Question

MAKE YOUR OWN QUESTION Consider a Solow economy with a Cobb-Douglas production function with capital share \alpha=\ldots_{Text {_. }} . TFP is normalized to one. The labor force is L=1. The supply of capital at time zero is K 0=…. The saving rate is s= . The depreciation rate is d= Compute the stock of capital in the economy at time t=1(K1) and at time t=2(K2). You can pick your own parameter values to answer this question. (In the exams, the parameter values will be chosen randomly.) Explain precisely the different steps to compute the capital stock at different dates.

Answer 1

To compute the capital stock of a Solow economy with a Cobb-Douglas production function at different times, the following steps can be followed:Step 1: Define the parameters: In this step, we have to define the parameters of the Solow economy. Given that we can choose our parameter values, we can select the following values for the parameters: α = 0.5, L = 1, s = 0.2, d = 0.1 and K0 = 1.Step 2: Calculate output per effective worker: To calculate the output per effective worker, we can use the Cobb-Douglas production function equation, which is: Y = A(K^αL^(1-α)).Here, we have A = 1 (normalized to one) and L = 1. Thus, we get: Y = K^α(1)^(1-α) = K^α.Step 3: Calculate the capital-output ratio: The capital-output ratio (K/Y) can be found using the equation: K/Y = 1/(y/k), where y = Y/L. Thus, we get: K/Y = 1/[(K/L)^α] = (L/K)^α.Step 4: Calculate the savings per effective worker: Savings per effective worker (sY/L) can be calculated using the savings rate (s) and output per effective worker (Y/L). Thus, we get: sY/L = (0.2)Y/L = 0.2K^α.Step 5: Find the investment per effective worker: Investment per effective worker can be calculated as the savings per effective worker minus the depreciation rate (d) times the capital stock per effective worker (K/L). Thus, we get: I/L = sY/L - δK/L = 0.2K^α - 0.1K/L.Step 6: Calculate the new capital stock: The new capital stock (Kt) can be found by adding the investment per effective worker (I/L) times the labor force (L) to the previous capital stock (Kt-1). Thus, we get: K1 = K0 + I/L × L = K0 + 0.1K0 = 1.1K0K2 = K1 + I/L × L = K1 + 0.1K1 = 1.21K0Therefore, the stock of capital in the economy at time t=1(K1) is 1.1K0 and at time t=2(K2) is 1.21K0.