2006ps_solutions

2006ps_solutions - Econ 702, Spring 2006 Problem Set 10 -...

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Econ 702, Spring 2006 Problem Set 10 - Suggested Answers Problem 1 Write the problem as max P t =0 β t u ( c t ) s.t: c t = Y t I t , k t +1 =(1 δ k ) k t + sI t , H t +1 δ H ) H t +(1 s ) I t , where s is the percentage of total investment dedicated to physical capital. Adding up the last two constraints, and writting the problem in recursive form gives us V ( k,H )= max k 0 ,H 0 £ u ¡ k θ H 1 θ δ k ) k ¢ δ H ) H k 0 H 0 + βV ( k 0 ,H 0 ) ¤ The f rst-order conditions are: { k 0 } : u 0 ( c k ( k 0 0 )( 1 ) { H 0 } : u 0 ( c H ( k 0 0 2 ) and the envelope conditions: V k ( u 0 ( c ) £ θk θ 1 H 1 θ +1 δ k ¤ ,and V k ( u 0 ( c ) £ (1 θ ) k θ H θ δ H ¤ . Using the ECs in (1) , (2) we get: u 0 ( c βu 0 ( c ) θ ³ k 0 H 0 ´ θ 1 δ k ¸ (3) u 0 ( c 0 ( c ) (1 θ ) ³ k 0 H 0 ´ θ δ H ¸ (4) Assume for simplicity that δ k = δ H . Then, dividing (3) by (4) yields: 1
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³ k 0 H 0 ´ =(1 θ ) ³ θ 1 θ ´ θ , which implies that ¡ k H ¢ = θ θ (1 θ ) 1 θ (5) is the condition we are looking for. Finally, in order to obtain the growth rate along he balanced growth path, go back to (3) or (4) , and using CARA preferences guess (and verify) that consumption grows at rate γ . Under this assumption (3) implies γ σ = β h θ θ (1 θ ) 1 θ i ,or γ = h β ³ θ θ (1 θ ) 1 θ ´i 1 . Problem 2 We have seen that for this model, the solution to the Social Planner’s prob- lem yields a greater growth rate than the solution to the decentralized market. Hence, here we’re going to work with the SPP. If we can show that there is no long run growth in this model, then we know that th same will be true for the market economy (plus solving the SPP is always a bit easier). There are several ways of showing that there is no long run growth in this economy. Here we show that this economy has a steady state. It can be easily veri f ed that this steady state is globaly stable. Hence, no matter what the initial condition is for this economy, sooner or later it will reach the steady state capital and stay there for ever. This means tha long run growth cannot exist. The production function here is Y t = A _ K η k θ n 1 θ = Ak η + θ n 1 θ , since we solve the problem of the Social Planner who realizes that _ K = k . The recursive problem is V ( k )=max k 0 £ u ( Ak η + θ k 0 )+ βV ( k 0 ) ¤ , Since leisure does not appear in the utility function, we know that at the optimum it is always n =1 . he FOC for this problem is: u 0 ( c )= 0 ( k 0 ) , 2
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and the EC is V 0 ( k )= u 0 ( c )( η + θ ) Ak η + θ 1 Using the EC in the FOC we have: u 0 ( c βu 0 ( c 0 η + θ ) A ( k 0 ) η + θ 1 , which implies that there exists a unique steady state capital given by: k SS =[ β ( η + θ )] 1 / (1 η θ ) . This steady state is globaly stable and so long run growth cannot take place in this economy. Intuitevly, this happens because with η< 1 θ , the returns to scale of the reproduvable factors (here only capital) are decreasing. In the model we studied on class, we had η =1 θ , which implied constant returns for the reproducable factors, and hence long run growth.
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2006ps_solutions - Econ 702, Spring 2006 Problem Set 10 -...

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