and causes capital to depreciate faster. Using capital more intensively also 18 Oil price shocks come in mind immediately. 16
22 increases energy costs. Because of wear and tear, the machinery is not as energy efficient, so that a higher rate of capital utilization also increases energy usage faster. Hence, we have k t+1 = [1 δ (h t )] k t + i t δ (h t ) = hω t ω ω 1 This is the law of motion with endogenous depreciation (i t is investment at date t). Energy costs per unit of capital are given by e t = a (h t )= hν t k t ν ν 1 Hence, the household s maximization problem is ( + ) X MaxE 0 β t (Lnc t + γln(1 l t )) t=0 s.t. w t l t +(1 τ) r t k t h t = c t + i t + x t + p t e t where c t is consumption, x t is a lumpsum tax and p t is the exogenous energy price. τ is a tax on capital income. Government: The economy is subject to exogenous government purchases g t, and the government budget balances every period: g t = τr t k t h t + x t Stochastic nature of the economy: Lnz t+1 = Lnz t + Lnz + u z,t+1 Lng t+1 = ρ g Lng t + 1 ρ g Lng + ug,t+1 Lnp t+1 = ρ p Lnp t + 1 ρ p Lnp + up,t+1 g t = g t z t 0 < ρ g,ρ p < 1 u t = u z,t u g,t E (u t ) = 0 u p,t 17
23 Where u is governed by a Markov process Φ (u t+1 u t ). Hence Lnz isthemeangrowthofz t, Lng is the mean of Lng t,andlnp is the mean of Lnp t. Since z t is not stationary and g t is stationary, g t is also not stationary (it grows with the size of the economy). Movements in z t generate permanent movements in g t, but movements in g t only generate temporary fluctuations in g t. Defining the recursive competitive equilibrium: The firms chooses l t and (k t h t )suchthat: w t = f 1 (z t l t,k t h t ) r t = f 2 (z t l t,k t h t ) Using dynamic programming to solve the household s problem, we have to define the problem s state and control variables: Individual state variables: k t Aggregate control variables: K t, u t Control variables: l t, h t, k t+1 The Bellmann s equation can be written as: V (k t,k t,u t )= Max {u (c t,l t )+βe t [V (k t+1,k t+1,u t+1 ) u t ]} l t,h t,k t+1 Definition: A recursive competitive equilibrium is a list of aggregate laws of motion L t (K t,u t ), H t (K t,u t ), K t+1 (K t,u t ), individual decision rules l t (k t,k t,u t ), h t (k t,k t,u t ), k t+1 (k t,k t,u t ),factorpricesw t (K t,u t ), r t (K t,u t ) and lumpsum taxes x t (K t,u t ), such that: 1) Firms maximize and markets clear: w t (K t,u t )=z t f 1 (z t L t (K t,u t ),K t H t (K t,u t )) r t (K t,u t )=f 2 (z t L t (K t,u t ),K t H t (K t,u t )) 2) Household maximize, i.e. Bellmann s equation is satisfied by l t (k t,k t,u t ), h t (k t,k t,u t ), k t+1 (k t,k t,u t ) 3) Consistency of individual and aggregate behavior: l t (K t,k t,u t )=L t (K t,u t ) h t (K t,k t,u t )=H t (K t,u t ) k t+1 (K t,k t,u t )=K t+1 (K t,u t ) 4) The government budget balances, i.e.: g t = x t (K t,u t )+τr t (K t,u t )K t H t (K t,u t ) 18
24 Once the equilibrium has been defined, one can look at the first order conditions. The first order condition with respect to labor, l t, capital utilization h t,andk t+1 are: w t u 1 (c t,l t ) = u 2 (c t,l t ) (FOC[l t ]) Marginal benefit of consumption = Marginal disutility of working 1 more hour (1 τ) r t k t = δ 0 (h t ) k t + p t k t a 0 (h t ) (FOC[h t ]) Aftertax marginal return to an increase in h t = Marginal depreciation+marginal energy cost u 1 (c t,l t )=βe t {u 1 (c t+1,l t+1 ) [(1 τ) r t+1 h t+1 +1 δ (h t+1 ) p t+1 a (h t+1 )]} (FOC[k t+1 ]) This last condition states that the marginal benefit of current consumption is equal to the marginal benefit of consumption next period, with returns from delaying consumption. Remark that, in the usual case, the last FOC would be: u 1 (c t,l t )=βe t {u 1 (c t+1,l t+1 ) [(1 τ) r t+1 +1 δ]}. In summary, the equilibrium is defined by 13 equations: 1) Firm s labor efficiency condition 2) Firm s capital services efficiency condition 3) Household s labor efficiency condition 4) Household s capital utilization efficiency condition 5) Household s capital accumulation efficiency condition 6) Domestic resource constraint 19 7) Production function equation 8) Law of motion of capital 9) Energy usage constraint 10) Government budget constraint 11) Stochastic equation governing technological shocks 12) Stochastic equation governing energy price shocks 13) Stochastic equation governing government spending shocks Now that the equilibrium equations have been established, we can see how energy shocks are diffused through the economy. However, it is VERY IMPORTANT to recognize that, given the general equilibrium nature of the model, all the effects of a particular type of shocks are taking place simultaneously. Hence, it may be difficult to disentangle these effects. Nonetheless, it is interesting to figure out all the channels through which a shock affects the economy. Ultimately, it will be necessary to simulate the model to account for the general equilibrium effects. Diffusion of a positive energy price shock: 19 Obtained by combining the household s budget constraint and the governemnt budget constraint: y t p t e t = c t + i t + g t. 19
25 Assume that there is positive energy price shock, i.e. that: p t This creates a negative income effect (from the domestic resource constraint). Consumption and leisure being normal goods, this implies that c t l t The capital utilization efficiency condition, and the fact that depreciation and energy costs are convex functions of utilization, imply that h t Also, due to the fall in capital utilization, y t is reduced, which enhances the income effect 20.Inthatsense,a positive energy price shock is equivalent, in a way, to a negative technology shock. In conclusion, positive energy price shocks tend to reduce capital utilization. Diffusion of a positive government spending shock: Assume a positive government spending shock. This again results in a negative income effect and c t l t g t If labor increases, the marginal productivity of capital services increases and from the capital utilization efficiency condition and the convexity of depreciation and energy costs h t Finally, with h t and l t increasing, total output produced y t tends to increase, which has a countereffect on the negative shock due to increased government spending 21. In conclusion, positive government spending shocks tend to increase capital utilization. The objective of the project is to determine a true technology shock, i.e. a measure of an economy s ability to produce output from a given quantity of inputs, regardless of energy prices or government spending. The usual calculations of Solow residuals would give us LnSR t =[ Lny t θ Lnl t (1 θ) Lnk t ] /θ 20 Notice that, due to the general equilibrium effects, the marginal productivity of labor diminishes and the household s labor efficiency condition implies that households substitute labor for leisure, hence that l t tends to decrease (mitigating the increase in l t previously mentioned). 21 Again to account for the general equilibrium effects, with labor increasing, the marginal productivity of labor decreases (mitigated by increase in h t ) and from the substitution effect, l t decreases {w t is the price of leisure, os that if w t decreases, leisure increases). 20
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