Suppose a price taking firm’s production function is given by q= √L, where is labor. When the wage rate is 1.5, and the output price changes from 2 to 4, the welfare gain of the firm (gain in the producer surplus) is
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Suppose a price taking firm’s production function is given by q= √L, where is labor. When the wage rate is 1.5, and the output price changes from 2 to 4, the welfare gain of the firm (gain in the
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- 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).Consider two firms that produce a single output good,y, using two inputs :Capital, K , and labor, L, the prices of each unit of capital and labor are r and w,respectively. The output good y sells for $p per unit. Firm A's production function is y = fa(K,L) = K1/4L1/4. The profit function is equals to : K1/4L1/4 - rK -wL. a) FInd the profit maximizing levels of K and L as functions of r,w, and p. b) Suppose that r = w= $1 and p =$4 . What is the profit maximizing level of output,y?A competitive firm's production function is given by y = f(K, L) = 8K 1/2 + L 1/4 Assume that the price of the good produced is p = $100, the price of capital is r = $4, and the price of labor is w = $5. Find the profit- maximizing levels of capital and labor (K * and L * ). Find the profit-maximizing level of output (y * ). What is the maximum profit of the firm (π * )?
- 1. Consider a firm with production function given by f (x₁, x₂) = (x₁)¹/4 (x₂) ¹/2 Assume the prices of inputs 1 and 2 are w₁ and w2, respectively, and the market price of the product is p. (a) Find the levels of the inputs that maximize the profits of the firm (i.e., x† and xz). (b) Derive the supply function of the firm (i.e., y* = f (x1,x₂)).A firm produces a good using two inputs, capital (K) and labour (L). For every unit of output being produced, the ratio of capital to labour must be , where is a positive parameter. Meanwhile, the total cost of production must equal $1,000. If the rental rate of capital is $20 per unit of capital, and the hourly wage is $10 per unit of labour, how much capital does the firm use for it production? a) k=100a/(2/a+1) b) k=100/a(2a+1) c) k=100/(2/a+1) d) k=100/(2a+1)The cost function c(w1,w2,y) of a firm gives the cost of producing y units of output when the wage of factor 1 is w₁, and the wage of factor 2 is w2. Find the cost functions for the following firms: a. A firm with production function f(x1,x2)= min{2x1, 3x2} b. A firm with production function f(x1,x2)= 2x1 + 3x2
- 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?Please answer all three questions below (part a,b & c), thank you. Consider a firm with 1 input and 1 output and a production function given by f(x) = x1α with α < 1. The cost of the input is c and the price of the output is p. Please Find (a) the factor demand, x*(p, c) (b)the supply function, q*(p, c) (c) the profit function, π*(p, c).Jess owns a firm that uses labour (L) and capital (C) to sell widgets (X), according to the following production function: X = F(L,C) = ln(L) + ln(C) Jesse buys her factors and sells her output in perfectly competitive markets. The market prices for L, C and W are a, r and p, respectively. What is the firm’s profit function. What is the firm's marginal product of labour , marginal product of capital and marginal rate of technical substitution Does the firm exhibit a diminishing marginal product of capital? What is the firms demand functions for labour and capital. Say the central bank decides to increase interest rates, causing ? to go up. What effect will this have on the firm’s use of L and C? What effect will it have on output (X) and profits?
- 3. A firm has a production function given by 1 1 A) y = 4x³x² ; 1 1 B) y = 3x4x² ; a) What are the factor demand functions? b) What are the conditional factor demand functions? c) What is the cost function? d) What is the supply function? 1 1 C) y = 5x³x2; 11 D) y = 12xx².Suppose that a price-taking firm’s production function is given by q = (K ½ +L ½) a short run supply function of a firm is ?Janet has the production function z = f(x, y) = 2√x + √y where x and y are inputs and z is the output. Janet's production function is strictly concave and increasing in each input. You don't need to show this. (a) Find the cost minimizing bundle of inputs as a function of the input prices and output. (b) Find Janet's supply function when the input prices of both inputs are $1 per unit.