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^{0.5}L^{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.

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