HOMEWORK #3 This homework assignment is due at NOON on Friday, November 17 in Marnix Amand s mailbox.

Transcription

1 Econ 50a second half) Yale University Fall 2006 Prof. Tony Smith HOMEWORK #3 This homework assignment is due at NOON on Friday, November 7 in Marnix Amand s mailbox.. This problem introduces wealth inequality into a two-period economy with production like the one discussed in lecture on Friday, November 0. a) Suppose that there are two types of consumers distinguished by their initial endowments of capital. In particular, type- consumers who comprise fraction θ of the population) are richer than type-2 consumers who comprise fraction θ of the population): type- consumers are endowed with k 0 units of capital and type-2 consumers are endowed with k 2 0 units of capital, where k 0 > k 2 0. The two types of consumers are identical in all other respects. Each consumer takes prices as given in particular, each consumer takes the aggregate, or total, capital stock in period as given) when making savings decisions in period 0. The equilibrium or consistency) condition is that the total savings of the two types of consumers in period 0 must equal the aggregate capital stock that consumers take as given when deciding how much to save. Assume that each consumer s utility function takes the form uc 0 ) + βuc ), with uc) = logc). The production technology available to firms is: y = k α n α, with 0 < α <, where y is the firm s output and k and n are the services of capital and labor, respectively. Consumers do not value leisure; if each consumer s endowment of time is normalized to one, then n = in each time period.) Derive the equilibrium aggregate capital stock in period as a function of primitives i.e., the parameters α, β, and θ and the initial capital stocks k 0 and k 2 0). b) Use your answer to part a) to show that changes in k 0 and k 2 0 that keep aggregate capital in period 0 i.e., θk 0 + θ)k 2 0) constant have no effect either on equilibrium aggregate savings or on equilibrium prices. This is a version of an aggregation theorem for this economy: holding the total amount of capital in period 0 constant, the behavior of the aggregates in this economy does not depend on the distribution of capital in period 0. c) Suppose that the felicity function takes the form: uc) = σ) c σ ), where σ > 0. Does an aggregation theorem like the one described in part b) hold

2 for this economy? Explain why or why not. Solution a) In this question we have two types of consumers: there are a fraction θ of type consumers, who are endowed with k 0 units of capital in period 0. The rest of the consumers, a fraction of θ, are endowed with k 2 0, where k 0 > k 2 0. Then, given the log utility function, the problem that a consumer of type i =, 2 must solve is: max logc i 0) + β logc i ) c i 0,ci which implies the FOC: st c i 0 = r 0 k i 0 + w 0 k i c i = r k i + w r k i + w = βr r 0 k i 0 + w 0 k i ) i =, 2 On the other hand, each period i = 0,, the firm must solve: which implies the competitive prices: α max Kα K i, N i N i K i r i N i w i i w = α) K α and r = α K α Also, as we saw in class, the CRS production technology implies that: ) r 0 K0 + w 0 N0 = y 0 = K α 0 N α 0 Finally, the equilibrium conditions are: Ki = θki + θ)ki 2 for i =, 2 and as labor is inelastically supplied, Ni = for i = 0,. Now we are ready to solve for the equilibrium. First, multiplying each type s FOC equation by its fraction in the population and summing up both equations, we get: r θk + θ)k 2 ) + w = βr [r 0 θk 0 + θ)k 2 0) + w 0 θk + θ)k 2 )] Using the equilibrium conditions and eq.): then, replacing the competitive prices, α K α r K + w = βr [ K α 0 K ] K + α) K α = βα K α [ K α 0 K ]

