Question

Consider the Solow growth model that we developed in class. Output at time t is given by the production function Y AK Lt, where A is total factor productivity, Kt is total capital at timet and L is the labour force. Total factor productivity A and labour force L are constant over time. There is no government or foreign trade and Y, + 1, where Ct is consumption and I is investment at tim. Every agent saves s share of his income and consumes the rest. Therefore, G = (1-s)Y, and S = sy, Each period, savings equal investment: I-St. Cap evolves according the transition equation K+(1-d)K+1 where d is the depreciation rate. 1. Combine the production function and the transition equation for capital to express K+ as a function of K and the parameters of the model. 2. Express the transition equation in per worker terms, letting kdenote capital per worker Suppose that A = 5, L 1, s= 0.2, d=0.1. Furthermore k.-8 per worker using the above production function. Calculate output per worker at time 3. Let y denote output per worker. Express output per worker in terms of capital 4. Calculate how much capital (per worker) depreciates at time t. Calculate investment per worker) at time t. Calculate the level of capital per worker in t1. Did capital per worker increase? 5. Suppose that k -64. Calculate how much capital depreciates (per worker) at time t Calculate investment (per worker) at time t. Calculate the level of capital per worker in . Did capital per worker increase? Compare with Q4 and comment on the differences.

Answer 1

1. Production function Y_t = AK_t^1/3L^2/3

^{Transition equation = K_t+1= (1-d)K_t - I_t}

^{We know , Y}t^{= C_t+ I_t   I_t= Y}t ^{- C_t}

Combined Transition equation for K_t+1 = (1-d)K_t  -AK_t^1/3L^2/3 - (1-s)Y_t

On further calculation K_t+1 = (1-d)K_t  -2AK_t^1/3L^2/3 - sAK_t^1/3L^2/3

2. Transition equation in per worker terms K_t+1 = (1-d)K_t - I_t

K_t = k_t X L = 8

K_t+1 = (1-0.1)K_t - sY_t [Since I_t = S_t and St = sY_t]

K_t+1 = 0.9X8- sAK_t^1/3L^2/3

K_t+1 = 7.2- 0.2X5X(8)^1/3X(1)^2/3

^K_t+1= 7.2- 0.2X5X2

K_t+1 = 5.2

3. Production function Y_t = AK_t^1/3L^2/3

Dividing L on both sides

Y_t/L = (AK_t^1/3/L) ^₍L/L)^2/3

^{y = A((K/L)_t)1/3}

Output per worker at t = ^{A((K/L)_t)1/3}

For 4th part should we be using values of 2nd part.