1.Find the firm's demand for inputs (as a function of output and the input prices) 2. Find the cost function of the firm. 3. For w1= 1 and w2= 1, find the firm's supply
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- A perfectly competitive rm produces output q with capital K and labor L according to the production function: q = f(K, L) = 4K 1 4L 1 4 The price of labor is w = 4, and the price of capital is r = 4. The rm has xed costs equal to 8. The current market price is 8. (a) Prove that this production function has decreasing returns to scale. (b) Find the optimal (cost-minimizing) ratio of capital to labor inputs K L for any level of output (use scale expansion path). (c) For any q, nd the cost minimizing inputs as functions of output: i.e., derive functions L(q) and K(q).A competitive firm’s production function is given by y= f(x1,x2)= 4x11/2 + 10x21/2 a) The price of factor 1 is 1, the price of factor 2 is 1, and the price of output is 2. Find the profit-maximizing quantities of x1 and x2? What is the profit-maximizing quantity of output? b) Redo part (a), this time by first deriving the firm’s factor demand functions and the supply function, and then substituting the prices in these functions.Consider the following shortrun production function: Q =100L - L2 where Q is the output level and L is labour input. If the price of output in the market is K.sh 50 and labour costs K.sh 1200 per hour, how many hours would the firm use to maximize profits. What is the profit maximizing level of output?
- A firm's production function is: q = 16L1/2K1/3 where q is the firm's hourly total product, L is the quantity of labor employed, and K is the quantity of capital employed. Assume that the quantity of capital employed is fixed at 125 units per hour. The labor market is perfectly competitive, and the current wage, w, is $10 per hour. The firm sells its product in a perfectly competitive market and the price is $5 per unit. a. What is the firm's value of marginal product of labor? b. What is the firm's profit maximizing quantity of labor?A firm produces basketballs with two variable inputs - labour (L) and plastic (K) - and has the following production function: f (L , K) = 6L1/3K1/6 If the price of output is 4, the price of labour is $2 and the price of plastic is $3, calculate the a) profit-maximizing level of each input b) the total production and c) the profit.The manager of Don Teeta Company Limited hires labour (L) and rents capital equipment (K) ina very competitive market. Currently, the wage rate of labour is GH¢2 per hour and capital isrented at GH¢5 per hour, the unit price of the product is GH¢0.75 and total cost of production isGH¢1,000. Suppose the firm’s production function (Q) is as follows:? = 14?0.5?0.5 + 10Determine the optimal input usage and the maximum profit.
- Exercise 3 A firm has a long-run cost function Cl(y) = y 3 − 10y 2 + 30y. (1) Derive the firm’s long-run average cost function. (2) Derive the firm’s long-run marginal cost function. (3) Find the level of production with the lowest average cost. (4) What is the long-run supply function for this firm? (5) If the market price is p = $18, how much would the firm decide to produce?Only typed answer A firm has the production function Q = K^0.8L^0.2. The wage is W and the rental rate on capital is R. Derive the long-run demand functions for capital and labor. Using Lagrangian is optional.A perfectly competitive rm produces output q with capital K and labor L according to the production function: q = f(K, L) = 4K 1 4L 1 4 The price of labor is w = 4, and the price of capital is r = 4. The rm has xed costs equal to 8. The current market price is 8. (d) Find cost functions T C(q), AT C(q), and MC(q), i.e. as functions of output. At what value of q does the minimum of AT C occur? (e) Find total and marginal revenue as functions of output, i.e. T R(q) and MR(q). (f) Find the pro t maximizing output q ∗ and use that to nd T R(q ∗ ), T C(q ∗ ), π(q ∗ ), L(q ∗ ), and K(q ∗ ).
- To maximize profit, a price taker will expand its output as long as the sale of additional units adds more to revenues (marginal revenues) than to costs (marginal costs). Therefore, the profit-maximizing price taker will produce the output level at which marginal revenue (and price) equals marginal cost. In a price-taker market, if a business produces efficiently (i.e., that is, where marginal revenues = marginal costs), the firm will be able to make at least a normal profit. True of False. Explaina) Find Bob the builder’s total cost function. Use the cost minimization conditions to find Bob the builder’s conditional demand for labor L^(q) and equipment K^(q). b) Substitute the conditional demand for labor and equipment and the current wage rate and rental cost of equipment in the expression TC = wL^(q) + rK^(q). c) Suppose Bob charges $4,000 for each veranda, how many verandas does he build? How many workers does he hire? How many pieces of equipment does he rent?Suppose a beverage company is profit-maximising. It has one factor of production, which is the amount of labour (l) it hires. For each hour of labour, the firm pays a wage 'w'. The production function is given by f(l) = l^1/2.(Assume that the price is equal to 1).a. Suppose that in equilibrium, the wage rate is fixed at w = 7. Solve for the firm’s optimal choice of how much labour to hire.b. Imagine the government votes to increase the minimum wage to w= 10. What happens to employment in the firm?c. Suppose the firm instead chooses to minimise the cost of producing a specific amount of q. Explain how this helps the firm maximise profits.
