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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- 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…13) You begin to work at a factory that produces Aguardiente (Firewater, a type of alcoholic beverage), the function of production its represented by the technology: Q(L, K, Z) = L³ K+z! Where with offers of input perfectly elastic A) Salary w is the price of labor L. B) Interest r is the price of capital K. C) Rent q is the price of drinkable alcohol Z. Calculate The function of demand for each input The function of offer for the Firewater The function of total costs for the Firewater The maximum benefit.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…
- 8. A firm's total cost function is given by TC = 3Q; +2Q,Q, + 7Q: Where Q, and Q. denotethe number of it ems of goods 1 and 2, respect ively that are produced. Using the substitution met hod, find the values of Q, and Q. which minimize costs if the firm is committed to producing 40 goods of eit her type in total.4. For production functions i) q = K0.25L0.25, ii) q = K0.5L0.5, iii) q = KL a) Assume w= $20 and v = $20, what are the long-run MC and AC equations? b) Tum the MC functions into firmlong-run supply equations. c) Graph the MC and AC functions. Assume that the price of output is $20, explain how the q = = q(p = $20) might or might not be a profit-maximizing outcome and whether the SOC's will be satisfied.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.
- 100 90 2 10 PRICE (Dollars per scooter) 8 8 70 2 20 10 D 0 MC ATC AVC 10 20 30 40 QUANTITY (Thousands of scooters per day) 50 00 70 80 80 100 Profit or Loss In the short run, given a market price equal to $45 per scooter, the firm should produce a daily quantity of scooters. On the preceding graph, use the blue rectangle (circle symbols) to fill in the area that represents profit or loss of the firm given the market price of $45 and the quantity of production from your previous answer. Note: In the following question, enter a positive number regardless of whether the firm earns a profit or incurs a loss. The rectangular area represents a short-run thousand per day for the firm.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?2. Firm A makes and sells motor cycles. The total cost of each cycle is the sum of the costs of frames, assembly and engine. The firm produces its own engines according to the cost equation C₁=250,000 + 1,000Q +5Q² The cost of frames and assembly is $2,000 per cycle. Monthly demand for cycles is given by the inverse demand equation: P = 10,000-30Q Calculate the MC of producing an additional engine.
- ASK YOUR TEACHER The function gives the weekly profit, in thousand dollars, that an airine makes on its fights from Boston to Washington DC when the ticket price spelers 2nterpret the f () (60)-S When the ticket price is s (b) (60)-4.5 When the ticket price is s () (90)--2 [SA When the ticket price is s the weekly profit to the airline on fights from Boston to Washington is s MY NOTES the weekdy profit to the airtine on fights from Boston to Washington is by the weekly profit to the airine on fights from Bestion to Washington bedby2. The market for a slice of pizza in Oakland is highly competitive. The market demand for a slice of pizza can be summarized by the function: D(p) = 1200 — 200p. Consider each pizza shop to be an individual firm with the 1 same cost function C(q) = FC + - q2, where FC represents fixed cost of renting a space from which to sell pizza, and the variable costs reflect the cost of the labor involved in producing and selling pizza. a. In the short-run (e.g. the six months following the signing of a lease for a space from which to sell pizza) determine the quantity each pizza shop would produce for any price in the market. b. In what way does your answer to part (a) depend on the fixed cost? Explain your answer in no more than two sentences. c. Suppose initially there are 100 pizza shops selling pizza in Oakland. Derive an expression for the short-run market supply for pizza. d. Find the short-run equilibrium price and quantity in the Oakland pizza market. How does this equilibrium depend on…The following graph plots daily cost curves for a firm operating in the competitive market for demin overalls. Hint: Once you have positioned the rectangle on the graph, select a point to observe its coordinates. PRICE (Dollars per overalls) 50 45 40 35 15 10 5 0 0 2 MC ATC AVC 10 12 4 8 14 16 QUANTITY (Thousands of overallses per day) 18 20 In the short run, given a market price equal to $15 per overalls, the firm should produce a daily quantity of The rectangular area represents a short-run Profit or Loss On the preceding graph, use the blue rectangle (circle symbols) to fill in the area that represents profit or loss of the firm given the market price of $15 and the quantity of production from your previous answer. Note: In the following question, enter a positive number regardless of whether the firm earns a profit or incurs a loss. of $ overallses. thousand per day for the firm.