Macroeconomics II Problem Set III Solutions PDF

Problem Set III Macroeconomics II Solutions Stefano Maria Corbellini 1 Solution : Ramsey model, General Equilibrium Conditions Consider a a neoclassical growt...

Problem Set III Macroeconomics II Solutions Stefano Maria Corbellini 1 Solution : Ramsey model, General Equilibrium Conditions Consider a a neoclassical growth model (Ramsey model) with infinitely lived agents. The representative household has utility such that u′ (·) > 0 and u′′ (·) < 0 and each period has an endowment of 1 unit of labor. The representative firm has a constant return to scale production function. 1. Write down the problem of the representative household and derive the optimality conditions. Solution: The household solves: ∞ X Intertemporala de la beta la t max β t u(ct ) (1) t=0 s.t kt+1 = kt (1 + rt − δ) + wt − ct + πt (2) T ′ lim β u (cT )kT +1 = 0 (3) T →∞ The Lagrangian writes: transversality condition is an optimality condition. ∞ X household will ask for this aprticiular condition. β t [u(ct ) + λt (kt (1 + rt − δ) + wt + πt − ct − kt+1 )] (4) t=0 The first order conditions with respect to consumption and capital are: u′ (ct ) = λt (5) λt = β(1 + rt+1 − δ)λt+1 (6) From which the Euler equation follows: u′ (ct ) = β(1 + rt+1 − δ) (7) u′ (ct+1 ) The three otpimality conditions of the household are the uler equation, the budget constraint and the transversality condition: u′ (ct ) = β(1 + rt+1 − δ)u′ (ct+1 ) (8) kt+1 = kt (1 + rt − δ) + wt − ct + πt (9) lim β T u′ (cT )kT +1 = 0 (10) T →∞ where (8) is the euler equation, (9) the law of motion of capital and (10) the transversality condition of the household. 2. Write the problem of the representative firm and derive the optimality conditions Macroeconomics II Problem Set III Solution: The firm simply maximizes its profit function choosing the input factor needed for production: maxKt ,Lt f (Kt , Lt ) − wt Lt − rt Kt. This yields equilibrium factory payments plus profits: wt = fL (Kt , Lt ) deriv lui f in raport cu L (11) rt = fK (Kt , Lt ) deriv lui f in raport cu K (12) πt = f (Kt , Lt ) − wt Lt − rt Kt (13) profitul 3. Write the market clearing conditions Solution: cifra afaceri - profit From the market clearing condition we can pin down Lt and Kt : Lt = 1 scoatem L si K (14) Kt = kt (15) 4. Combine conditions recovered in points 1,2,3 to derive the two core equations that characterize the dynamics of the economy Solution: From (9), (11), (12), (13), (14) and (15) we can find the resource constraint of the economy: kt+1 =kt (1 + rt − δ) + wt − ct + πt =kt (1 + rt − δ) + wt − ct =kt (1 − δ) + f (kt , 1) − ct (16) Where in the second lines I used πt = 0 and in the third one πt = 0 together with (13). From (8), (12), (14) and (15) we can rewrite the euler equation of the representative household as the second core equation: u′ (ct )=β(1 + rt+1 − δ)u′ (ct+1 ) =β(1 + fK (Kt+1 , Lt+1 ) − δ)u′ (ct+1 ) u′ (ct )=β(1 + fk (kt+1 , 1) − δ)u′ (ct+1 ) (17) The general equilibrium is then defined by the inital k0 together with (16), (17) and the transver- sality condition from (10). For a given k0 the choice for c0 pins down k1 through the resource constraint, which pins down c1 through the euler equation, and so on. This yields {ct , kt+1 }∞ t=0 , which pins down the remaining variables rt , wt ,.... However, note that the initial choice for c0 is not free but must satisfy the transversality condition from (10). 2 Solution : Ramsey model, General Equilibrium: Steady state Consider the infinitely lived representative agent model analyzed in 1. Suppose that utility is of the CRRA type, c1−σ t −1 u(ct ) = , 1−σ and production is characterized by a Cobb-Douglas function, f (Kt , Lt ) = Ktα L1−α t , 2/12 Macroeconomics II Problem Set III or, in per capita terms, f (kt , 1) = ktα. 1. Derive and plot the steady state resource constraint and the steady state Euler equation (with k on the horizontal axis of the diagram and c on the vertical axis). Solution: we remove the kt+1 = (1 − δ)kt + f (kt , 1) − ct time indexes in the steady u (ct ) = β(1 + fk (kt+1 , 1) − δ)u′ (ct+1 ) ′ state index Evaluated at the steady state values these conditions simplify to the following to equations: c̄ = f (k̄, 1) − δ k̄ (18) 1 = β(1 + fk (k̄, 1) − δ) (19) For the production function and utility function given in the question, (18) and (19) become c̄ = k̄ α − δ k̄, 1 = β(1 + αk̄ α−1 − δ). for everyone in consumption, a single k' The plot of the two nullclines is shown in figure 1. leading to a songle vertical line Figure 1: Nullclines 2. Solve for the steady state values of consumption and capital (the modified-golden-rule capital stock). Solution: From (19) we solve for the steady state capital k ss 1 = β(1 + fk (k ss , 1) − δ) 1 = β(1 + α(k ss )α−1 − δ) I divide by β   1 ss 1 − β(1 − δ) α−1 k = (20) αβ 1/β=1+αk- 3/12 Macroeconomics II Problem Set III We can use the result from (20) together with (18) to find the steady state consumption level: css = (k ss )α − δk ss   α   1 ss 1 − β(1 − δ) α−1 1 − β(1 − δ) α−1 c = −δ (21) αβ αβ 3. Derive the golden-rule capital stock. Show that the modified-golden-rule capital stock is necessarily smaller than the golden-rule capital stock. Solution: The golden rule capital stock is defined as the capital stock that maximizes steady state consump- tion: we maximize f(k)-delta k k gr = arg max css k argument of maximum... = arg max f (k) − δk k = arg max k α − δk solution to the optim problem k Taking the first order conditions of the expression above, we obtain αk α−1 − δ = 0, that yields
