For each of the following production functions and quantity wished to produce, given a fixed capital amount equal to 1, what is the amount of labor that minimizes costs?

Question

For each of the following production functions and quantity wished to produce, given a fixed capital amount equal to 1, what is the amount of labor that minimizes costs?

(Answer as a whole number, no decimals included; if impossible, answer NA)

a) q = K + L, 10: ___

b) q = min {20K, 10L}, 10: ___

c) q = min {20k, 10L}, 40: ___

d) q = K^.5 L^.5, 10: ___

Answer 1

(a) q = K + L, given that K = 1, and q = 10,

then L = q - K = 10 -1 = 9

(b) q = min {20K, 10L}, when q = 10,

In this production function, the quantity (q) is the minimum value between 20 times the capital (K) and 10 times the labor (L) inputs. 

10 = min { 20*1, 10*L}

10 = min { 20, 10L}

We need the minimum value of 20 and 10L equal to 10 to minimize costs. 

Since 20 is greater than 10, the minimum value will be 10L. 

Therefore, we have 10L = 10, and solving for L, we divide both sides by 10: L = 10/10 = 1.

The amount of labor that minimizes costs is 1.

(c) q = min {20K, 10L}, when q = 40,

In this production function, the quantity (q) is the minimum value between 20 times the capital (K) and 10 times the labor (L) inputs. 

40 = min { 20*1, 10*L}

40 = min { 20, 10L}

We need the minimum value of 20 and 10L equal to 40 to minimize costs. 

Since 20 is less than 10, the minimum value will be 20. 

Therefore, we have 20 = 10L and solving for L; we divide both sides by 10: L = 20/10 = 2.

The amount of labor that minimizes costs is 2.

(d) q = K^0.5L^0.5, when q = 10,

10 = K^0.5L^0.5 = 1^0.5L^0.5

10 = L^0.5

L = 10² = 100

I hope this is helpful.