Consider the following economy. Output is produced according to the production f
ID: 1147302 • Letter: C
Question
Consider the following economy. Output is produced according to the production function Capital and labour are supplied inelastically and are K = 100 and 1-400. The marginal product of capital and labour are given by: MPK =-' , and MPL = 2K 2L On the demand side, consumption is given by C' = 250 + .75(Y-T), and investment is given by 1 = 1000-50r. In addition, C-1000 and T-1000. Use the above information to answer the following: 1. Does the production function exhibit constant returns to scale? Explain. 2. Find national income, Y. What is the real wage and the real rental rate of capital? How is capital shared between capital and labour? 3. Find consumption, C. What is the marginal propensity to consume? 4. Find the equilibrium interest rate, r, and the level of investment, I, that prevails at this rate. 5. Suppose the government reduces taxes by $100. Without resolving the entire model, explain how this policy change affects (a) disposable income, (b) consumption, (c) investment, (d) and the equilibrium interest rateExplanation / Answer
Production function is given by Y = 25K^0.5L^0.5 where K is fixed at 100 and L is fixed at 400.
a) Returns to scale are constant when the sum of shares of capital and labor in the output is equal to
1, so that any given proportionate change in both the inputs by the same fraction, increases output by
the same. Here the sum of 0.5 for labor and 0.5 for capital is 1. Hence there are CRS.
b) National income is Y = 25*100^0.5 x 400^0.5 = 5000. Capital share in income is 0.5*5000 = 2500
and labor share is 0.5*5000 = 2500. Wage rate = MPL = 5000/2*400 = 6.25 and rental price of capital
= MPK = 5000/2*100 = 25.
c) Consumption C = 250 + 0.75(5000 – 1000) = 3250. MPC is the slope of consumption function and
so it MPC = 0.75
d) At the equilibrium, Y = C + I + G.
5000 = 3250 + 1000 + 1000 – 50r
-250 = - 50r
r* = 5% and so investment I = 1000 – 50*5 = $750
e) When taxes are reduced by $100, the income is increased by multiplier times. Tax multiplier is
-MPC/1-MPC = -0.75/0.25 = -3. Hence income is increased by $300. The new income is $5300 and
so the disposable income is now (5300 – 900) = $4400. The disposable income was previously
($5000 - $1000) = $4000.
Consumption is now C = 250 + 0.75*(4400) = $3550. Investment is now 1000 – 50*5 =$750 remains
unchanged and so is the equilibrium interest rate.