1 k gr = αδ α−1. Now let’s compare the two capital levels: ? k gr > k ss   α−1 1   1 δ ? 1 − β(1 − δ) α−1 > α αβ  1  1−α   1−α 1 1 − β(1 − δ) ? δ > αβ α 1 − β(1 − δ) ? >δ β ? 1 − β(1 − δ) > βδ 1>β that holds, as β ∈ (0, 1). 4. Why does steady state consumption fall short of consumption at the modified-golden-rule capital stock (although the equilibrium is Pareto efficient)? Solution The consumer does not maximize lifetime consumption but rather lifetime utility. Because future consumption is discounted at rate β the consumer would be better of, if she consumed some of the extra capital rather then investing it. This will increase her current consumption at the cost of higher future consumption, which is optimal from a utility perspective. 3 Solution : Ramsey model, General Equilibrium: Phase diagram and model dynamics Consider the neoclassical growth model from exercise 2. 1. Show the dynamics within the phase diagram and draw the saddle path. Solution: 4/12  Macroeconomics II Problem Set III The resource constraint from (18) indicates the curve, for which capital stays constant, i.e. it depicts the nullcline for capital. Equivalently, the euler equation from (19) defines the level of capital for which consumption is constant, i.e. the nullcline for consumption. These optimality conditions allow us to study the dynamics of the economy when it is not in the steady state, for example after a technology shock. From the resource constraint and the euler equation we can derive the dynamics of consumption and capital, which in turn will give us all remaining variables given an inital level of capital. From the aggregate resource constraint: kt+1 = (1 − δ)kt + f (kt , 1) − ct Hence: kt+1 f (kt , 1) ct = − +1−δ (22) kt kt kt From the Euler equation: u′ (ct ) = β(1 + fk (kt+1 , 1) − δ)u′ (ct+1 ) Then: c−σ t α−1 = β(1 + αkt+1 − δ)c−σ t+1 And finally: ct+1 α−1
1 = β(1 + αkt+1 − δ) σ (23) ct kt+1 ct+1 Realise that kt = 1 (i.e. ∆k = 0) and ct = 1 (i.e. ∆c = 0) if and only if capital and consumption are at their steady state levels. For all other combinations of {kt , ct } the economy is dynamic and moving along some optimal path. To figure out the dynamics of this economy we can study the two equations and find out how the economy develops. The simplest way to do this, is with the help of a phase diagram shown in figure 2. The little arrows, indicate the direction and speed at which the economy is changing. We can Figure 2: Phase Diagram de repretat acasas dynamic graphs quantitative advanced macroeconomcis no parameter change could mean that the steady state (=1) would be equal to the initial state divide the graph into four different areas around the steady state to study the evolvement of the economy, where we refer to the values that lie on the nullcline as k ∗ and c∗ : kt+1 North-east: This means that kt > k ∗ and ct > c∗ , from (22) and (23) we know that kt k ∗ and ct < c∗ , from (22) and (23) we know that kt >1 ct+1 and ct 1 ct+1 and ct >1 Together these observations help us to find out where the saddle path must lie. The saddle path is a (unique) path that shows how the economy converges to the steady state. The saddle path is pinned down by the transversality condition and the equations (22) and (23). All other paths violate the transversality condition and are therefore not optimal. This implies that for a given k0 consumption c0 is a jump variable and chosen such that the bundle {k0 , c0 } lies on the saddle path. 2. Suppose that the economy is in the steady state. How do kt+1 , ct , wt and Rt respond to the following, somewhat model-inconsistent shocks? Draw the adjustment path. Moreover, explain the adjustments from the household’s point of view. (a) An earthquake destroys some of the initial (steady state) capital stock. Solution: The destruction of capital stock reflects a loss of wealth and additionally it lowers the income because less capital can be rented out to firms. Further, the marginal product of labor de- creases, which lowers the wage. Although capital can be rented out at a higher price, this does not make up for the losses and the household drops the consumption level. As capital can be rented out a high price, rt is large, the consumer accumulates capital through saving. Addi- tionally, the high rental rate allows the consumer to spend more on consumption. This implies that both capital and consumption is increasing. As the capital stock is slowly recovering the rental rate is decreasing, and the wage is increasing, until eventually the economy is back at the steady state and ct+1 = ct , kt+1 = kt. In figure 3 we see the dynamics of the economy after the shock. Note the jump in consumption immediately after the economy is hit by the shock. 