3 which implies K = αβ K α 0 + αβ Thus, as period aggregate savings only depends on aggregate period 0 savings, and not on how much has each consumer, if this aggregate is not changed, K won t change either. And as prices depend also only on aggregates, changes in the distribution of capital endowments, that leave the aggregate equal, won t affect them. b) So now we just have to change the consumers utility function. Each type s problem is: c i 0) σ + β ci ) σ σ σ which implies the FOC: max c i 0,ci st c i 0 = r 0 k i 0 + w 0 k i c i = r k i + w r 0 k i 0 + w 0 k i = βr ) /σ r k i + w ) The firm s problem has not changed, so we have the same equations for prices. Following the same steps of part a), we get that: 2) r 0 K0 + w 0 K = βr ) /σ r K + w ) then, remembering that wages and interest rates only depend on aggregates, from eq.2), which implicitly defines K, we can conclude that if K0 does not change then K won t change also, and thus, the aggregation results holds here too. 2. Consider a neoclassical growth model in which consumers have time-separable preferences given by: t=0 βt uc t ). Let the aggregate production or resource) function take the form: f k, n) = A k α n α + δ) k, where δ is the rate of depreciation of capital. The parameters satisfy: 0 < β <, A > 0, 0 < α <, and 0 < δ. Consumers are endowed with one unit of time in each period but do not value leisure so that n = ). In this problem, you will solve explicitly for the recursive competitive equilibrium of this economy under the assumptions that uc) = logc) and δ =. Assume too that the economy is decentralized in the manner that we have discussed in class.)

4 Solution a) Suppose that aggregate capital evolves according to k = G k) = sf k, ). You will verify the validity of this conjecture below.) Find explicit formulas for the value function vk, k) and the decision rule k = gk, k) of a small or typical) consumer who takes the law of motion for aggregate capital as given. The functions v and g depend on s as well as on primitives of technology and preferences. Hint: Guess that vk, k) = a + b logk + d k) + e log k) and then find expressions for the unknown coefficients a, b, d, and e in terms of the structural parameters α and β and the behavioral parameter s.) b) Find the competitive equilibrium value of s by imposing the consistency condition G k) = g k, k). Verify that the resulting law of motion for aggregate capital solves the planning problem for this economy. Display v and g for the equilibrium value of s. c) How does an increase in aggregate capital affect the savings behavior and the indirect) utility of a typical consumer holding fixed the consumer s own holdings of capital)? i. Given the equivalence of competitive equilibrium and central planning problem here, we can conjecture the transition function of aggregate state as k = sa k α. Now we can define a recursive competitive equilibrium for this economy explicitly. A Recursive Competitive Equilibrium for this economy is a set of functions: price function : r k), w k) policy function : k = g k, k ) value function : v k, k ) transition function : k = sa k α such that: ) k = g k, k ) and v k, k ) solves consumer s problem: v k, k ) = max ln c) + βv k, k ) {c,k } s.t. 2) Price is competitively determined: c + k = r k) k + w k) k = sa k α r k) = F k, ) = αa kα w k) = F2 k, ) = α) A kα

5 3) Consistency: sa k α = g k, k) We solve for the recursive competitive equilibrium by the guess and verify method. We conjecture that the value function is of the following form: v k, k ) = a + b lnk + d k) + e ln k) Now we start to solve for the solution by plugging in our conjecture. Bellman equation becomes v k, k ) = max ln αa k α k + α) A k α k ) +β a + b lnk + d k ) + e ln k ) ) {k } or s.t. k = sa k α v k, k ) = max ln αa k α k + α) A k α k ) +β a + b lnk + dsa k α ) + e lnsa k α ) ) {k } Take F.O.C. with respect to k, we have = αa k α k + α) A k α k k + dsa k α k + dsa k α = αa k α k + α) A k α k ) k = + αa k α k + α) ds A k α + Plug back into Bellman equation, we have v k, k ) ) ln + αa k α α) + ds k + A k α + ) +β a + b ln + αa k α α + ds) k + A k α ) + e lnsa k α ) + [ ) ] αa + ) ln + βa + ln) + βe lnsa) + + ) α) + ds + ) ln k + k + [ + ) α ) + αβe] ln k) α Since the LHS = a + b lnk + d k) + e ln k), we have the following equations: ) a = ) αa + ) ln + βa + ln) + βe lnsa) + 2) b = + ) 3) d = α) + ds α 4) e = + ) α ) + αβe

6 Solve for these four equations, we get the solutions a = [ ) β β) 2 ln αa β)) + β ln + β b = β d = α α s e = α αβ) β) ] α ) β αβ lnsa) This gives us the form of value function. Plug these coefficients into F.O.C., we get k = + αa k α k + = αβa k α k + α) ds A k α + α) αβ s) A k α α s Therefore, we have our value function and policy function as v k, k ) = a + b lnk + d k) + e ln k) gk, k) = αβa k α α) αβ s) k + A k α α s with coefficients as defined above. ii. Imposing consistency condition, we have G k) = g k, k) sa k α = αβa k α k α) αβ s) + A k α α s α s) s = α s) αβ + α) αβ s) s 2 + αβ) s + αβ = 0 s = αβ and s 2 = Consider the context of this economic problem, obviously only s = αβ will be a solution to this problem. Consequently the resulting recursive competitive equilibrium is v k, k ) = a + b lnk + d k) + e ln k) gk, k) = αβa k α k G k) = αβa k α

