Graduate Macroeconomics 2 Problem set Solutions

Transcription

1 Graduate Macroeconomics 2 Problem set Solutions Question 1 1. AUTARKY Autarky implies that the agents do not have access to credit or insurance markets. This implies that you cannot trade across states at a given point in time and the economy is not endowed with a storage technology to transfer resources across time. Hence, the agents maximization problem is the following: max c i 1,ci,j 2 u(c i 1) + β π j u(c i,j s.t. c i 1 y i 1 i, c i,j 2 y j 2 j Since u( ) is strictly increasing and any unconsumed income would be lost (due to the lack of a storage technology), the agent will consume all his income in every period, in every state. Hence, at the optimum, both the constraints will be binding. This implies that c i 1 = y1 i and c j 2 = yj 2. Consumption in the second period will be independent of the realized income in the first period. Therefore, the expected lifetime value of an agent, U A, under autarky is given by the following expression: U A = u(y i 1) + β U A = u(y1) i + β π j u(y j N π j u(y j Since, we get that, u(y1) i = π j u(y j = E[u(y)] U A = (1 + β)e[u(y)] 2. SELF INSURANCE Given that the agent can only save in this economy (and not borrow), at a rate R = 1 β, his/her maximization problem is modified in the following manner: 1

2 max c i 1,ci,j 2 u(c i 1) + β π j u(c i,j s.t. c i 1 y i 1 i c i,j 2 = y j 2 + R(yi 1 c i 1) i, j Substituting the second constraint in the utility function and taking account of the fact that the agent cannot borrow, we get the following Lagrangian function which the agent maximizes with respect to c i 1: L = u(c i 1) + β π j u(y j 2 + R(yi 1 c i 1)) + λ i (y1 i c i 1) L c i 1 = u (c i 1) βr N u (c i 1) = βr π j u (c i,j λ i = 0 π j u (c i,j + λ i This implies that (remember, βr = 1): u (c i 1) π j u (c i,j The above equation states that the marginal utility of consumption today has to be greater or equal to the expected marginal utility of consumption tomorrow. When the agent saves, the third constraint is not binding. In this case λ i = 0, which implies that we get the standard Euler equation. In those states of the world, when the agent wants to borrow from the future to smooth his consumption but he cannot, the marginal utility of consumption today is strictly greater than the expected marginal utility of consumption tomorrow. In this case, the third constraint is binding, i.e. λ i > 0. You can intuitively think that in some states of the world, the agent s income would be low which would imply that his consumption is also quite low (since he cannot borrow). Hence, his marginal utility from an additional unit of consumption would be high. The inability of the agent to borrow restricts him from reaching his optimal consumption path. 3. COMPLETE MARKETS 2

3 The Lagrangian of the decentralized economy is: L = The FOCs are given by: {c i 1} : u(c i 1) + β π j u(c i,j + λ L c i 1 {c i,j 2 } : L c i,j 2 (y1 i c i 1) + β π j (y j 2 ci,j = u (c i 1) λ = 0 u (c i 1) = λ i = βπ j u (c i,j βπ j λ = 0 u (c i,j = λ i, j Hence u (c i 1) = u (c i,j i, j implying that ci 1 = c i,j 2 = c. Consumption is perfectly smoothed across periods and states. Complete markets therefore allow full insurance. The complementary slackness condition gives the budget constraint holding with equality. Plugging c i 1 = c i 2 = c into the BC gives: (y i 1 c) + β y1 i c + β π j (y j 2 c) = 0 N π j y j 2 β c = 0 Using that y1 i = π j y j 2 = E[y] We get that: E[y] c + βe[y] β c = 0 c =E[y] Consumption in each period equals expected income per period. We have seen in Lecture 11 that the social planner s optimum also features full insurance when there are complete markets. Consumption is therefore completely smoothed across periods and states and we get: c i 1 = c i 2 = c P i, j. We have to determine the value of U 0 such that the planner and decentralized economy solutions correspond, i.e. the expression for U 0 for which c = c P. To do this, we 3

