Question

good weekend! Problem 1 Assume an economy is populated by L workers with total capital stock K. Production of this economy is organized by Y = K L.. Suppose household's saving rate s = 0.6, and firm's depreciation rate of capital 8 = 0.1. The rule for accumulation of captial in per worker terms is of the time-to-build type: Ak=¿- Sk. Standard Transformation of the Production Function a. Show that the production function is constant return to scale (CRS). b. Rewrite the production function in per worker terms. c. Define the steady state in your own words. Computing the Steady State Variables d. What is capital per worker in the steady state? e. What is the corresponding output per worker, consumption per worker, and break-even investment per worker in the steady state? Comparative Statics acro88 Steady States f. Assume households becomes more inclined to save in this economy, i.e., 8 = 0.7, what would be the corresponding capital per worker and consumption per worker in the new steady state? Provide intuitions for your results. 6. Assume capital depreciates in a faster pace, i.e., 8 0.2, what would be the corresponding capital per worker in the new steady state? Provide intuitions for your results. Golden Rule h. What is the steady state capital per worker, output per worker, and consumption per worker under Golden Rule? (Hint: under Golden Rule, a hypothetical social planner will maximize consumption per worker by choosing the optimal steady state capital per worker.) Transitional Dynamics 1. Assuming initial saving rate 0.6, and depreciation rate equals to 8 0.1. Use graphs to show the transitional process of consumption per worker, investment per worker, and output per worker over time. Q Search or type URL

Answer 1

Ans )

Y = K^1/2L^1/2

^(a)

^{Let us increase the capital and labor by a proportion equal to 'a'.}

^So,

Y(aK, aL) = (aK)^1/2(aL)^1/2 => aK^1/2 L^1/2

^{As the output also increases by a proportion of 'a'.So, the production functiin exhibits constant returns to scale.}

^{(b) Production function in per-worker terms is written as:}

  Y/L = (K^1/2 L^1/2)/L => (K/L)^1/2  => k^1/2

Thus,

y = k^1/2

(c) Steady state is that point in the production process when the capital per worker remains constant.In other words, the investment in capital exactly equals the depreciation of capital.

(d) From the definition of steady state we know that,

Change in capital = 0 , in the steady state

Investment = 0.6(y) => 0.6k^1/2

Depreciation = 0.1k

  

Change in capital = investment - depreciation = 0

0.6k^1/2 - 0.1k = 0

0.6k^1/2 = 0.1k

k/(k^1/2) = 0.6/0.1

k^1/2 = 6

k = (6)² = 36

So, the capital per worker in the steady state is equal to 36 units.