Answer to Question #238401 in Microeconomics for Kuds

Question

Consider a firm with the following production function: 𝑄 ≡ 𝐹(𝐿,𝐾) = 𝐿 ଴.ହ𝐾 ଴.ହ . The price of labour (L) is GHS 4 per unit while the price of capital (K) is 5 per unit. For parts (a) – (e) assume that this firm has 25 units of capital it cannot change in the short-run a) If the price of output is GHS 10 per unit, how many units of labour should the firm employ to maximize profit? b) At this quantity of labour (from part (a)), what quantity of output will the firm produce and how much profit will the firm make? c) Derive the firm’s short-run total cost function. d) Derive the firm’s marginal and average cost functions. e) What is the total cost of producing 100 units of output? For the rest of this question, assume that the firm is now free to choose any level of capital and labour. f) How many units of capital and labour will it choose to minimize the cost of producing 100 units of output?

Answer 1

a)

$a) Production\space function as: Q = F(L, K) = L^{0.5}\times K^{0.5}$

So, marginal product of labor$MPL = \frac{dQ}{dL} = \frac{1}{2}\times L^{\frac{1}{2}}-1\times K^\frac{1}{2} = 0.5\times(\frac{K}{L}) ^{\frac{1}{2}}$

Marginal revenue product of labor$= price\times MPL = 10\times 0.5\times (\frac{K}{L})^{1}{2},$ and wage rate that is the price of labor is given as 4. So with K = 25, we must have

$10\times 0.5\times (\frac{25}{L})^{\frac{1}{2}} = 4\\ 5\times \frac{5}{L^{\frac{1}{2}}}= 4\\ ^{\frac{1}{2}}\\, so\\ L = (\frac{25}{4})2 = 39.0625 \\39 units \space approximately$

b)Quantity of output that the firm will produce$= F(39.0625, 25) = \frac{39.0625}{2}\times \frac{25}{2} = 6.25\times5 = 31.25 units$

Profit = total revenue - total cost

$Profit = 10\times 31.25 - (4\times 39.0625 + 5\times 25)\\ Profit = 312.5 - 156.25 - 125 = \$31.25$

c)Short-run total cost = price of labor$\times$ quantity of labor + price of capital$\times$ quantity of capital

$As \space Q = K^{\frac{1}{2}}\times L^{\frac{1}{2}}\\ Q = 25^{\frac{1}{2}}\times L^{\frac{1}{2}} = 5\times L^{\frac{1}{2}}\\ So,\space L = (\frac{Q}{5})^2\\ Short-run\space total \space cost = 4\times (\frac{Q^2}{25}) + 5\times 25\\ SRTC(Q) = 0.16\times Q^2 + 125$

d)

$MC =\frac{ ∂TC}{∂Q}\\=\frac{0.16Q^2+125}{Q}\\=0.16Q$

e)

$Q=0.16\times Q^2+125\\=0.16\times 10^2+125\\=0.16\times 100+125\\=141$

f)

$L=\frac{Q^2}{25}=\frac{100\times 100}{25}=400\\K=\frac{Q}{\sqrt{L}}=\frac{100}{\sqrt{400}}=\frac{100}{20}=5$