A competitive firm charges a price of 50 PHP per output produced, given by Q = L0.25 K0.25. The input costs of labor and capital are 2 PHP and 2 PHP respectively. However, the firm has a required input capacity of only L + K = 20. • Compute for the profit-maximizing levels of labor and capital using the Lagrangean method • Verify that the second-order sufficient condition for a local maxima is met • Compute for the maximum profit
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- Making dresses is a labor-intensive process. Indeed, the production function of a dressmaking firm is well described by the equation Q = L − L2∕800, where Q denotes the number of dresses per week and L is the number of labor hours per week. The firm’s additional cost of hiring an extra hour of labor is about $20 per hour (wage plus fringe benefits). The firm faces the fixed selling price, P = $40. Over the next two years, labor costs are expected to be unchanged, but dress prices are expected to increase to $50. What effect will this have on the firm’s optimal output? A- Increase B- Decrease C- No EffectThe manager of Don Teeta Company Limited hires labour (L) and rents capital equipment (K) in a very competitive market. Currently, the wage rate of labour is GH¢2 per hour and capital is rented at GH¢5 per hour, the unit price of the product is GH¢0.75 and total cost of production is GH¢1,000. Suppose the firm’s production function (Q) is as follows: Q = 14K0.5L0.5 + 10 Determine the optimal input usage and the maximum profit.A firm has production function F(K, L) = 1/4 (K1/2 + L1/2) . The wage rate is w = 1 and the rental rate of capital is r = 3. (a) How much capital and labor should the firm employ to produce y units of output? (b) Hence find the cost of producing y units of output (the firm’s cost function). (c) Differentiate the cost function to find the marginal cost, and verify that it is equal to the value of the Lagrange multiplier
- Suppose the input combination currently used by a producer is such that the TRS is equal (in absolute value) to the ratio of the input prices. Then, we can conclude that the producer is maximizing her profits. Group of answer choices True FalseA 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?rofit is found as total revenue (TR) minus total cost (TC). With that in mind, let us now consider the possible profits for the firm that we looked at in assignments 7 and 8, where the firm’s capital is fixed at 2 units and where the cost of capital is $40 per unit per period, while the cost of labor (or wage rate) is $30 per unit of labor per period and the firm has the following short run production function: Labor: 0 1 2 3 4 5 6 7 8 9 Output: 0 6 24 60 120 170 210 240 260 270 Now, let us further assume that the firm is perfectly competitive (i.e., it is a price taker) and that it can sell each unit of the output that it produces for $3 [remember, for price takers, the price (P) is also the marginal revenue (MR)]. (a) Use this information set up a diagram (i.e., an excel chart) that shows total cost (TC) and total Revenue (TR) of the firm per period in the short run with the level of output on the horizontal axis. (b) Also, use…
- A 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?Functions of several variables and partial differentiation, Unconstrained Optimization with two independent variables, Stationary points, Sufficient conditions for optimisation and saddle point Question 2: A firm has the production function Q=f(L,K)=100L0.5K0.5, where L is labour and K is capital. The wage rate is $5 per hour and the rental price of capital is $4 per hour and the price of the good is $1. 1. Find the marginal product (MP) of labour and capital (that is, fL,fK ) 2. Show that there are diminishing marginal productivities, that is, fLL<0,fKK<0 3. Find the cost function. 4. Find the amount of labour and capital the firm should hire to maximize profits. [Hint: write the profit function in terms of L and K and then take partial derivatives. Use the concepts involved in Question 1 above to determine the profit is maximised.] (Question 1: Let f(x,y)=2x2+x2y+y3+12y2. Find the stationary points and for each stationary point determine whether it is a maximum, minimum,…Let the production function be Q = 4* K1/4 L1/4 assume that both factors are variable. (a) Derive the contingent demand functions for K and L (b) Substitute the contingent demand functions in the total cost that you minimized in part a) to obtain the total cost function. (c) Find the amount of K and L necessary to produce Q=8 when v=16 and w=1 (with minimal possible cost). (d) Find the average and marginal cost functions.
- A firm's production function is given as: q = K1/3 L1/3 where the rental rate (price) of capital r = N$60 and wage rate w = N$54 What is the cost-minimizing capital to labor ratio?If the production function of a firm is given by Q=K^2L^3 and the input prices are r= 8 per unit and w= 2 per unit find the levels of labor and capital that maximize the level of output for a total outlay of 240 birr?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.