Suppose that a firm had a production function given by: q=2L0.5K0.5. The rental rate for the firm is $10 and the wage is $5. Solve the optimization condition for K and then fill in the value that appears in front of L.K = Answer Question 2[input]L(Round to the nearest 2 decimal places if necessary.)

Question

BlackTom AI · Accepted Answer

We start by restating the problem in our own words and outlining the optimization setup. The firm uses a Cobb-Douglas production function q = 2 L^{0.5} K^{0.5}, with input prices w = 5 for labor L and r = 10 for capital K. In a typical profit-maximization or cost-minimization framework, the firm equates the marginal rate of technical substitution (MRTS) to the ratio of input prices to determine the optimal combination of inputs.

First, compute the marginal products:
- MPL = ∂q/∂L = 2 * (0.5) * L^{-0.5} * K^{0.5} = K^{0.5} / L^{0.5} = √(K/L).
- MPK = ∂q/∂K = 2 * (0.5) * L^{0.5} * K^{-0.5} = L^{0.5} / K^{0.5} = √(L/K).

Next, determine the marginal rate of technical substitution (MRTS), which is the rate at which K can substitute for L while keeping output constant. MRTS is MPL/MPK:
- MRTS = (√(K/L)) / (√(L/K)) = √[(K/L) * (K/L)] = K/L.

Under cost-minimization (or profit-maximization with price normalization), we set MRTS equal to the ratio of input prices w/r:
- MRTS = w/r = 5/10 = 0.5.

Thus,
- K/L = 0.5, which implies K = 0.5 L.

Interpreting the result, the coefficient in front of L in the optimal K is 0.5. This relationship tells us how K scales with L under the given input prices for this production technology.

If you want to verify consistency, you could also check the proportionality by noting that with the given Cobb-Douglas form, the optimal input mix is determined by equating the proportion of the square roots of inputs to the price ratio, which yields the same K = 0.5 L result.

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