4 replace c i 1 and c i 2 by c in the planner s constraint, which is binding in equilibrium: With c = E[y], we have: U 0 = u( c) + β u(e[y]) + β π j u( c) = U 0 U 0 = (1 + β)u(e[y]) π j u(e[y]) Recall that in autarky, we had: U A = (1 + β)e[u(y)] The agent is risk averse. In this case, the utility function is concave. We have u(e[y]) > E[u(y)] which implies U 0 > U A : total consumption smoothing is better for the agent than random consumption. Question 2 1. If the natural borrowing constraint is binding in period t in this economy and in period t + 1 the realization of income is y 1 then the consumer needs all his income to repay his debt, hence he can only consume 0. If the standard Inada condition which states that as c 0, the agent s utility u(c) holds, then the agent will choose never to hit the borrowing constraint in order to prevent u(c) =. Obviously if the realization of income y 1 is impossible in period t + 1, then the event described above cannot occur. We need to assume additionally that this is not ruled out: P r(y t+1 = y 1 y t ) > 0 for all y t. The other way to solve this problem is to redefine the natural borrowing limit in a more natural way which is φ t = 1 r min i I{y i P r(y t+1 = y i y t ) > 0}. 2. If the agent does not face any borrowing constraints then the Euler equation for this problem is u (c t ) = β()e t (u (c t+1 )). Assume that β() 1. Then this implies that u (c t ) E t (u (c t+1 )). 4

5 If we also assume that u is decreasing and convex, then by Jensen s inequality we have that u (c t ) E t (u (c t+1 )) u (E t (c t+1 )), and by monotonicity c t E t (c t+1 ). Using the budget constraint: a t+1 a t 1 (y t E t y t+1 + E t a t+2 a t+1 ). Iterating this forward, and assuming that the no-ponzi condition holds, we have that a t+1 a t j=0 If y t E t y t+1 0 then a t+1 > 0. 1 () j+1 (E ty t+j E t y t+j+1 ). Hence the conditions that we need to impose are: 1. β() 1 2. u is concave and u is convex 3. y t E t y t+1 4. the No-Ponzi condition holds: lim t 1 (1+r) t a t = 0 5. a 0 0 These conditions are quite intuitive: the agent is prudent and his stream of income is constantly shrinking so he has to save in order to smooth his consumption. An alternative sufficient condition for the borrowing constraint not to be binding is similar to part 1: as long as there is a positive probability in each period that y t = 0 and the utility function satisfies the Inada condition, the constraint will not be binding. Question 3 1. If c t is lognormal, then c γ t is lognormal as well. To see this observe that a lognormal c t with parameters µ t and σ t (in the exercise v t = σ 2 t is the variance parameter) can be written as c t = e µt+σtzt, where z t is a standard normal random variable. If z t is standard normal, then z t = z t is also a standard normal variable. 5

6 Now express c γ t as c γ t = (e µt+σtzt ) γ = e γµt γσtzt = e γµt+γσt zt which shows that c γ t is lognormal with parameters γµ t and γσ t. Now apply what we know about the mean of a lognormal distribution for both c t and c γ t : E t (c t ) = e µt+ 1 2 σ2 t ln E t (c t ) = µ t σ2 t = E t (ln c t ) v t E t (c γ t ) = e γµt+ 1 2 γ2 σ 2 t ln E t (c γ t ) = γµ t γ2 σt 2 = γe t (ln c t ) γ2 v t Having established this, let us look at the Euler equation for this model: c γ t = βre t (c γ t+1 ). Taking the log of this expression we get ln c γ t = ln βr + ln E t (c γ t+1 ). Now plug in what we know for ln E t (c γ t+1 ) to get: ln c γ t = ln βr γe t (ln c t+1 ) γ2 v t. Subtract E t (ln c t+1 ) from both sides to get: E t ln c t+1 = 1 γ ln βr γv t. 2. In this setup, we are just told that consumption is a lognormal variable, i.e. it is a black box, and we do not know the source of uncertainty of future consumption. Let s assume that the higher v t is the result of higher uncertainty in the income stream, y t, (keeping the mean constant). Then a higher v t leads to a lower consumption c t hence higher savings in period t. This means that the agent saves for precautionary reasons. and 3. According to the permanent income hypothesis c t is a martingale sequence, c t+1 = c t+1 c t = r ( ) j j = 0 1 (E t+1y t+1+j E t y t+1+j) 6

