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- 2. Consider a two person pure exchange economy with two divisible goods: : a consumer can consume any positive amount of any good The goods are; x1 and x2. The utility function are u' (x1, x2) = x1+Vx2, and u?(x1, x2) = x1 + x2, and the initial endowments are el Pi = 1, compute the competitive equilibrium for this economy. It is to say that you need to find the vector of prices, and allocations that sustain the Walrasian equilibrium. (25, 75) and e? = (75, 25). AssumingFall 2023/24) Adrienne and Stephen consume pizza, Z, and cola, C. Adrienne's utility function is and Stephen's is Us = (23) 05 (₂) 05 Their endowments are ZA-10, CA = 60, Zg=10, and C=140. What are the competitive equilibrium prices, where one price is normalized to equal one? is S $$ (Enter) UA =ZA CA Determine p, the competitive price of Z, where the price of C is normalized to equal one. Enter your response rounded to two decimal places) The competitive price, p.9. Consider an Edgeworth box economy with two consumers, whose utility func- tions and endowments are u'(x},x}) = (x}Xx})} (5,5) In the following, use the normalization p2 = 1. (a) Find the competitive equilibrium price. (b) State the first fundamental theorem of welfare and verify that it holds in this economy. (e) Consider the allocation ĩ = (x',) = (2,3), (8, 7). Show whether this allo- cation can supported as an equilibrium with transfers. (d) State the second fundamental theorem of welfare, and briefly discuss whether the result in part (c) conform with or violate this theorem. 3.
- Neha and Lorenzo need to decide which one of them will take time off from work to complete the rather urgent task of digging postholes for their new fence. Neha is pretty good with a post auger; she can dig the holes in 30 minutes. Lorenzo is somewhat slow; it takes him 5 hours to dig the holes. Neha earns $110 per hour as a personal trainer, while Lorenzo earns $25 per hour as a clerk. Keeping in mind that either Neha or Lorenzo must take time off from work to dig the holes, who has the lower opportunity cost of completing the task? a) Neha and Lorenzo face identical opportunity costs b)Neha C)LorenzoThe garment producer faces the inverse demand function PG = 300 - 2G, where PG is the price of garment and G is the quantity of garment. For simplicity, assume that the garment producer has no other cost of production, other than the price per unit of electricity it pays to the electricity producer. 2 Suppose that the electricity producer and the garment producer interact as fol- lows. First, the electricity producer sets a price PE that the garment producer must pay for each unit of electricity. Then the garment producer decides how much elec- tricity to purchase from the electricity manufacturer and how much garment G to produce and sell. (i) Suppose that the marginal cost of a unit of electricity is $10 per unit and every unit of garment requires two units of electricity. Find the profit maximizing prices and quantities of electricity and garment, PE, PG, E and G. What is the profit of each firm? What is the total profit? (ii) Now suppose that the two firms merge together, i.e., the…Firm A and firm B are battling for market share in two separate markets.Market I is worth $30 million in revenue; market II is worth $18 million.Firm A must decide how to allocate its three salespersons between themarkets; firm B has only two salespersons to allocate. Each firm’srevenue share in each market is proportional to the number ofsalespeople the firm assigns there. For example, if firm A puts twosalespersons and firm B puts one salesperson in market I, A’s revenuefrom this market is [2/(2 1)]$30 $20 million and B’s revenue is theremaining $10 million. (The firms split a market equally if neitherassigns a salesperson to it.) Each firm is solely interested in maximizingthe total revenue it obtains from the two markets.a. Compute the complete payoff table. (Firm A has four possibleallocations: 3–0, 2–1, 1–2, and 0–3. Firm B has three allocations: 2–0,1–1, and 0–2.) Is this a constant-sum game?b. Does either firm have a dominant strategy (or dominated strategies)?What is the…
- 5. Suppose there is a Chinese firm that could produce a "widget" at a cost of 9qw, where qw is the number of widgets. It can then ship these widgets to a U.S. firm at a transport cost of $1 per unit and for a price of pw. The U.S. firm can then turn one widget into one car at a cost of $10. Cars are then sold on the world market, where inverse demand for cars is given by: P = 500-2Q. (a) If the Chinese firm is a perfect competitor, what is P = TU.S. = Q = qw= Pw= Chinese = (b) If the Chinese firm is a monopolist, what is P = = TU.S. Pw= qw= TChinese =(4) Resource Allocation. Consider a combinatorial auction with three items {a,b,c} and three players (I, II, III. The valuations each player has for each subset of a b с ab bс ас abc I 1 2 1 6 4 11 II 3 5 3 3 5 III| 4 5 0 7 5 7 the items is shown below. What is the allocation produced under the VCG (Vickrey-Clarke-Groves) mechanism and what are the VCG payments of the players?8. (a) Prove that competitive equilibria are always Pareto efficient. (b) Consider two-persons, two-goods exchange economy. Person A has an endowment of (1, 0) and preferences U^ =Cf C$. Person B has an endowment of (1, 2) and preferences U* = Min [2Cf 3C ]. Using an Edgeworth Box draw their offer curves. Find competitive equilibrium prices and quantities.
- 7. Sheila and Bruce are going tubing. They brought with them boxes of cereal (x) and bags of pretzels (y). Their utility functions and endowments are given in the table: Sheila Bruce Utility Function In xs+ in ys 0.5ln x³+ ln y Endowment (20, 10) (15,45) Suppose that Sheila and Bruce set up competitive markets for cereal (x) and pretzels (y). Let Px be the price of cereal and Py the price of pretzels. a) Write the two equations that define Bruce's best bundle of cereal and pretzels. Solve for his best bundle (i.e. his Marshallian Demands) b) Use his Marshallian Demands to write Bruce's net demands for cereal (x) and pretzels (y). c) Do the same for Sheila (find her Marshallian Demands and her net demands for cereal and pretzels) d) Use Sheila and Bruce's net demand for cereal (x) to find the price ratio that clears the market for cereal. Show that this same price ratio clears the pretzels market. e) The price ratio represents the number of units of the y good that an individual must…4. Consider a two-consumer, two-good exchange economy. Utility functions and endowments are u'(x1, x2) = (x1x2)² and e' = (18, 4),u²cx1, x2) = In(x1) + 2 In(x2) and e? = (3, 6). (a) Characterize the set of Pareto-efficient allocations as completely as possible. (b) Characterize the core of this economy. (c) Find a Walrasian equilibrium and compute the WEA. (d) Verify that the WEA you found in part (c) is in the core.QUESTION 10 Mary is the only one in town who can make red velvet cupcakes. Each cupcake costs her $4 to make (i.e. her marginal cost is constant at $4 per cupcake). Currently, Mary charges $4.50 for a cupcake. Below is the table of potential prices she could charge and the corresponding quantities. Price ($/cupcake) 6.50 6.00 5.50 5.00 4.50 4.00 Quantity (cupcakes) 30 42 56 72 92 105 Answer the following questions: a. What is Mary's marginal revenue when the price is $4.50 per cupcake? Answer to the nearest two decimal places. $ per cupcake. b. At a price of $4.50 per cupcake, should Mary decrease or increase the number of cupcakes sold to maximise profit? Type D for Decrease or I for Increase. c. Calculate Mary's profit-maximising quantity of cupcakes to sell. Answer to the nearest whole number (with no decimal places). cupcakes.