Question

Suppose the production function is written as follows: 0.5 0.5 ?=? ?

Suppose that saving rate (s) is 0.3, population growth rate (n) is 0.05, and capital depreciation rate (d) is 0.05. (note: when you write your equations, be careful to distinguish capitalized characters and non- capitalized characters!) 3-1.

(5%) Derive the per-capita production function (i.e. ? = ? and ? = ?). ?? 3-2.

(5%) Write down the key equation of the Solow model on capital accumulation per capita. Then, impute key parameter values. (you do not have to derive the key equation)

(15%) Draw a key graph for the Solow model. (hint: ? on y-axis and ? for x-axis. Then draw per- capita production function. You also need to draw two additional curves derived from the capital accumulation equation) 3-4.

(5%) What is the steady-state level of per-capita capital? Solve the model and obtain the number. What is the economic growth rate when the economy is under the steady-state? 3-5. (10%) Suppose at the first period of the economy, ? = 4. What happens to the economy? Use a graph to show your answer.

(10%) Suppose the economy now has higher population growth rate. What happens to the steady state? What happens to the economic growth rate? Discuss with a graph

Answer 1

Given :

s= 0.3, n = 0.05 and d= 0.05

a) In per capita terms, we divide the equation by L, and we get,

where y: per capita output and k: per capita capital stock

Per capita production function:

b) Solow growth model: Change in per capita capital stock k_t+1 -k_t = sy_t - (n+d)k_t

Δk_t = 0.3k_t^0.5 - (0.05 + 0.05)k_t = 0.3k_t^0.5 - (0.1)k_t

Law of motion of capital equation: Δk_t = 0.3k_t^0.5 - (0.1)k_t

At steady state: Δk_t = 0 i.e. cange in per capita capital stock isz zero.

Therefore, 0.3k*^0.5 - 0.1k* = 0

0.3k*^0.5 = 0.1k* or k*/ k*^0.5 = 0.3/0.1 = 3

=> k*^0.5 = 3

=> k* = (3)² = 9

y for second equation is saving function, y for third equation is depreciation

d) steady state per capita capital stock is equal to 9 units.

e) Output will grow at the rate of population growth rate which is equal to 5%.