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.
K= 25 (given)
MPL is the first derivative of q with respect to L.
MPL = 40(0.5) K0.5 L-0.5
MPL = 20(25)0.5/ L0.5
= 20(5) / L0.5
MPL = 100/L0.5
APL = q/L = 40K0.5 L0.5 /L
= 40 (25)0.5 / L0.5
= 40(5)/ L0.5
APL = 200/ L0.5
Now, by inserting th different values of L , we get the different values of MPL and APL:
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.
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