6/12 Macroeconomics II Problem Set III Figure 3: Evolution of the economy after destruction of capital stock MP_K decreases because it is inversely re;ated to capital 7/12 Macroeconomics II Problem Set III (b) There is a onetime, permanent increase in technology a, i.e. the production function changes from f (kt , 1) to a · f (kt , 1), with a > 1. Solution: The unexpected increase in technology leads to a new steady state in consumption and capital. With technology a in the production function, the two equations to study the dynamics become: kt+1 = (1 − δ)kt + af (kt , 1) − ct u′ (ct ) = β(1 + afk (kt+1 , 1) − δ)u′ (ct+1 ) Substituting for the Cobb-Douglas formulation: kt+1 = (1 − δ)kt + aktα − ct u′ (ct ) = β(1 + aαkt+1 α−1 − δ)u′ (ct+1 ) So in steady state: 1 1 ! α−1 β − (1 − δ) k= (24) aα c = ak α − δk (25) α 1 1 ! α−1 1 ! α−1 β − (1 − δ) β − (1 − δ) =a −δ = (26) aα aα  α 1  1 ! α−1 1 ! α−1 1 β − (1 − δ) β − (1 − δ) = a 1−α  −δ  (27) α α For a changing from 1 to a level greater than 1, we have greater capital and consumption in steady state. Because firms, have access to an improved technology in the production process, the steady state capital is larger than the previous level. The increase in the capital steady state, makes labor more productive which leads to a higher wage. Together, this increases the income of the household, which allows it to spend more on consumption and increases the steady state level of consumption. Immediately after the improvement in the technology we see either a jump (wealth effect dominating) or a fall (substitution effect dominating) in consumption. Expecting higher wages and capital in the future makes the house more wealthy and drives it to consume more today (wealth effect). However, the higher productivity of capital drives the household to save more and consume less (substitution effect). The effect that dominates is the one that actually determines whether consumption initially jumps or falls. As labor income and rental rate go up, the households experience an increase in income which allows it to spend more on consumption. Due to the high return on capital, consumers save some of the additional income and invest in a higher capital stock. As they accumulate capital, the return on capital is slowly decreasing. However, the additional capital makes labor more productive which increases the households labor income. Over time, we see that capital is accumulated until the rental rate has returned to its previous level. In fact the steady state rental rate of capital does not depend on a: 1 r = aαk α−1 = − (1 − δ) (28) β where the second equality is obtained by substituting k for (24). The steady state wage instead 8/12 Macroeconomics II Problem Set III depends positively on a: α 1 ! α−1 α α−1 α 1 β − (1 − δ) w = f (k, 1) − kfk (k) = ak − kaαk = ak (1 − α) = a 1−α (1 − α) (29) α Figure 4 and 5 report the dynamics of the economy according to the dominating wealth effect or dominating substitution effect case. Dynamics of capital and consumption are in red, initial nullclines and saddle path are in black and new nullclines and saddle path are in blue. Figure 4: Evolution of the economy after a permanent technology shock. Wealth effect dominating. Figure 5: Evolution of the economy after a permanent technology shock. Substitution effect dominating. (c) There is the announcement that, at certain time T in the future, there will be an increase in technology a. The increase indeed realizes at time T. Solution: The dynamics of the economy are illustrated in Figure 7 (wealth effect dominating) and 8 (substitution effect dominating). At the time of the announcement, the dynamics are still determined by the old nullclines (the black ones). Again, consumption jumps upwards or downwards according to the wealth or substitution effect dominating. However, it jumps in order to arrive onto a divergent trajectory such to bring the economy on the new saddle path when the productivity increase will be realized. The divergent trajectory features, in the case of wealth effect dominating, a gradual disinvestment in capital in favor of current consumption, which decreases the marginal productivity of labor (the wage) and increases 9/12 Macroeconomics II Problem Set III Figure 6: Evolution of the economy after a permanent technology shock (substitution or wealth effect dominating). Pattern of rental rate of capital and wage. the marginal productivity of capital (the rental rate). The opposite holds in the dominant subsitution effect case, where accumulation of capital lowers consumption and rises the wage, while depressing the rental rate. From period T - when the shock realizes - the economy is on the new saddle path: from that time onwards, the description about the evolution of kt+1 , ct , wt , Rt coincides with the case of the unexpected productivity shock. 10/12 Macroeconomics II Problem Set III Figure 7: Evolution of the economy after the announcement of a permanent technology shock, which actually realizes in the future.. Wealth effect dominating. you want to save more and consume less if the substitution effect because the new saddle path shifts to right 11/12 Macroeconomics II Problem Set III Figure 8: Evolution of the economy after the announcement of a permanent technology shock, which actually realizes in the future. Substitution effect dominating. 12/12