Question

Assume that the following functions exist and describe production, savings, investment and capital behavior.

$Y=K^\frac{2}{3} * N^\frac{1}{3}, \alpha =\frac{2}{3}, (1-\alpha )=\beta =\frac{1}{3}$

$I=S+T-G$

$S=sY$

$K_{t+1}=(1-\delta )K_{t}+l_{t}$

a) Solve for output per worker in terms of capital per worker. [Y/L=something in the form of K/L]

b) Assume the government is running a balanced budget, what does this mean for the investment function?

c) Solve for investment in terms of output.

d) How does Capital per Labor change over time? (Give the equation for the Dynamics of Capital and Output). (Keep everything in terms of Capital/Labor ratios). Explain in 1-2 sentences what each term means.

e) Find the Steady State Capital per worker and Output per worker, and Consumption per worker.

f) Now assume the federal government has a surplus. Verbally discuss how this would change your answers above.

Answer 1

Solution:

a)

Output per worker is derived as

Y = K^2/3N^1/3

Y/N = (K/N)^2/3(N/N)1/3

y = k^2/3

Here y = output per worker and k is capital per worker

b)

When budget is balanced, tax revenue equals the spending by government so T = G. Hence I = S, investment equals saving

c)

I = S

I = sY

This is how investment is related to output.

d)

We have been the given the dynamic investment function. This implies that

kt+1 - kt = It - δkt

kt+1 - kt = sy - δkt

kt+1 - kt = sk2/3 - δkt

This is how capital per labor changes over time. The change in capital per worker over time is the difference in the saving in previous period and depreciated capital for that period.

e)

Steady state has kt+1 = kt. This implies

sk2/3 - δk= 0

k* = (s/δ)3

This is steady state capital per worker. Output per worker y* is k2/3 or (s/δ)2/3*3 = (s/δ)2. Consumption per worker = (1-s)y = (1-s)(s/δ)2