Question

7. A firm has the production function Q=LK. The firm initially faces input prices w = $1 and r = $1 and is required to produce Q=100 units. Later the price of labor w goes up to $4. Find the optimal input combinations for each set of prices and use these to calculate the firm's price elasticity of demand for labor over this range of prices.

Answer 1

Consider the production Q = LK and input prices as w and r.

This implies that MRTS = w/r becomes MPL/MPK = w/r or K/L = w/r.

Now K = (w/r)L

Use this in production function to get L*(w/r)L = Q or L = (Qr/w)^0.5 and K = (Qw/r)^0.5

Case 1

w = r = 1 and Q = 100

This gives L = (100)^0.5 = 10 and K = 10

Case 2

w = 4 and r = 1 but Q = 100

This gives L = (100*1/4)^0.5 = 5 and K = (100*4/1)^0.5 = 20

Price elasticity of demand for labor = % change in units of labor demanded / % change in price

= (5 - 10)*100/10 divided by (4 - 1)*100/1

= -50%/400%

= -0.125