3. [5 points] Suppose the economy has the production function, P(K, L) -Kil. The

Question

3. [5 points] Suppose the economy has the production function, P(K, L) -Kil. The economy begins with 100 units of capital and 100 units of labor. a. Show that this production function is a Constant Returns to Scale production function. b. How much output does the economy produce? c. Suppose output price P-1. What are the wage and rental price of capital? d. If the plague kills half the population (so now there are only 50 units of labor), what is the new level of output? e. What is the new wage and rental price of capital?

Explanation / Answer

F(K,L)=K^0.5L^0.5, K=100 and L=100, P=1

(a) F(tK,tL)=(tk)^0.5(tL)^0.5=tK^0.5L^0.5=tF(k,L)

Thus the production function follows constant returns to scale.

(b) Substitue K=100, L=100 in the production function.

Y=100^0.5*100^0.5=10*10=100

(c) Wage=Price*Marginal product of Labor

Marginal product of labor=dY/dL=0.5K^0.5/L^0.5

Wage=1*0.5*10/10=0.5

Rental=Price*Marginal product of capital

Marginal product of capital=dY/dK=0.5L^0.5/K^0.5

Rental=1*10/10=0.5

(d) K=100, L=50

Y=100^0.5*50^0.5=70.7

(e) Wage=1*0.5*10/50^0.5=0.7

rental=1*0.5*50^0.5/10=0.3

60.5

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