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Consider an industry in which firms produce undifferentiated commodity for which demand is given by X = 100 - P. A number of firms produce the item in question, all of whom have production function
where Y is the level of output and a is some constant from (0, 1), K is the amount of capital employed, and L is the amount of labor employed. The prices of capital and labor are fixed at $1 throughout. All firms have fixed costs of $16.
(a) Suppose precisely six firms are in this industry, all of which maximize profits taking prices as given. What is the equilibrium in this case?
(b) Suppose there is free entry into this industry, with all of the firms having the production function given above. What is the equilibrium in this case?
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- A firm operates in the short run with the production function Q = 10 K0.5 L 0.5, where Q is output, K is capital, and L is labor. The price of capital is $100 per unit, and the wage rate is $50 per unit of labor. How many units of capital and labor should the firm use to minimize its total cost while producing 400 units of output?Miguel and Jake run a paper company. Each week they need to produce 1,000 reams of paper to ship to their customers. The paper plant's long-run production function is: q= 4K^L+ where q is the number of reams produced, K is the quantity of capital rented, and L is the quantity of labor hired. K For this production, MP, =| L and MP, = 34 K K The weekly cost function for the paper plant is C=10K+2L where C is the total weekly cost How much capital and labor will Miguel and Jake need to rent and hire in order to produce 1,000 reams of paper each week, and how much will hiring these inputs cost them?Widget factory Inc. in Wisconsin has the following production function: F(L,K) = 2L1/2 K1/2 L represent the number of labour hours. Workers at this factory are paid an hourly wage of $30 and they rent capital at $25/hour. Since this is a competitive market, the factory output the factory gets per is output is $50 per unit. Let's pretend the firm operates in the short run with capital fixed at 900, how many factory workers would Widget Factory Inc employ? What is their profit rate?
- Consider a firm with 1 input and 1 output and a productionfunction given by f(x) = x1α with α < 1. The cost of the input is c and the price of the output is p. Write down the profit function and the first-order conditions associated with profit maximization. Find the factor demand, x*(p, c), the supply function, q*(p, c), and the profit function, π*(p, c).1) Miguel and Jake run a paper company. Each week they need to produce 1,000 reams of paper to ship to their customers. The paper plant's long-run production function is: q = 4K L where q is the number of reams produced, Kis the quantity of capital rented, and L is the quantity of labor hired. For this production, MP, = and MP, = The weekly cost function for the paper plant is C= 10K +2L where C is the total weekly cost What ratio of capital to labor minimizes Miguel and Jake's total costs? (Hint: Find the Marginal Rate of Technical Substitution (MRTS) for capital and labor.) b. How much capital and labor will Miguel and Jake need to rent and hire in order to produce 1,000 reams of paper each week? How much will hiring these inputs cost them?Consider a firm that produces widgets according to the following Cobb-Douglas production function: Q = A * L^α * K^β where: Q is the quantity of output, L is the quantity of labor, K is the quantity of capital, A is a scale parameter (total factor productivity), α and β are the output elasticities of labor and capital respectively. Given that A = 1, α = 0.6, β = 0.4, L = 16 and K = 9, a) Calculate the quantity of output Q. b) If the firm increases the quantity of labor (L) to 20 while keeping the quantity of capital (K) constant, what will be the new quantity of output?
- Consider a firm that has production function f(L,K)= 3L2/3K1/3. What is the expression for this firm’s Marginal Product of capital? MPK(L,K)= 3L2/3/K1/3. MPK(L,K)= 3L2/3/K2/3. MPK(L,K)= L1/3/K1/3. MPK(L,K)= L2/3/K2/3. MPK(L,K)= 2L2/3/K1/3.Consider the production functions of three different Firms utilizing inputs labor (L) and capital (K) in producing goods X, Y, and Z given below. The three firms face the same fixed price for labor and capital at 5 per unit and 10 per unit, respectively. X = KL2 – L3; Y = 10K1.5L0.5; Z = K0.5L0.5 a. Derive the short-run cost function of Firm Z if 25 units of capital are employed. Suppose that good Z is sold at a perfectly competitive price of 10 per unit, calculate Firm Z’s profit and discuss if the Firm Z should continue to operate.b. Derive the long-run cost function of Firm YSuppose the production function for a competitive firm is y=f(x1,x2)= x₁2x₂2/3. The cost per unit of the first input is w₁ and the cost of the second input is w₂. a. What are the returns to scale of this production function? Find the cheapest input bundle, x₁ and x2, that yields the given output level of y. b. Write down the formula of the firm's total costs as a function of y. Are the average costs increasing, constant or decreasing in y? Are the marginal costs increasing, constant or decreasing in y? Why?
- Suppose that the market for apples is perfectly competitive. Production of apples requires two inputs:workers (L) and land (K). The production function for apples is:F(K,L)=3K^(1/3)* L^(2/3) A) Suppose that W=4, R=16, and in the short-run, K=27. Find the firm’s short-run cost function SRC(q)and short-run marginal cost function SRMC(q).B) What is the optimal production of apples if P=4? What are consumer surplus and profit?C) Find the firm’s short-run supply of apples Q(P) when W=4, R=16 and K=27.D) Find the firm’s long-run cost function LRC(q) and long-run marginal cost function LRMC(q) when W=4and R=16. How do these compare to the firm’s short-run cost function and marginal cost function?E) Find the firm’s long-run supply of apples q(P) when W=4 and R=16.Trotter Inc is a polling company. Trotter uses both labor, L, (measured in hours worked) and capital, K, (measured in phone lines) to conduct surveys. The production function of Trotter’s is given by Q=10L^0.25 K^0.25 where Q is measured in surveys completed per hour. The price of a unit of L is $16 per hour and the price of a unit of K is $1 per hour. Trotter’s has additional fixed costs of $227 per hour. For parts (a) – (b) below assume that the number of phone lines is fixed in the short run. In particular, Trotter's has 16 phone lines (so K = 16). This results in additional fixed costs in the short run of $16 (ie in addition to the 227). a) In the short run what is Trotter’s (compensated) demand curve for labor? What is the variable cost curve? What is the total cost curve? b) In the short run, what is the marginal cost curve of Trotter’s? What is the average cost curve? What is the optimal size of the firm? Illustrate the short run marginal and average cost curves. For parts (c)-…A firm uses labor (L) and capital (K) to produce output (q) according to the function: q=1/2 ⋅ L1/2 ⋅ K In the short-run, the firm's level of capital is fixed at two units (K=2) and each unit of labor must be paid a wage of $3 (w=3). What must be the price (p) the firm can charge for each unit of output if the firm maximizes short-run profits by producing q=4 units of output? a.) p = ? dollars