2. A seller's production function is q = f(k,1) = (kl), w = v = 1, and the competitive output price is P. For all parts going forward, assume we are in the long run and all inputs are variable. e. Find the profit-maximizing q. f. Find unconditional demand for labor. g. Find the profit function.
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- B. Suppose that your production process is characterized by the production function x = f(e) = 100 In ( + 1). For purposes of this problem, assume w > 1 and p > 0.01. a. Set up your profit-maximization problem. b. Derive the labor demand function. C. The labor demand curve is the inverse of the labor demand function with p held fixed. Can you demonstrate what happens to this labor demand curve when p changes? d. Derive the output supply function. e. The supply curve is the inverse of the supply function with w held fixed. What happens to this supply curve as w changes? (Hint: Recall that In x = y implies e' = x, where e ≈ 2.7183 is the base of the natural log.) f. Suppose p = 2 and w profit will you make? = 10. What is your profit-maximizing production plan, and how mucha) Graph the TP, MP and AP data below. Clearly identify the 3 stages of production by drawing dotted lines vertically at the appropriate labor levels (increasing returns, decreasing returns, negative returns). productA small specialty cookie company, whose only variable input is labor, finds that the average worker can produce 100 cookies per day, the cost of the average worker is $32 per day, and the price of a cookie is $1.00. Is the firm maximizing profit? The firm A. is not maximizing profit because the marginal revenue product of labor is greater than the wage. B. is not maximizing profit because the marginal revenue product of labor is less than the wage. C. is maximizing profit because the marginal product of labor is greater than the wage. D. is not maximizing profit because the price of the output is not equal to the wage. E. is not maximizing profit because the marginal product of labor is greater than the wage.
- A firm uses labor (L) and capital (K) to produce rocking chairs (Q) with the following production function Q=LK. The wage (w) is $10 and the rate of capital (r) is $20. The target number of rocking chairs to produce is 800. It is the short run and the amount of K is fixed at 5. What the optimal values for L* and K* in the short run? Enter the number for the the optimal amount of L in the short run? Enter the number for the the "optimal" amount of K in the short run?Suppose the long-run production function for a competitive firm is f(x1,x2)= min {x1,2x2}. The cost per unit of the first input is w1 and the cost of the second input is w2. .a. Find the cheapest input bundle, i.e. amount of labor and capital, that yields the given output level of y. .b. Draw the conditional input demand functions for labor and capital in the x1-y and x2- y spaces. .c. Write down the formula and draw the graph of the firm’s total cost function as a function of y, using the conditional input demand functions. What is the relationship between the returns to production scale and the behavior of the total costs? .d. Write down the formula and draw the graph of the average cost function, as a function of y. .e. Write down the formula and draw the graph of the marginal cost function, as a function of y.Marginal Revenue Product = 40 - Q/10. The cost of Labor WL = 15. The production function is Q = 25L. What is the profit maximizing quantity of Labor to hire? Answer is integer.
- XYZ Co. operates in a competitive market. Its production function is q = L^α K^β . The firm takes the wage, rental rate and price as given. a) Derive the firm's conditional labor and capital demand functions. b) How does the firm react to an increase in the wage rate? c) Derive the firm's cost function.Suppose that labor is the only input used by a perfectly competitive firm. The firm’s production function is as follows: Number of Workers: 0, 1, 2, 3, 4, 5, 6, 7 Units of Output: 0, 7, 13, 19, 25, 28, 29, 29 a. Calculate the marginal product for each additional worker. b. Each unit of output sells for $10. Calculate the value of the marginal product of each worker. c. Compute the marginal profit if the wage is $100 a day. 2. Your enterprising uncle opens a sandwich shop that employs 7 people. The employees are paid $12 per hour, and a sandwich sells for $6. If your uncle is maximizing his profit, what is the value of the marginal product of the last worker he hired? What is that worker’s marginal product?Consider the following shortrun production function: (Q=100L- L*L), 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?
- Please answer according to the picture: What are the parameters of the problem? Find the conditional factor demand functions. Label them l*(w, r, y) and k*(w, r, y). Find the cost function: c(w, r, y). What is its interpretation?3) Suppose a production function is y = x,"² x,", and w, is the price of x, and w, is the price of X2, and P is the price of output. a) Find the conditional factor demands. b) Find the cost function. c) Show that dx,/dw, = dx,/dw,.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.
