Solutions to Macro Final 2006

Transcription

1 Solutions to Macro Final 6 th December 6 1 Problem Part A Rewrite the utility function as U = ln(n) + ln (c) γ ln ( c) Notice that since the agent taes c as a constant, it will not factor into the optimization, which implies we are really just maximizing U = ln(n) + ln (c) which implies the agent will tae half his time as leisure and half as labor, or n = 1 and c = Part B The social planner nows that in the end c = c, so the plugging this in at the beginning yeilds ln(n) + (1 γ) ln (c) Again, by the standard homothetic utility function, the agent will spend 1 γ on labor, yeilding n = γ, c = (1 γ) γ 1.3 Part C of her time on labor and 1 γ γ You should realize that dierent values of γ have dierent economic meanings. More specically, the value of gamma measures the direction and intensity of the externality corresponding to consumption. Case 1: γ = If γ =, this implies that there is no externality to consumption, and therefore the social planner's solution and the individual's solution coincide Case : γ < This case corresponds to a positive externality of consumption. We may thin of this as people gaining positive utility from seeing their neighbors enjoying high consumption, or high utility from the fact that there is low poverty in society, or being happy for their neighbor's good fortune, etc. Notice that the social planner choses more consumption in this case than the individual would on her own. Case 3: γ > This case corresponds to a negative externality of consumption. We may thin of this as people gaining negative utility from seeing their neighbors enjoying high consumption, or being jealous of others, habit formation, or trying to eep up with the Jones'. Notice that the social planner choses less consumption in this case than the individual would on her own. 1

2 Problem In order for the woman to be indierent between the two wage proles, it must be the case that her lifetime utility under both proles be equal. The rst thing to notice is that if she could borrow, she would prefer to frontload consumption. However, since she is constrained, the best she can do is to consume her wage at each point in time. Therefore, w t = c t for all t. We now that for either wage prole, w t = w e gt. The lifetime utility of this path of consumption is U = c 1 σ t 1 σ e θt dt = This implies that to be indierent, we must have 1 c t e θt dt = 1 w A(θ + g A) = 1 w B(θ + g B) w A w B w A (θ + g A ) = w B (θ + g B ) = θ + g B = θ + g A. +.4 = e (g+θ)t 1 dt = w w (θ + g) This maes sense that w A > w B, since A grows slower that B, and therefore the agent must be compensated with a higher starting wage. 3 Problem 3 Intuitively, a higher β implies a larger adjustment cost. It is important to realize that the = and q q = are the same for each country. To see this, notice that q q = r F q, since δ = and τ = initially. Setting this equal to will give us the q q = locus. q q = r F q = q = F r Since β does not eect r or F, the q q = locus must be the same for the two countries. As for the = locus, remember that In this problem = φ(q) δ = φ(q) = δ q = 1 + c ( ) i = 1 + β ( i φ(q) = i ( ) 1 q 1 = β ) This implies that the = locus is a horizontal line at the q which solves ( q 1 β ) 1 = δ q = 1 + βδ

3 Generally, if β is dierent for two countries, then q will be higher in countries with higher β. In this problem, however, δ =, so we have that q = 1 for all countries. Since the locii are the same for both countries, they have the same steady state both before and after the change in τ. Therefore, if there is a dierence in the size of the jump at time t, it must be because of a dierence in the stable arms in the two countries. In order to determine the properties of the stable arm, we must compare the speed of adjustment of both and q. First consider the speed of adjustment in q: q q = r F q Notice that β does not eect q q, so given any starting point, the vector of force acting upon q will be the same for both countries. Now we consider the speed of adjustment for : = φ(q) δ = ( q 1 Now higher β decreases the absolute magnitude, and the vector of force acting upon will be of lower magnitude for countries with higher β. Putting the two together, it implies that the stable arm will be steeper for a country with high β. Therefore, since they have the same steady state, the stable arm for country A must be below the stable arm for country B on the left of the steady state, and above to the right. Finally, when τ increases, it shifts the q q = locus to the right. Since capital cannot jump, we must have a jump in q to get us onto the new stable arm. Since the stable arm for A is below B, country A must jump down more. 4 Problem Solving Intuitively Notice that asset A is the same thing as two asset Cs and one asset B. Therefore, by a simple arbitrage condition, we must have that the price of A is the same as the price of two asset Cs and one asset B. Normalizing the price of asset A to 1, and calling x the price of B, and y the price of C, we have β ) 1 1 = y + x y = 1 x Notice that this is independent of the probaility of the states. 4. Solving Mathematically We solve the problem max γ ln (A + C) + (1 γ) ln (B + C) s.t. 1 + x + y = A + xb + yc A,B,C where γ is the probability of the rainy state. Setting up the Lagrangean we have L = γ ln (A + C) + (1 γ) ln (A + B) + λ (1 + x + y A xb yc) which yeilds FOC's L A = γ A + C + 1 γ A + B λ = L B = 1 γ A + B λx = L C = γ A + C λy = In equilibrium, we now that A = B = C = 1 since individuals will not trade since they are all identical. Therefore we have 3

