A price-taking firm has a production function given by y = 3(x3)¹/3 (max{x₁,8x2})¹/3 where 1, 2 and 3 are inputs and y is output. Let w₁, w2 and w3 denote input prices, and let P denote the output price. Solve the profit maximization problem and the corresponding cost minimization problem.
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- Q1.The following is a Cobb-Douglas production function: Q = 1.75K0.6L0.5. What is correct here? * -This production function displays constant returns to scale -This production function displays increasing returns to scale -A one-percent change in L will cause Q to change by one percent -This production function displays decreasing returns to scale Q2. For studying demand relationships for a proposed new product that no one has ever used before, what would be the best method to use? * -consumer surveys, where potential customers hear about the product and are asked their opinions -double log functional form regression model -ordinary least squares regression on historical data -market experiments, where the price is set differently in two marketsQ.No.3. Consider the production function: (3) Y = 0.75X + 0.0042X2 – 0.000023X3 (a) At what level of X, the output will be maximum? (b) If input price is 0.15$ and output price is 4$ then at what level of X, profit will be maximum?What is the cheapest way of producing 850 units of output if a firm operates with theproduction function 0.5 0.5 Q = 30K L and can buy input K at 75 $ per unit and L at 40 $per unit?
- Given the following cost function, determine the underlying production function. 1 2C(m, w, y) = 10mwy, where y is the output and m and w, are the prices of two inputs x1 and x2 respectively.Minimization CT = W1X1 + W2X2 Hold it: Y = X11/3 X21/6 where X1 and X2 are the productive factors and W1 and W2 are their prices, respectively. Determine: If X2 = 27 and W2 = 2 a) The short-run cost functionA purely competitive firm has a single variable input < (labor), with the wage rate. W0 per period. Its fixed inputs cost the firm a total of F dollars per period. The price of the product is P0. (a) write the production function, revenue function, cost function, and profit function of the firm. (b) what is the first-order condition for profit maximization ? Give this condition an economic interpretation. (c) What economic circumstances would ensure taht profit is maximized rather tahtn minimized?
- A purely competitive firm has a single variable input L (labor), with the wage rate W0 per period. Its fixed inputs cost the firm a total of F dollars per period. The price of the product is P0. (a) write the production function, revenue function, cost function, and profit function of the firm. (b) What is the first-order condition for profit maximization? Give this condition an economic interpretation. (c) What economic circumstances would ensure that profit is maximized rather thatn minimized?E1 Suppose that production for good X is characterized by the following production function, Q = K0.5L0.5, where K is the fixed input in the short run. If the per-unit rental rate of capital, r, is $15 and the per-unit wage, w, is $25, then the average fixed cost of using 9 units of capital and 81 units of labor is $75. incalculable since there is insufficient information to determine the average fixed costs. $80. $5.A firm has two variable factors and a productionfunction f(x1, x2) = 6x1/21X21/3. The price of its output is 3, the price of factor 1 is 3, and the priceof factor 2 is 2.– What is the optimal production output level?– What is the maximum profit-level?
- Suppose that the production function takes the form X = min(10L, 5K) and that a competitive firm faces a wage rate of £60 per week and a weekly capital rental of £32. (a) How much must the firm spend to produce 100 units of output, and what is the average cost of production when X = 100? (b) What is the incremental cost of producing the 101st unit of output? (c) What happens to the cost of producing 100 units of output if the wage rate and the rental cost of capital rise by 25 per cent each? What happens to the average and marginal cost? (d) What happens to the cost of producing 100 units of output if the wage rate increases by £1, or if the cost of capital increases by £1?A firm engaged in the manufacture of RTWS faces the short-run production function Q = 250L - 5L², where L is the number of units of labor and Q is the number of RTWs produced annually. d.) How many RTWS can be produced by the firm in a year if there are 10 units of labor? e.) Compute the marginal product of the 40th unit of labor. f.) How many RTWs can be produced by the firm in a year if there are 40 units of labor? g.) Sketch the graph of the production function.Define Q to be the level of output produced and sold, and assume that the firm’s cost function is given by the relationshipTC = 20 + 5Q + Q2Furthermore, assume that the demand for the output of the firm is a function of price P given by the relationshipQ = 25 - Pa. Define total profit as the difference between total revenue and total cost, and express in terms of Q the total profit function for the firm. (Note: Total revenue equals price per unit times the number of units sold.)b. Determine the output level where total profits are maximized.
