Consider the following production function for a firm: y=(x1-1)(x2-1), if x1≥1 and x2≥1; and y=0, if either x1<1 or x2<1: Find the firms supply function and verify the law of supply.
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Consider the following production function for a firm: y=(x1-1)(x2-1), if x1≥1 and x2≥1; and y=0, if either x1<1 or x2<1:
- Find the firms supply function and verify the law of supply.
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- Juan Valdez owns a coffee farm in Colombia. His production function is: f(x1,x2)=(x1−1)^0.25 x2^0.5 Assume the price of input 1 is r and the price of input 2 is w. (a) Write down an expression for the technical rate of substitution. (b) Find Juan's demand for inputs conditional on the quantity y of coffee Juan wants to produce. (c) Find Juan's cost function. (d) What is the supply function of Juan's firm?The demand and supply functions for coal are given as: (In pictures) what is the inverse demand function? what is the inverse supply function?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.
- Consider the general supply function: Qs = 1,000 + 20 P - 9 PI +25 F Qs = quantity supplied P = price of the commodity PI = price of a key input in the production process F = number of firms producing the commodity Derive the equation for the supply function when PI = $480 and F = 60.A firm has the following production function Y=(x_1^α+x_2^α )^(1/2α) Obtain the conditional demand functions for x1 and x2 given a level of output q.Say the the aggregate inverse demand function is D(Q) = 100 - Q, where Q = q1+q2. Both firms have the same cost c(q)=cq. Only firm q1 is allowed to participate. Therefore, q2 = 0. Find the function that maximizes welfare. What is the demand and price when welfare is maximized?
- Firm A and Firm B sell identical goods The total market demand is:Q(P) = 1,000-1.0P The inverse demand function is therefore: P(QM) = 10,000-10QM QM is total market production (i.e., combined production of firm’s A and B). That is: QM = QA + QB As a result, the inverse demand curve for each firm is: P(QA,QB) = 10,000-10QA-10QB The difference between this example and the example in class is that the two firms have different costs. Firm A has the same cost as in class, but firm B has a different cost function: TCA(QA) = 5000QA TCB(QB) = 5000QB Using the demand function and the cost functions above, what is firm A’s profit function? Using the profit function above and assuming that firm B produces QB, calculate what firm A’s best response is to firm B’s decision to produce QB. (Note: Firm A’s best response should be a function of QB) Using the demand function and the cost functions above, what is firm B’s profit function? Using the profit function above and assuming that firm A…A firm can manufacture a product according to the production function Q=3K1/2L1/2, and capital is fixed at 4. a) When the firm hires 16 units of labor. The average product of labor is......... b) when the firm hires 16 units of labor, the marginal product of labor is....... C) if the firm can sell its output at a price of $ 10 per unit and can hire labor at $ 10 per unit , it should hire?............units of labor maximize the profits.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.
- Let the production function of a firm is given as q=(x0.5 +y0.5)2 Where x and y are inputs and wx is the price of input x and wy is the price of input y. a) Assume the firm has a limited budget to spend on buying input. Find the cost-conditional input demand function for each input. b) Now, assume the firm has no budget restriction but it has a production quota. Find the output-conditional input demand function for each input. c) Find the cost function of the firm. d) Assume the firm has no budgetary or production restrictions and set up the profit maximization problem. e) Write the conditions that need to be met to find a non-zero output that maximizes profits.A profit-maximizing firm produces a good using two inputs through the production function f(x1, x2) = (x1 + x2)0.5. Find the profit function π(p, w1, w2) and the supply function y(p, w1, w2)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.