Question

Find the steady state level of capital for this economy.

Exercise 1 (50 Points]: The Solow Growth Model Consider an economy where there are 1,354 agents, each of them having one unit of available time to labor. There is no government. There is a representative firm that has a Cobb-Douglas production technology that creates output (or income) following the form of Y = 2KN- The depreciation rate is 4.9%, the population of agents grows at 1.2% and the capital share of income is given as (a = 0.314). This economy saves 29.8% of its income and total factor productivity is 4.2. (a) (2 Points) How many agents are there next period? (b) (15 Points) Suppose there are 7,600 units of capital. 1. How much investment took place? 2. How much depreciation took place? 3. What is the level of capital next period? 4. What is the level of output this period? 5. What is the level of output next period? 6. What is the level of capital per agent in this period? 7. What is the level of capital per agent next period? 8. What is the level of GDP per capita this period? 9. What is the level of GDP per capita next period? 10. Calculate the growth rate for GDP per capita. (c) (15 Points) Instead, suppose there are 76,000,000 units of capital. 1. How much investment took place? 2. How much depreciation took place? 3. What is the level of capital next period? 4. What is the level of output this period? 5. What is the level of output next period? 6. What is the level of capital per agent in this period? 7. What is the level of capital per agent next period? 8. What is the level of GDP per capita this period? 9. What is the level of GDP per capita next period? 10. Calculate the growth rate for GDP per capita. (d) (9 Points) (i) Find the steady state level of capital per agent for this economy. (ii) Now do this if TFP decreases by 50%. (iii) Instead, suppose TFP has its original value but the depreciation rate increases by 5 percentage points (add 5% to original percent value), find the steady state level of capital per agent. (e) (9 Points) (i) Find the steady state level of capital for this economy with the original values for the number of agents, TFP and the depreciation rate. (ii) Now do this if TFP decreases by 50%. (iii) Instead, suppose TFP has its original value but the depreciation rate increases by 5 percentage points (add 5% to original percent value), find the steady state level of capital.

please help with D.

Answer 1

d)

i) TFP = z = 4.2

Y = z*K^0.314N^0.686

Y = 4.2*K^0.314N^0.686

Y/N = 4.2*K^0.314N^0.686 / N

y = 4.2K^0.314 / N^{1- 0.686}

y = 4.2*K^0.314 / N^0.314

y = 4.2*(K/N)^0.314

y = 4.2*k^0.314

At steady state capital :

sy = (d + n + g)k

Here,

s = 29.8%

d = 4.9%

n = 1.2%

g = 0%

Put value, we get

29.8* (4.2*k^0.314) = (4.9+1.2)*k

125.16*k^0.314  = 6.1*k

125.16/6.1 = k/k^0.314

20.52 = k^0.686

k = 81.81 --- steady state capital per agent

ii) IF TFP reduce by 50%

z = 4.2(1-0.5) = 2.1

y = 2.1k^0.314

Now put this in steady state,

29.8* (2.1*k^0.314) = (4.9+1.2)*k

62.58*k^0.314  = 6.1*k

62.58/6.1 = k/k^0.314

10.26 = k^0.686

k = 29.78

iii) Now, d = 4.9% + 5% = 9.9

Put value, we get

29.8* (4.2*k^0.314) = (9.9+1.2)*k

125.16*k^0.314  = 11.1*k

125.16/11.1 = k/k^0.314

11.28 = k^0.686

k = 34.2