Question

2. Consider the following production functions, to be used in this week’s assignment:

(A) F(L, K) = 20L^2 + 20K^2

(B) F(L, K) = [L^1/2 + K^1/2]^2

a (i) Neatly draw the Q = 2,000 isoquant for a firm with production function (A) given above, putting L on the horizontal axis and K on the vertical axis. As part of your answer, calculate three input bundles on this isoquant. (ii) Neatly draw the Q = 10 isoquant for a firm with production function (B) given above. As part of your answer, calculate three input bundles on this isoquant.

b For each of production functions (A) and (B) given above, do the following steps.

(i) Calculate the marginal product of labor MPL(L, K) = ∂F(L, K) / ∂L.

(ii) Calculate the marginal product of capital MPK(L, K) = ∂F(L, K) / ∂K.

(iii) Calculate the absolute value of the technical rate of substitution as the ratio of marginal products and simplify as far as possible: |TRS(L, K)| = MPL(L, K) / MPK(L, K).

(iv) Determine whether the absolute value of the technical rate of substitution is constant, diminishing, or increasing in labor along an isoquant. For most production functions, if we move along an isoquant by increasing labor, we must also decrease capital to keep output fixed. Justify your determination. You may recall we did a similar exercise using arrows to determine the shape of indifference curves.

Answer 1

a (i) The isoquant for F(L, K) = 20 L² + 20 K² are drawn below:

As, 20 L² + 20 K² = 2000 => L² + K² = 100

If L = 5 , then K² = 100 – 25 = 75 => K = 8.66

If L= 6 then, K² = 100 – 36 = 64 => K = 8

If L = 7 then, K² = 100 – 49 = 51 => K= 7.14

Figure 1

(ii)

Figure 2

Q = F(L, K) = L^1/2 + K^1/2]^2= 10

If L = 1 , then K₁ = 10^(1/2) – 1^(1/2) = 3.16 – 1 = 2.16

If L= 6 then, K₂ = 10^(1/2) – 2^(1/2) = 3.16- 1.414 => 1.746

If L = 7 then, K² = 10^(1/2) – 3^(1/2)=> K= 1.428

B) For the function F(L, K) = 20L^2 + 20K^2

MPL(L, K) = ∂F(L, K) / ∂L = ∂( 20L^2 + 20K^2) / ∂L = 40L

MPK = ∂F(L, K) / ∂K = 40K

For the function F(L, K) = [L^1/2 + K^1/2]^2

MPL =  ∂( L^1/2 + K^1/2]^2) / ∂L = 2( L^1/2 + K^1/2] * 1/2 L ^{1/2 -1} = (L^1/2 + K^1/2) / L^1/2 = L + (K/L)^1/2

MPK = ∂( L^1/2 + K^1/2]^2) / ∂K = 2( L^1/2 + K^1/2] * 1/2 K ^{1/2 -1} = K + (L/K)^1/2