Consider a firm with production function given by 𝑄(𝐾, 𝐸) = 𝐾^1/3𝐸^2/3 K denotes units of capital and E denotes units of labour. Capital is fixed at 27 units. The firm’s output decision has no effect on the price of output or the wage (w). (a) The firm is profit maximizing and can sell output for $12 per unit. Solve for the firm’s labour demand function. That is, express E as a function of w. Note that for this production function, the marginal product of labour (MPL= MP_e) is MP_E = 𝜕Q(𝐾, 𝐸)/𝜕E = 2/3 𝐾^1/3𝐸^−1/3 Show your formulations and derivation.
Question
Consider a firm with production function given by 𝑄(𝐾, 𝐸) = 𝐾^1/3𝐸^2/3 K denotes units of capital and E denotes units of labour. Capital is fixed at 27 units. The firm’s output decision has no effect on the price of output or the wage (w). (a) The firm is profit maximizing and can sell output for $12 per unit. Solve for the firm’s labour demand function. That is, express E as a function of w. Note that for this production function, the marginal product of labour (MPL= MP_e) is MP_E = 𝜕Q(𝐾, 𝐸)/𝜕E = 2/3 𝐾^1/3𝐸^−1/3 Show your formulations and derivation.
Solution
To solve for the firm's labour demand function, we first need to understand that a profit-maximizing firm hires labour up to the point where the marginal product of labour (MPL) equals the real wage (w).
Given the production function Q(K,E) = K^(1/3)E^(2/3), the MPL is given by:
MP_E = ∂Q(K,E)/∂E = 2/3 K^(1/3)E^(-1/3)
Setting this equal to the real wage, we have:
w = 2/3 K^(1/3)E^(-1/3)
We are given that K is fixed at 27 units, so we can substitute this into the equation:
w = 2/3 * 27^(1/3)E^(-1/3)
Solving for E, we get the labour demand function:
E = [(2/3 * 27^(1/3))/w]^3
This is the firm's labour demand function, expressing E as a function of w.
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