Problem set 1 - · PDF fileProblem set 1 1. Solow-Swan model Suppose the production function...
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Problem set 1 1. Solow-Swan model Suppose the production function takes the following form: Y =F (K,AL) Where Y : output; K :capital; A :Technology or effectiveness of labor; L :labor k = K L ; w = dF (K,AL) dL ; r = dF (K,AL) dK - δ; A = (1 + g)A; L = (1 + n)L; K = K + sY - δK Assume that both labor and capital are paid their marginal products (a) Show that w = A [f (k) - kf (k)] (b) Show that if both capital and labor are paid their marginal products, constant returns to scale imply that the total amount paid to the factors of production equals net output. (c) The return to capital r is roughly constant over time, as are the shares of output going to capital and to labor. Does a Solow economy on a blanced growth path exhibit these properties. What are the growth rates of w and r on a balanced growth path? (d) Suppose the economy begins with a level of k less than k*. As k moves toward k*, is w growing at a rate greater than, less than, or equal to its growth rate on the balanced growth path? What about r? (e) Assume that the production is Cobb-Douglas (i) Express k as a function of k (ii) What is k*? (iii) Find the rate of convergence of k to k * . 2. Do excercise 2 in lecture notes, Introduction to Dynamic Programming Theory, p.21 1
Transcript of Problem set 1 - · PDF fileProblem set 1 1. Solow-Swan model Suppose the production function...
Problem set 11. Solow-Swan modelSuppose the production function takes the following form:Y =F (K,AL)Where Y : output; K :capital; A :Technology or effectiveness of labor;
L :labork = K
L ; w = dF (K,AL)dL ; r = dF (K,AL)
dK − δ; A′ = (1 + g)A; L′ = (1 + n)L;K ′ = K + sY − δK
Assume that both labor and capital are paid their marginal products(a) Show that w = A [f (k)− kf ′ (k)](b) Show that if both capital and labor are paid their marginal products,
constant returns to scale imply that the total amount paid to the factors ofproduction equals net output.
(c) The return to capital r is roughly constant over time, as are the sharesof output going to capital and to labor. Does a Solow economy on a blancedgrowth path exhibit these properties. What are the growth rates of w and r ona balanced growth path?
(d) Suppose the economy begins with a level of k less than k∗. As k movestoward k∗, is w growing at a rate greater than, less than, or equal to its growthrate on the balanced growth path? What about r?
(e) Assume that the production is Cobb-Douglas(i) Express k′as a function of k(ii) What is k∗?(iii) Find the rate of convergence of k to k ∗ .2. Do excercise 2 in lecture notes, Introduction to Dynamic Programming
Theory, p.21
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