Question

Bonus Question. Suppose the production function for a firrm is Q(K,L) = K1/2L1/2, so the marginal product of labor is MPL = 1 2 K1/2L−1/2 and the marginal product of capital is MPK = 1 2 K−1/2L1/2.

a) Find the equation of the isoquant for Q = 1. That is, when Q = 1, find L as a function of K or K as a function of L to obtain an equation for the isoquant.

b) Find K1, K2, L3, and L4 so that all four of the following points are on the isoquant for Q(K,L) = 1: (K1,1) (K2, 1/4) (9,L3) (16,L4)

c) Find the marginal product of labor and marginal product of capital for all four points. You may want to first nd the capital to labor ratios, K/L.

d) Find the marginal rate of technical substitution for all four points.

e) Use the definition of the elasticity of substitution to calculate σ for points 1 and 2.

f) Repeat part (e) for points 3 and 4. Comparing your answer for (e) and (f), can we say that the production function is a CES function

Answer #1

The production function is , and the MPL is or .

**(a)** For Q=1, the isoquant will be
or
or
.

**(b)** All the stated points would satisfy the
isoquant equation
. In other words, for any point
on isoquant, would satisfy the equation
or
.

As (K_{1},L_{1}=1) satisfies the isoquant, we
have
or
or
.

As (K_{2},L_{2}=1/4) satisfies the isoquant, we
have
or
or
.

As (K_{3}=9,L_{3}) satisfies the isoquant, we
have
or
or
.

As (K_{4}=16,L_{4}) satisfies the isoquant, we
have
or
or
.

**(c)** The marginal product of labor would be as
or
. The values would be as below.

- For (K
_{1},L_{1}), it would be . - For (K
_{2},L_{2}), it would be . - For (K
_{3},L_{3}), it would be . - For (K
_{4},L_{4}), it would be .

The marginal product of capital would be as or . The values would be as below.

- For (K
_{1},L_{1}), it would be . - For (K
_{2},L_{2}), it would be . - For (K
_{3},L_{3}), it would be . - For (K
_{4},L_{4}), it would be .

**(d)** The MRTS would be
or
or
. The values would be as below.

- For (K
_{1},L_{1}), it would be . - For (K
_{2},L_{2}), it would be . - For (K
_{3},L_{3}), it would be . - For (K
_{4},L_{4}), it would be .

**(e)** The definition of elasticity of
substitution is that it is the ratio of rate of change in capital
labor ratio and rate of change in MRTS.

The change in capital labor ratio is (similar to midpoint elasticity formula) or or 1.7647. The change in MRTS is or or 1.7647. Hence, the elasticity of substitution is 1.7647/1.7647 or 1.

**(f)** The change in capital labor ratio is
or
or 1.0386. The change in MRTS is
or
or 1.0386. The elasticity of substitution is hence 1.0386/1.0386
or 1.

Thus, YES : We can definitely say that the production function is a (type of) CES function, since changing the capital labor combinations did not change the elasticity of substitution.

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