The production function of firm is given by y=√√√x₁+√√√√x₂ where x1, x2 inputs are priced at w₁ and w₂ resp. The output price is p. (a) Derive the firm's unconditional input demand functions, and x. [Solve the firm's profit maximization problem] (b) Derive the output function and profit function, II(p, w₁, W₂) 811 (c) Verify Hotelling's Lemma: that др = y
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only part (a) please, thank you.
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- 1. A firm’s production function is , where Ldenotes the size of the workforce. Find the value of MPLin the case when: (a) L=1, (b) L=10, (c) L=100, (d) L=1000 Does the law of diminishing marginal productivity apply to this particular function? 2. Show that the price elasticity of demand is constant for demand functions of the form where A and n are positive constants. 3. The demand and total cost functions of a good arerespectively and a) Find expressions for TR, (profit) , MR, and MC in terms of Q. b) Solve the equation and hence determine the value of Q which maximizes profit. c) Verify that, at the point of maximum profit, MR=MC. 4. The cost of building an office complex, x floors high, in a prime location in Accra is made up of three components: (a) GH¢10 million for the land (b) GH¢1/4 million per floor (c) Specialized costs of GH¢10000x per floor. How many floors should the office complex contain if the average cost per floor is to be minimized? 5. The supply…Prove it! 2. If the demand function for products A and B is in the order Pa = 36 - 3Qa and Pb = 40 - 5Qb and the common cost function is TC = Qa' + 2QaQb + 3Qb². With the above information, then calculate: a. Production values A and B at the maximum or minimum profit levels. b. The maximum or minimum profit value Did the profit reach its maximum or minimum value? Prove it c.A company manufacturing laundry sinks has fixed costs of $100 per day but has totalcosts of $2,500 per day when producing 15 sinks. The company has a daily demand functionof q = 360 − p, where q is the number if laundry sinks demanded and p is te price ofa laundry sink. ) If production increases continuously, what is likely to be the average cost per sink? How many laundry sinks will the company need to produce in order to maximise it′s profits? What is the maximum profit?
- 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.5. Suppose that a firm has a Cobb-Douglas production function Q = KLB and that it faces output price p and input prices w and r, respectively. Solve the first-order conditions for profit maximizing input bundle. Use the second-order conditions to determine the values of parameters a, ẞ, p, w, and r for which this solution is global maximum.A firm has production function f(X1,X2)=X1^(1/4) X2^(1/4). The prices of the inputs wland w2.` 1.Find the firm's demand for inputs (as a function of output and the input prices) 2. Find the cost function of the firm. 3. For w1= 1 and w2= 1, find the firm's supply function.
- 6. Each of the 10 firms has the production function q = √KL. The wage is 5 and the rent on capital is 10. Assume we are in the long run, so firms can vary both factors. Write an expression for one firm's cost minimization problem. Use whichever method you prefer to minimize cost and derive an expression for one firm's total cost: TC (9₁) c. Compute one firm's marginal cost and derive the inverse supply curve for one firm and the inverse supply curve for the market. What is the elasticity of supply? a. b.7. An analyst has found that Tom's Transmissions Company cost function is: C(x) = 40,000 + 500x – 0.55x² for, where x is the number of transmissions. a) Determine the marginal Cost function C'(x). b) Determine when the Cost function is maximized. c) Determine the marginal cost function at the 100th item proe Explain significance of the marginal cost at this point. ed and sold. d) Determine the cost difference between 101st and 100th boat ( C(101) - C(100)). Is this value the same or different from the marginal cost from part c? Explain.2. Assume that a firm has the following production function; q = L0.5K0.5. Its capital is fixed at K = 1. It faces a rental rate of r = $90 and a wage rate of W = $10. d. Determine the average, average variable, and average fixed cost functions. e. Determine the marginal cost function. f. Calculate the lowest cost, and associated quantity, that the firm can produce this good for in the short run.
- Consider total cost and total revenue given in the following table: TABLE IN IMAGE Calculate profit for each quantity. How much should the firm produce to maximize profit?(ii) Calculate marginal revenue and marginal cost for each quantity. Graph them. (Hint: Put the points betweenwhole numbers. For example, the marginal cost between 2 and 3 should be graphed at 2½.) At what quantitydo these curves cross? How does this relate to your answer to part (a)?(iii) Can you tell whether this firm is in a competitive industry? If so, can you tell whether the industry is in along-run equilibrium? (8.75)3. Assume that ACME Products operating in perfectly competitive input and outpat markets produces output with the following production function: (potentially helpful: For y = In(x), ==. Also In(ex)=x and e(In(x)) dy 1 dx X =X) q In(K) + In(L) with TC = wL+ vk a. Derive the firm's cost minimizing input demand functions. b. Explain, in words, why it would not be appropriate to use this labor demand function as the short run labor demand function. C. From this, derive the firm's supply function d. Derive the firms profit maximizing input demand functions. e. Use the profit function to derive the firm's supply function.Q1) Complete the following table and what production technique should this firm use: Assuming the price of capital (k) is $5 and the price of labor (L) is $10, and the price of commodity (O) is $3. Technique TC Profit(7t) A 50 10 6 B 50 9 8 C 50 5 10 D 50 12 8 E 50 2 10 F 50 7 12