Ecn 101 Problem Set 1

University of California, Davis ECN 101 Problem Set 2

Answer to Problem 1: The trick is to always divide both sides with the term L. ¯ ¯ a) For Y = AK 1/3 L2/3 , dividing both sides with L, we get y = Ak 1/3 . For Y = ¯ 3/4 L1/4 we get y = Ak 3/4 . ¯ AK b) y = k. ¯ c) y = k + A ¯ d) y = k − A All the graphs can be found in the separate file uploaded on the web site with the title “Figures for Problem 1”.

Answer to Problem 2: ¯ a) i) Production function: Y = AK 3/4 L1/4 . ii) Rule for hiring capital (3/4)(Y /K) = r. iii) Rule for hiring labor (1/4)(Y /L) = w. ¯ iv) Capital demand equals capital supply K = K. ¯ v) Labor demand equals labor supply L = L. b) The method used here is exactly the same as in your book and the lec¯ ¯ ¯¯ ¯ ture. Of course, K ∗ = K and L∗ = L. Therefore, Y ∗ = AK 3/4 L1/4 . Finally, ∗ = (3/4)A(L/K)1/4 and w ∗ = (1/4)A(K/L)3/4 . ¯ ¯ ¯ ¯ ¯ ¯ r ¯ ¯ ¯ c) Let A = 1, K = 100, L = 1000. With these numbers it is easy to see that ∗ = 0.044, r ∗ = 1.33, Y ∗ = 177.82. w ¯ d) If now L = 1500, we have w∗ = 0.032. Clearly, the real wage went down. This makes sense since the demand did not change, but the supply of labor went up. ¯ ¯ ¯ e) Consider again general values for A, K, L. The equilibrium value of output per ∗ = Y ∗ /L∗ = A(K/L)3/4 . ¯ ¯ ¯ person will be given by y

Problem 3: The (steady state) equilibrium value of output per person in the Solow growth model (Chapter 5) is given by ( )1/2 ¯ y ∗ = A3/2 s/d ¯ ¯ . (1) The equilibrium value of output per person in the production model (Chapter 4) is given by ¯¯ y ∗ = Ak ∗ 1/3 . 1 (2)

a) In (2) it is clear that y ∗ depends on equilibrium capital per person, k ∗ , but the same is true for (1). In the Solow model we have “endogenized” (made endogenous as opposed to exogenous) the capital stock. Hence, when we write ( )1/2 ¯ y ∗ = A3/2 s/d ¯ ¯ , part of this equation includes the per person (steady state) capital stock. b) This is a deeper question. The answer is that in the Solow model the...