7 with coefficients as a = b = d = e = β) 2 β α α β) α αβ) β) [ ln αa β)) + β ln ) ] β α ) β + β αβ lnαβa) Now let s check whether G k) solves the central planning problem. The recursive formulation of central planning problem is v k) = max ln A k α k ) + βv k ) k Guess that v k) = a + b ln k. Plug into Bellman equation, we have v k) = max ln A k α k ) + β a + b ln k ) k F.O.C. is A k α k = k k = + A k α Plug back into Bellman equation, we have α + ) ln k v k) = max ln A k α k ) + β a + b ln k ) k a + b ln k = ln A k α ) )) + A k α + β a + b ln + A k α [ ) ] a + b ln k = + ) ln A + ln + βa + ln + + ) + Again since this is an identity, we have ) a = + ) ln A + ln + βa + ln + + ) and b = α + ) Solve for this, we get a = ) αβ ln A + ln αβ) + ln αβ β αβ αβ b = α αβ

8 Therefore, the solution for central planning problem is v k) = ) αβ ln A + ln αβ) + ln αβ β αβ αβ k = αβa k α + α ln k αβ This is the same aggregate state evolution law as in the recursive competitive equilibrium. iii. Take derivative with respect to k, we have and g k, k ) = αβa k α k g 2 k, k) = αβ α ) A kα 2 k < 0 v k, k ) = a + b lnk + d k) + e ln k) v 2 k, k) = bd k + d k + ē k v 2 k, k) = α α) k k) α β) k + α) k) αβ) k v 2 k, k) > 0 if k > k = 0 if k = k < 0 if k < k Note that here we do not consider equilibrium behavior, therefore the sign is ambiguous. 3. This problem studies a neoclassical growth model with an externality in production. Leisure is not valued and the representative) consumer has time-separable preferences with discount factor β 0, ). Consumers, who own the factors of production, are endowed with k 0 units of capital in period 0 and with one unit of time in each period. There is a large number of identical profit-maximizing firms each of which has the following production technology: fk, n, k) = Ak α n α kγ + δ)k, where k is the amount of capital rented by the firm, n is the amount of labor hired by the firm, k is the aggregate capital stock, δ is the rate of depreciation of capital. The parameters satisfy: 0 < γ < α, 0 < α <, and 0 < δ. Thus there is a productive externality from the rest of the economy: a higher aggregate capital stock increases the productivity of each firm. A typical small) firm takes the aggregate capital stock as given when choosing its inputs.

9 a) Carefully define a sequential competitive equilibrium for this economy. b) Carefully define a recursive competitive equilibrium for this economy. c) Find a second-order difference equation that governs the evolution of the aggregate capital stock in competitive equilibrium. Hint: Find a typical consumer s Euler equation and then impose equilibrium conditions.) Use this equation to find an expression for the steady-state aggregate capital stock in competitive equilibrium. d) Display the Bellman equation for the social planning problem in this economy. The planner internalizes the externality in production: his production technology is h k, n) f k, n, k) = A k α+γ n α + δ) k. Is the competitive equilibrium allocation Pareto optimal? Hint: Compare the planner s Euler equation to the second-order difference equation that you found in part c).) e) Now introduce a government that subsidizes savings at a proportional rate τ and finances these subsidies by means of a lump-sum tax on consumers. The investment subsidy is constant across time but the lump-sum tax varies over time so as to balance the government s budget in every period. Define a recursive competitive equilibrium for this economy. f) For what subsidy rate τ is the competitive equilibrium steady-state aggregate capital stock equal to the steady-state aggregate capital stock in the planning problem? Solution Define the function F as F k, n, k) = Ak α n α kγ, and then fk, n, k) = F k, n, k) + δ)k We will use this notation all through this question. i. A sequential competitive equilibrium for this economy is a sequence {R t, w t, c t, k t+, n t } t such that )Given {R t, w t } t 0, {c t, k t+, n t } t 0 solves the consumer s problem: {c t, k t+, n t } t 0 = argmax t 0 β t uc t ) s.t. c t + k t+ = k t R t + w t n t k 0 given

