ECON 60B Problem Set Suggested Solutions Fall 0 September 7, 0 Optimal Consumption with A Linear Utility Function (Optional) Similar to the example in Lecture 3, the household lives for two periods and the life time utility function is linear in consumption both in first and second periods, U = c today + β c future. The present value of the life time wealth is W. The discount factor is β and the interest rate is r.. Suppose ( + r) β >, find the optimal consumption plan for today and future. Explain the intuition of the result. Solution: The maximization problem is: max c today U = c today + β c future subject to c today + c future + r = W () We can transform the problem as follows: max U = c today + β ( + r) ( W c today ) c today The First-Order condition is: The Euler equation is: β ( + r) = 0 β ( + r) = However, since ( + r) β >, and the marginal utility of consumption is constant and equals, then the cost of deferring one unit of consumption good today (which equals ) is less than the benefit of deferring (which equals ( + r) β > ). Therefore, the optimal consumption plan is to defer the consumption to the future. That is, c today = 0, and c future = ( + r) W.
. Does the Euler equation hold in this case? Why or why not? Solution: According to part, the Euler equation does not hold in this case. That is because the utility function is linear, which means the marginal utility of consumption is a constant. Given that ( + r) β is not equal to, the consumers cannot equalize the present values of consumption between today and tomorrow, resulting in a corner solution. 3. Would your answers to the sub-questions. and. change, if the utility function is concave in both c today and c future? Solution: If the utility function is concave in both c today and c future, the consumer can choose c today and c future to ensure that the present value of future consumption equals consumption today. Thus the answers to the sub-questions. and. will change. Marginal Propensity to Consume In Lecture 3, we saw an example where the household only lives for two periods. Now suppose that the household lives for N periods. Therefore, the life time utility function is U = N β t u(c t ) and u (c) > 0, u (c) < 0. The income stream over the N periods is y t, where t =,, 3 N. The interest rate is r.. If N =, write down the Euler equation between Period t and t, where t t. (Hint: You may find the Lagrange method useful in this case.) Solution: The objective function for the household is: max c t β t u(c i ) t= The inter-temporal budget constraint for the household is: Note that c + c + r + + c N ( + r) N + = y + y + r + + y N ( + r) N + c t (+r) t is the present value of consumption in Period t. Left hand side gives the present value of life time consumption, while the right hand side gives the present value of life time income. Then the Lagrangian function is: t= L = t= β t u(c t ) + λ [y + y + r + + y N ( + r) N + (c + c + r + + c N + )] ( + r) N For any t {,,, N}, the First Order Condition is Thus, for any t {,,, N}, FOC(c t ): β t u (c t ) = λ FOC(c t ): β t u (c t ) = λ ( + r) t ( + r) t
Therefore, or β t t u (c t ) = u (c t ) ( + r) t t u (c t ) = u (c t ) β t t ( + r) t t () (3) Think of the special case where t = t +. The general version of Euler equation becomes u (c t ) = u (c t+ ) β ( + r) (). If N <, and for simplicity, we assume that the discount factor β = and interest rate r = 0. Solve for the optimal consumption plan for the N periods, c t, where t =,, 3, N. Solution: With β = and r = 0, we arrive at the conclusion that for all t {,,, N} u (c t ) = u (c t+ ) which implies, c t = c t+ (5) Combining equ (5) with the household s inter-temporal budget constraint, Using equ( 5), we know the following, c + c + + c N = y + y + + y N N c t = y + y + + y N we can get the optimal consumption plan c t = N N t= y t where t =,,, N 3. What s the marginal propensity to consume in Period t =? (You can explain in words.) Solution: Since we know in Period t =, N t= c = y t N which means he or she consumes a fraction ( ) of permanent income in Period. If permanent income N increases by one unit, c increases by N. So the marginal propensity to consume is N.. What s the marginal propensity to consume in Period t =? (You can explain in words.) (Optional) Solution: The marginal propensity to consume in Period t = is N. 3
Similar to part 3, we can obtain that N t= c = y t + s N which means that the agent consumes a fraction ( N ) of permanent income in Period, where s denotes the savings from Period. If permanent income increases by one unit, c increases by N. So the marginal propensity to consume is N. 5. Now suppose N = 3 and y = y 3 < y. The consumer is borrowing constrained in the first period. Find the optimal consumption plan for the three periods, c t, where t =,, 3. (You don t have to go through the math and it is sufficient to justify your answers in words.) Does the Euler equation hold in this case? Why or why not? Solution: Since y < y + y + y 3, we know that in Period, the consumer wants to borrow. However, 3 the consumer is borrowing constrained, that s c y, therefore he chooses c = y. Since y > y 3, the consumer saves and therefore, c = c 3 = y + y 3. Since y = y 3 < y, c < c = c 3. Then the Euler equation no longer holds in this case due to the borrowing constraint. 3 Precautionary savings Suppose an individual lives for two periods. There is no discounting and the interest rate is zero. The individual s objective function is to maximize: U(c ) + E(U( c )) subject to c = y c + ỹ Suppose that ỹ is a normal random variable with mean y and variance σ y. The utility takes the following CARA form: U(c) = a e ac (Hint: A random variable X is log-normally distributed, if log(x) is normal. If log(x) is a normal random variable with mean µ and variance σ, then E(X) = exp(µ + σ ), where exp( ) is the exponential function.). Derive a closed-form expression for c. What is the level of precautionary saving? How does the variance of second period income affect the level of precautionary saving? How does the risk aversion parameter affect precautionary saving? (Hint: recall that precautionary saving equals certainty equivalent consumption minus actual optimal consumption. To find certainty equivalent consumption, solve the problem by assuming y equals y with certainty.) Solution: Restate the problem, we have c = arg max c U(c ) + E(U( c )),
subject to, c = y c + ỹ (6) where U(c ) + E(U( c )) = a e ac + E( a e a c ) = a e ac a E(e a c ) Further we have: log(e a c ) = a c = a (y c + ỹ ) Given ỹ N(y, σ y), log(e a c ) N( a (y c + y), a σ y). Then by the fact given we have the following: E(e a c ) = e a (y c+y)+ a σ y Therefore, the problem transforms to the following: c = arg max c a a σ y e ac a e a(y c+y)+ FOC generates: e ac = e a(y c +y)+ a σ y Or ac = a(y c + y) a σ y Rearrange the FOC, we have: c = y+y aσ y If y = y with certainty, it is easy to show the consumption allocation in the certainty case, So the precautionary saving would be: c = c = y + y c c = y + y = aσ y y + y + aσ y So if the variance of the second-period income increases, the precautionary saving would increase. Further, if the risk aversion parameter increases, the precautionary saving would increase too. 5
. How do the level of precautionary saving and the marginal propensity to consume out of first-period income vary with the level of y? Solution: Given Precautionary Saving equals to aσ y, we could see that it is independent of the level of y. Further, given c = y+y, we could easily compute out the marginal propensity to consume out of first-period income: aσ y MP C y = dc dy = i.e. the marginal propensity to consumption of first period income is a constant. 6