Question

Kool (producer of hi–tech fans) production function is Q = KL0.5. The company over the years uses input rates of 80 and 20 for capital and labour respectively. The price of capital is $100 and labour is $25 per fan.

a) Is there efficiency in production? Why?

b) If $25,000 is allocated to production, what is the maximum output?

Answer 1

a) Output is Q = KL^0.5

Also w = 25 and r = 100

From production function MRTS = MPL / MPK or 0.5K/L

Wage rental ratio is 25/100 = 0.25.

At K = 80 and L = 20, MRTS = 0.5*80/20 = 2

Since MRTS > wage rental ratio, the firm is not efficient when it uses K = 80 and L = 20

b) First use the fact that MRTS = 0.5K/L and wage rental ratio is 0.25.

At the optimum choice MRTS = w/r or 0.5K/L = 0.25. This gives L = 2K.

Now the budget constraint is 25000 = wL + rK or 25000 = 25L + 100K

Use L = 2K to get 25000 = 25*2K + 100K.

This gives K = 25000/150 = 166.67 units and L = 333.33 units

Maximum production Q = 3043 units.