Question

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

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.

q= 40K0.5L0.5

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:

 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.

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