Suppose the total cost function for a firm is given by C=qw²/31/3 where w and v are prices of a unit of labor and capital, and q is output (a) Use Shephard's lemma to compute the demand functions for each input I and k (as functions of q, v, w) [Hint: Partially differentiate C with respect to input price]
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- 8. The production function for a price-taking firm is given by q = 2.5k0.4L0.4. What is the supply function q(v,w,p)? [Show your work]Suppose that the long run cost function of a price taking firm is given byC(q)=12+3q2+3q. Solve the profit maximization problem of the firm and find the firm's supply function. Draw the firm's supply function and average variable cost function on the same graph. Clearly show which is which. Find the profit function of the firm. What is this a function of? (What are its arguments?)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.
- Suppose a firm engaged in the illegal copying of DVD’s has a daily short run total cost function given by: STC = (q^2)+25 If pirated DVD’s sell for $20, how many will the firm copy each day? What will its profits be? What is the firm’s short run producer surplus at P=20? Develop a general expression for this firm’s producer surplus as a function of the price of pirated DVD’s.Suppose that there is an industry with a big number of firms, each of them with the cost function ci(w₁,w₂,y)=(yi²+1)w₁+(yi²+2)w₂a.Find the average cost of each firm.b.Which is the supply function curve in the short run of each firm?c.Describe the necessary input set of a firm.A firm has a linear demand function for its product. When the price of the product isSh.220, the quantity demanded is 40 units. When the price increases to Sh.240, thequantity demanded becomes 30 units. In addition, the firm’s marginal cost function isgiven by:MC = 40q – 2q2 + 2Fixed cost = Sh.5 millionWhere q = quantity demanded, MC = marginal cost (Sh. million)Evaluate the level of output that maximizes profits.
- 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 prot maximizing output q ∗ and use that to nd T R(q ∗ ), T C(q ∗ ), π(q ∗ ), L(q ∗ ), and K(q ∗ ).Plz give answer ASAP. A price taking firm has the production function, q = f(l) = √ l where q is the output and l is the labor input. The price of labor input or wage is equal to $ 5, the output price is given by p, and the fixed cost is $80. a. Find the prices at which the firm makes a positive profit, and graph the prices along the curve, Hint find the profit function (SOLVE) b. Find the firm’s supply curve. (chek if correct) Q^2 = L TC or C(q) = wl = wq^2 The level of output wich firm maximizes profit is firms supply function We know profit function = pq – wq^2 dπ/dq = p – 2wq =0 p=2wq q(p,w) = p/2w thus supply function q=p/2w c. When p=10, since the firm makes a negative profit, it is better not to produce. True or False? Explain Hint part (a) we found positive values use (a) to demonstrate (SOLVE)Question one Assume that the firm's operation is subject to the following production function and price data: Q=3X+5Y-XY Px = $3; Py = $6 Where X and Y are two variable input factors employed in the production of Q. (a) In the unconstrained case, what levels of X and Y will maximize Q? (b) It is possible to express the cost function associated with the use of X and Y in the production of Q as TC = 3X+6Y. Assume that the firm has an operating budget of $250. Use the Lagrange multiplier technique to determine the optimal levels of X and Y. What is the firm's total output at these levels of input usage? (c) What will happen to the firm's output from a marginal increase in the operating budget?
- 1. Let a firm’s cost function be c(y1, y2) = F + αy1y2 + y21 + y22, where α is some constant.a) What restriction on α is required to guarantee that this cost function exhibits economies ofscope?b) What restriction on α is required to guarantee that this cost function exhibits cost complementarities in both goods? Note that you’ll need to verify cost complementarities for y1 andy2.2. Given the industry demand function X(p) = 100 − 2p, consider the following scenarios:• The market is a perfectly competitive market. Assume there are identical firms with marginalcost of 12 in this perfectly competitive market.• The market is dominated by one monopolist with a marginal cost of 12. This monopolist isable to achieve 1st degree pricing.• The market is dominated by one monopolist with a marginal cost of 12, but the monopolistis able to achieve only 2nd degree pricing. Assume the menu offers only 2 choices:(Q∗1 = 30, p∗1 = 35), and (Q∗2 = 60, p∗2 = 20).• The market is dominated by one monopolist…Assume that the marginal revenue equals rising marginal cost at 100 units of output. At this output level, a profit-maximizing firm's total fixed cost is $700 and its average variable costs are $5. If the price of the product is $4 per unit and the firm produces the profit-maximizing level of output, How much profit firm will earn ?Consider the production functions of three different Firms utilizing inputs labor (L) and capital (K) in producing goods X, Y, and Z given below. The three firms face the same fixed price for labor and capital at 5 per unit and 10 per unit, respectively. X = KL2 – L3; Y = 10K1.5L0.5; Z = K0.5L0.5 a. Derive the short-run cost function of Firm Z if 25 units of capital are employed. Suppose that good Z is sold at a perfectly competitive price of 10 per unit, calculate Firm Z’s profit and discuss if the Firm Z should continue to operate.b. Derive the long-run cost function of Firm Y