10 n t, t 0 lim t k t+ t j=0 R t 2){k t+, n t } t 0 solves the firm s problem: k t, n t ) = arg max k t,n t F k t, n t, k t ) + δ)k t R t k t w t n t t 3)Market clearing: labor market: n t = t good s market: c t + k t+ = F k t,, k t ) + δ)k t t ii. A Recursive Competitive Equilibrium for the economy is a set of functions: price function : r k), w k) policy function : k = g k, k ) value function : v k, k ) transition function : k = G k) such that: ) k = g k, k ) and v k, k ) solves consumer s problem: vk, k) = max uc) + βvk, k ) {c,k } s.t. 2) Price is competitively determined: c + k = r k) k + w k) k = G k) r k) = f k,, k) = αa k α+γ + δ) w k) = f 2 k,, k) = α)a k α+γ 3) Consistency: G k) = g k, k) iii. Solve for consumer s problem in the normal way, we get the Euler equation: βu c t+ ) u c t ) r k t+ ) = βu c t+ ) αa k α+γ u t+ + δ) = c t )

11 Imposing the equilibrium condition k = k and n =, we get that aggregate consumption in every period is equal to: c t = F k t,, k t ) + δ) k t k t+ Since all agents are identical, in equilibrium it will be the case that c t = c t for every t you can think of this as each individual s consumption will equal per capita consumption). If we substitute this in the individual s Euler equation we derived above, we get: βu c ) u c) αa k α+γ + δ) = βu F k,, k ) + δ) k k ) u F k,, k) + δ) k k αa k α+γ + δ) = ) βu F k,, k ) + δ) k k )αa k α+γ + δ) = u F k,, k) + δ) k k ) which is a second-order difference equation that governs the evolution of the economy s aggregates. iv. The recursive formulation of the planning problem is The Euler equation is v k) = max u c) + βv k ) {c,k } s.t. βu c ) u c) c + k = A k α+γ + δ) k A α + γ) k α+γ + δ ) = If we compare the above Euler equation to the one we found in the competitive equilibrium case we can see that they are different. Since planner s problem gives us the Pareto optimal allocation, the equilibrium in the competitive case cannot be Pareto optimal. The intuition behind this result is that the firms in the competitive equilibrium do not internalize the externality in the production, whereas the planner is able to do so.

12 v. A Recursive Competitive Equilibrium for the economy with taxation is a set of functions: price function : r k), w k) policy function : k = gk, k) value function : vk, k) taxation function : T k) transition function : k = G k) such that: ) k = g k, k ) and v k, k ) solves consumer s problem: vk, k) = maxuc) + βvk, k ) {c,k } s.t. c + τ)k δ)k) = R k)k + w k) T k) k = G k) where R k) = r k) δ). 2) Prices are competitively determined: r k) = f k,, k) = αa k α+γ + δ) w k) = f 2 k,, k) = α)a k α+γ 3) Government balances budget in each period: T k) = τ k δ) k ) 3) Consistency of transition function: G k) = g k, k) vi. We can solve for F.O.C. in the normal way as Envelope condition gives us The Euler equation is τ) u c) = βv k, k ) v k, k) = u c)r k) + τ) δ)) βu c t+ ) u c t ) τ R k) + τ) δ)) = βu c t+ ) u c t ) τ αa k α+γ + τ) δ)) =

13 In steady state, we have β τ αa kα+γ + τ) δ) ) = ) δ) αa k α+γ = τ) k τ = τ) β ) δ) β αa α+γ where k τ represents the steady-state aggregate capital stock in economy with taxation. From the Euler equation of the planner s problem we can find the steady state: A α + γ) ko ) α+γ + δ = β k o = β + δ α + γ) A ) α+γ where k o represents the optimal steady-state aggregate capital stock. Now set k τ = k o, we have ) k τ = k o τ) δ) β αa τ α = τ = γ α + γ γ α+γ α + γ α+γ = β + δ α + γ) A ) α+γ Therefore, for τ = the competitive equilibrium steady-state aggregate capital stock the same as the steady-state aggregate capital stock in the planning problem.