Question

2. Assume that a manufacturer faces a Cobb-Douglas production function, q=40K^0.5L^0.5

where q is output per period, L is labor, K is capital. The market price of labor (w) is $50 per unit and the price of capital (r) is $200 per unit.

a. Specify and illustrate graphically the short-run MPl and APl for L = 5 to 30 units (assume that the level of capital is 25; use increments of 5 units of labor). Is this firm operating in stage I, II, or III? Explain.

Answer 1

q= 40K^0.5L^0.5

K= 25 (given)

MPL is the first derivative of q with respect to L.

MPL = 40(0.5) K^0.5 L^-0.5

MPL = 20(25)^0.5/ L^0.5

= 20(5) / L^0.5

MPL = 100/L^0.5

APL = q/L = 40K^0.5 L^0.5 /L

= 40 (25)^0.5 / L^0.5

= 40(5)/ L^0.5

APL = 200/ L^0.5

Now, by inserting th different values of L , we get the different values of MPL and APL:

L	MPL	APL
5	44.64	89.68
10	31.64	63.29
15	25.84	51.67
20	22.37	44.74
25	20	40
30	18.28	36.56

By plotting these we get MPL curve and APL curve, It is shown in the figure below:

This firm is operating in the 2nd stage of production because 2nd stage implies where MPL and APL both are decrasing but are positive . And in this question they both are decreasing and are positive.