7 i.e. c t+1 only depends on the revision of permanent income from period t + 1 onwards. This implies that the y t should not have an affect on it. In our setup c t = E t c t+1 = e µt+0.5vt. In this case the PIH will hold. However if the correlation between income y t and v t is not zero, i.e. corr(y t, v t ) 0, then we will get α 1 α 1 0 in our regression since y t plays a role as a proxy for v t. Hence this test is not suitable for checking whether the PIH holds or not. This is due to the fact that for the PIH we need c t to be a martingale sequence, whereas here we are testing whether ln c t is a martingale. Question 4 Using the equations for the income process, we get y p t obtain: y t u t = y t 1 u t 1 + v t y t = u t + y t 1 u t 1 + v t. = y t u t. Using this we Using the change in consumption equation given in the lecture notes, we can express the change in consumption as a function of the permanent innovation v t and the transitory innovation u t. Rewriting the RHS of that equation we get: j=0 ( ) j [ 1 (E t E t 1) y t+j = (E t E t 1) y t + 1 ] y t Plugging in the above expression for y t in the first term of the RHS of the above equation we get: (E t E t 1 ) y t = E t y t E t 1 y t = y t (y t 1 u t 1 ) = u t + y t 1 u t 1 + v t (y t 1 u t 1 ) = u t + v t. In other words, the unexpected change in income y t (compared to the one-stepahead forecast E t 1 y t ) is the sum of the permanent and the transitory innovations at time t. In a similar way the second term in the summation can be expressed as: (E t E t 1 ) y t+1 = (E t E t 1 ) (u t+1 + y t u t + v t+1 ) = (E t E t 1 ) (u t+1 + (u t + y t 1 u t 1 + v t ) u t + v t+1 ) = (E t E t 1 ) (u t+1 + (u t + y t 1 u t 1 + v t ) u t + v t+1 ) = v t, 7

8 since all the terms indexed by t 1 drop out because E t (x t 1 ) = E t 1 (x t 1 ) = x t 1 and all the terms indexed by t + 1 drop out because E t (x t+1 ) = E t 1 (x t+1 ) = 0. It is easy to show that: (E t E t 1 ) y t+j = v t for all j > 1. In other words the forecast revision between time t 1 and t about income beyond time t equals the permanent innovation at time t. Now we can express the change in consumption as: c t = r [ = r u t + v t = r u t + v t. u t + v t + 1 ] v t +... ( ) j 1... Hence, households adjust their consumption responding to the annuitized change in income. This means that they will respond only weakly to purely transitory shocks (u t ), whereas they will respond one for one to permanent shocks (v t ). Indeed the former shocks have only a small effect on permanent income, while the latter change permanent income one for one. Question 5* j=0 Carefully explain why an efficient allocation must satisfy: c i(s) c i(s ) = c j(s) c j(s ) Proof by contradiction. Assume: c i(s) c i(s ) > c j(s) c j(s ) Consider the following allocation: transfer ɛ to consumer i from consumer j in state s and and ɛ in state s. Choose ɛ and ɛ such that consumer j is indifferent: c j (s) ɛ c j (s ) ɛ = 0 Then each consumer i will strictly prefer this allocation: 8

9 c i (s) ɛ c i (s ) ɛ = > > 0 c i (s ) c i (s ) ( Ui c i(s) c i(s ) c j(s) c j(s ) ɛ ɛ ) ɛ ɛ This demonstrates that if the condition is not satisfied, we can achieve a Pareto improvement. Question 6* Assume that there are no aggregate shocks and we are in an endowment economy, i.e. i ωi (s) = ω for every s. Show that in a Pareto optimal allocation, agents fully insure themselves, i.e. c i (s) = c i (s ) for every s, s. Proof by contradiction. A Pareto optimal allocation satisfies: Assume c i (s) < c i (s ). Since j we have: c i (s) c i (s ) = c j (s) c j (s ) (s) U i (c i (s)) π j (s) U j (c j (s)) = λi λ j we also have c j (s) < c j (s ). Given that this is true for all c j (s) < j j c j (s ) = ω In state s agents don t use all endowments. Pareto optimal. Therefore the allocation is not 9