Consider a firm with a production function given by y = a ln x1 + ß lIn x2 with a, ß E (0,1) which can sell its product for price p and by its inputs at input costs w1 and w2. Find the conditional input demand functions and the cost function for some given level of output y.
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- 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.To produce a recorded DVD, a firm uses one blank disk D and the services of a recording machine M for one hour. The production function in this case is given by: Q = min{D, M}(a) Using this production function, find the firm’s demand function for recording machine-hours M.(b) ) Draw the total product, average product, and marginal product of M curves for the production function identified(c) Let PM denote the hourly price of renting the recording machine M and PD denote the price of one blank disk. Use your answer from (b) to calculate the firm’s short-run total, average, and marginal cost functions.The Director of ABC Enterprise hires labour (L) and rents capital equipment (K) in a competitive market to produce mango juice. At the moment, the wage rate of labour is GH¢2 per hour and capital is rented at GH¢5 per hour. Also, the unit price of mango juice is GH¢0.75 and total cost of production is GH¢1,000. Suppose the firm’s production function (Q) follows a Cobb-Douglas specification given as: 0.5 0.5 ?=14? ? +10 Determine the optimal input usage and the maximum profit that ABC Enterprise would obtain at the optimal input levels.
- Suppose the long-run production function for a competitive firm is f(x1,x2)= min {3x1,2x2}. 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: 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? C: Write down the formulas and draw the graphs of the average cost and marginal cost functions, as functions of y.Suppose the production function for a competitive firm is y=f(x1,x2)= x1 1/4 x2 1/4 . 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? B: Find the cheapest input bundle, x1 and x2, that yields the given output level of y. C: Write down the formula of the firm’s total costs as a function of y. D: Are the average costs increasing, constant or decreasing in y? Are the marginal costs increasing, constant or decreasing in y?Let y = f(x1, x2)=x11/2 + x1x2 be a firm’s production function, where x1≥0, x2≥0. Write down the firm’s production possibility set, and its input requirement set. Is this production function concave, quasi-concave? Is this production function homogenous? Find its returns to scale when x1=1, and x2=1.
- A competitive firm’s production function is given by y= f(x1,x2)=x1ax2b If a=b=0.5 , the price of factor 1 is 12, and the price of factor 2 is 3, find the cost minimizing input combinations and the total cost of producing 40 units of output. Redo part (1), this time by first deriving the firm’s conditional factor demand functions and the cost function.Suppose that a firm’s production technology is described by theproduction function f(x1, x2) = (x1)^2x2, where x1 denotes the quantity ofinput 1 and x2 denotes the quantity of input 2. Let the price of input 1 be$1 and the price of input 2 be $4.a. Derive the conditional input demand functions for bothinputs.b. Derive the firm’s cost functionif a product function of a firm is given by Q=k^2L^3 & the input price r=8 w=2 A find the level of 'L' &'k' that maximizes output total outlay 240
- A firm is jointly owned by Juan and Roda. The firm’s production function requires two inputs: effort by Juan, denoted by x, and effort by Roda, denoted by y. Effort is only observable by the person who exerts it. The cost to Juan of a unit of his effort is c j = 2 and the cost to Roda for a unit of her effort is cr = 2. The price received for the goods is p = 2. The production of the firm is given by Q = 10(ln(x + 1) + ln(y + 1)). Assume that both Juan and Roda are risk-neutral rational agents.a) What are the socially optimal amounts of effort x* and y*? What is the total surplus in that case? (Hint: Solve the problem of a social planner that cares equally for Juan and Roda.)b) Suppose that Juan and Roda have a contract that specifies that Juan pays a fixed amount w = 15 to Roda and that Juan gets to keep and sell all the output. What is the total surplus now? How much of that surplus goes to Juan? To Roda?c) Now suppose that the contract between Juan and Roda specifies that the total…Consider a price-taking firm whose production function is given by q = 3 L1/5 K1/9 where L and K denote respectively the amount of labour and capital the firm uses to produce q units of output. Suppose the price of labour is w = 16, the price of capital is 24 and the price of the firm's output is p=225 . Find the firm's cost function. Then enter below the value of the firm's marginal cost at the point where q = 100.Suppose that the production function is given by y = 2x0.5 The price of x is $3 and the price of y is $4. Derive the corresponding VMP and AVP functions. What is MFC? Solve for the profit-maximizing level for input use x.