4 γ γ = λ 1 γ = λx γ 3 = λy Plugging the last two equations into the rst equation, we have 5 Problem Part A λy + λx = λ y + x = 1 y = 1 x From our standard rst order condition from consumption, we have U (c t ) U (c t+1 ) = 1 + r 1 + θ However, since c t+1 is uncertain, the problem changes so that in expectation the same condition holds: We let θ = for simplicity. Therefore, U (c t ) EU (c t+1 ) = 1 + r 1 + θ e ct = (1 + r)e(e ct+1 ) From the rule he gave us, we have e ct σ E(ct+1)+ = (1 + r)e e c σ c+ = (1 + r)e 1 = (1 + r)e σ r = e σ 1 5. Part B Taing the derivative with respect to σ : r σ = 1 e σ < This implies that the interest rate becomes more negative as the variance increases. Notice that in this context, the interest rate is essentially how much people are willing to pay to store consumption from one period to another. If tomorrow's consumption is risy, but on average I will consume the same today and tomorrow, then the ris averse agent would want to transfer consumption from today to tomorrow in order to insure against bad outcomes. This implies that individuals will want to save as a form of insurance. However, if someone wants to save, someone else must borrow. But if agents are identical ris averse agents, we have an economy full of savers and no borrowers. Therefore, the equilibrium interest rate must fall until no one wants to borrow or lend. The higher the uncertainty (σ ), the more people want to save, and the lower the interest rate must be to intice them to not save. 4

5 6 Problem 6 We will solve this problem using two dierent methods, and will illustrate the second method using two dierent congurations of the parameters. 6.1 Solving Using Consumption This problem can be thought of as a standard consumption problem if we can translate the information given into an appropriate interest rate. From the rst order condition of a CRRA utility function, we have that ċ c = 1 (r θ) σ What is r? Remember that r = f () δ n. In this case, we can thin that on average, planting one grain of wheat will give us a gross 4% return, implying that the interest rate is 3% after we adjust for population growth. Therefore ċ c = 1 σ (r θ) = 1 1 (.3.1) =. c t = c e.t At time, the agent has an initial stoc of wheat (assets) W. Therefore, the budget constraint is W = c t e rt dt = c e.t e.3t dt = c e.1t dt = 1c c = 1 W 1 This ratio will be preserved since consumption and assets are both growing at %. To put this into perspective, if we have 1 grains of wheat, we consume one, plant the other 99 and get approximately 13, one of which is used for population growth. Therefore, we start the second period with approximately 1 grains of wheat. (the numbers would be exact if we were using continuous time). 6. Solving Using Growth We can specify the production function in one of two ways: y =.4 where δ =, or where δ = 1. Either way, we get just as before. y = 1.4 r = f () δ n =.3 Method 1: δ = Setting up the equation, we have = sy (d + n) = (.4s.1) =.4s.1 Notice that from the production function we have ẏ y = We must also have ẏ y = ċ c = f () δ n θ =. 5

6 otherwise our solution would not be optimal. Putting it all together, we get. = ċ c = ẏ y = =.4s.1.4s =.3 s = 3 4 To put this into perspective, if we have 1 grains of wheat, we plant them all and get 14 grains of wheat. Since there is no depriciation, we actually have only produced four grains of wheat. Saving three-fourths is equivalent to consuming one-fourth of our production, i.e. consuming one grain of wheat. With the remaining 13 grains, one of them is used for population growth, and we start the next period with 1 grains. This is exactly the same as the solution using the consumption method. Method : δ = 1 In this case we have = sy (d + n) = (1.4s 1.1) = 1.4s 1.1. = ċ c = ẏ y = = 1.4s s = 1.3 s = If we have a hundred grains, then production is 14, and we save 13 and consume 1. Another 1 goes to population growth, and so we are left with 1 to start the next period. Again, the same solution. 6