Question

Ed’s building company has the following production function q = 10KL − 1/2KL^2 where q is the number of houses built, L is the quantity of labor Ed employs and K is the quantity of capital Ed uses. In the short run, K is fixed at K¯ = 2

a. Derive MPL and APL.

b. For what values of L is the MPL > 0?

c. For what values of L is the MPL diminishing?

In the long run, both L and K can be easily varied.

d. Derive the MRTS of L for K.

Answer #1

q = 10KL − (1/2)KL^{2}

K= 2 (given in short run)

q = 10(2)L − (1/2)(2)L^{2}

q = 20L − L^{2}

a. Derive MPL and APL:

MPL= Differentiation og q with respect to L= 20-2L

APL= q/L= (20L-L^{2} ) / L= 20-L

b.

MPL > 0

20-2L > 0

Add 2L on both sides:

20 > 2L

Divide both sides by 2:

10 > L

**It implies that for all value of L less than 10, MPL
will be positive.**

c.

MPL= 20-2L

Differentiate it with respect to L

dMPL/dL= -2

It implies that as the unit of labor increases by 1 unit it will cause MPL to decrease by 2 units. This will true for all the values of L.

**So for all the values of L, MPL is
diminishing.**

d.

q = 10KL − (1/2)KL^{2}

MRTS=MPL/MPK

MPL= partial differentiate q wrt L= 10K-(1/2)(2)KL

MPL= 10K-KL

MPK= partial differentiate q wrt K= 10L-(1/2)L^{2}

MRTS= (10K-KL) / [10L-(1/2)L^{2} ]= (10K-KL) /
[(20L-L^{2} )/2]

**MRTS= (20K-2KL) / (20L-L ^{2} )**

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