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Consider the production problem discussed in class in which 24 labor hours and 4 machine
hours are to be allocated between food (x) production and wine (y) production. Suppose the
production functions for x and y are as follows:
Also, suppose the preferences of the agents are given by:
Finally, suppose the MRT is given by
where x and y denote, respectively, the aggregate quantities of food and wine produced.
For each of the following allocations, determine whether they are (i) feasible and (ii) Pareto
efficient. If they are inefficient, explain intuitively how it is possible to make someone betteroff without making the other person worse-off.
a. ((14,2),(10,2),(5,10),(23,18))
b. ((8,2),(16,2),(5,14),(11,18))
c. ((14,2),(10,2),(14,21),(14,7))
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- Solve parts d,e and f Jen from Thunder Bay consumes housing (H) and food (F). Her studentstipend is $600/month. She has a utility function of U(H, F) = F1/3H2/3.The initial price of food is pF = 1 and the price of housing is pH = 10. (a) Determine how much food and housing she consumes.(b) Suppose the government decides to subsidize her housing by 50%.This has the effect of lowering the price she actually pays for housingto $5. If the price of food is $1 and the price of housing is $5, calculateJen’s utility maximizing bundle of goods.(c) How much does the Government spend on this program (just on Jen)?(d) Calculate her utility level using the utility function above and thequantities consumed.(e) Calculate Jen’s utility maximizing bundle and her resulting utilitylevel if the government chooses to simply give her the money theywould have spent on the program (rather than subsidize housing).How much utility does Jen get?(f) Given the two programs cost the same, which one is better for…2. Consider a small island economy where the government is contemplating imposing a tax on bananas to raise revenue for public projects. The islanders' utility from consuming bananas (B) and coconuts (C) is given by the Cobb-Douglas utility function: U(B, C)=B°C1-a where a = 0.3. (a) Write down the consumer's optimization problem and derive the demand functions for bananas and coconuts. (b) Suppose the initial prices are PB $2 and Pc = = $4, and the consumer's income is I = $100. Calculate the initial quantities of bananas and coconuts consumed. (c) Calculate the compensating variation (CV) and equivalent variation (EV) for a tax rate of t = $0.50. Interpret your results in terms of the impact on consumer welfare. (d) Determine the tax revenue per consumer after the tax is imposed, assuming supply is perfectly elastic so the tax falls entirely on consumers. (e) Compare the impacts on consumer welfare between a flat tax equal to the revenue found in part (d) and the per-unit tax.Consider an economy inhabited by George and Harriet, whose utility functions are Ug : (ac)² (bc)2 Он тан + 2bн The total quantities of ale and bread that can be produced by the economy are a and b, and they are constrained by the production function b = 2(10 – a)/2 There are infinitely many Pareto optimal allocations. In one of them, Harriet's utility is 8. a) An allocation in this economy is described by a list of four variables. What are these variables? b) What four equations describe the Pareto optimal allocation in which Harriet's utility is 8? c) Find this Pareto optimal allocation.
- Consider an economy in which two commodities doors (Q) and beds (Q2) are produced. The utility function of all consumers in the economy is given as U = q q, . The production functions for the production of doors and beds are given as q x, 3 1 and 92 where x, is the input used in the production of doors and x2 is input used in producing beds. (a) Determine the margina! rate of subsiitution (MRS) for all consumers in the economy. (3marks) (b) Suppose x, +x, 108, derive the Marginal Rate of Transformation (MRT) for this economy. (c) Compute the quantity of doors and beds produced and sold at which the entire economy attains equilibrium. (d) What is the equilibrium price of a unit of door and a unit of bed in the economy? (7 marks) (3 marks) (2marks)Dick, Jane, and their dog, Spot, share an apartment. Their utilities depend upon the cleanliness of the apartment and the hours spent doing housework. Dick and Jane are both averse to housework, though Dick finds it especially troublesome. Their utility functions are where c is an index of cleanliness and h represents hours of housework. The degree of cleanliness is determined by the “production function” a) Define the marginal rate of substitution MRSi as the increase in cleanliness that exactly compensates person i for one more hour of housework: and define the marginal rate of transformation as the number of hours of leisure that must must be given up to generate a one unit increase in cleanliness. Find the Samuelson condition for this economy. b) An allocation is a triplet (h D, hJ, c). What two conditions must a Pareto optimal allocation satisfy? Show that there are Pareto optimal allocations in which Dick, despite his aversion to housework, does more…Consider the problem: minf,(x, y) = x² +y² minf;(x, y) = y²= x + 5 st. -5 < x, y:<5 Solve by the constraint method: find only a few Pareto optimal: 2: solutions
- The market for tickets to this year’s football games consists of two groups of consumers. Let x1denote the demand for tickets by fans of one of the visiting teams, and let x2 denote the demand for tickets by home fans who enjoy the spectacle. Let the price of the ticket be denoted by px, and suppose that everyone who purchases a ticket pays the same price. The demand functions of the two consumer groups has been determined as follows. There are 15,000 seats available in the stadium (thus X=15). x1= 12 −px/2 x2= 6− px/4 (a) What is the market demand function for football tickets, as a function of px? (Remember that demand cannot be negative for either consumer group.) (b) If the school was required to sell tickets to visitors at the same price as home team Students, what would the price for tickets be? (c) Suppose that, at the ticket price chosen by the university, the price-elasticity of demand for…There are three individuals in an economy. The demand functions for these individuals are given as follows: X1 = 22 - 2P, X2 = 8 - P, X3 = 28 - 2P. What is the market demand when P = 10? (a) X = 30 - 3P (b) X = 50 - 4P (c) X = 28 - 2P (d) X = 36 - 3P (e) X = 58 - 5PConsider a two-agents, two goods economy, in which both agents, A and B, are represented by the following utility function: UA(1, 22) = rr2 UB(y1, Y2) = Y1y% There are w units of each good in the economy. 1. Characterize the set of Pareto optimal allocations and represent it in the Edgeworth box. The 3 units of each goods are initially share as follows: A has w units of good 1 and zero unit of good 2; B has zero unit of good 1 and w units of good 2. 2. Determine the Walrasian equilibrium 3. Represent the economy in the Edgeworth box.
- Let say you have 3 products: cake, candy, and bread. The revenue per cake is $8. The revenue per candy is $4. The revenue per bread is $5. Define the number of cake, candy, and bread you are about to produce per day as x, y, and z respectively. The manager wants that the number of cake accounts for at least 30% of the total production per day. What is the correct linear constraint to show this? Group of answer choices 0.3x – 0.7y – 0.7z ≥ 0 0.7x – 0.3y – 0.3z ≥ 0 0.7x – 0.3y – 0.7z ≥ 0 0.3x – 0.7y – 0.3z ≥ 0Consider the following utility function U(X,Y) = 2X + 3Y PX,PY, and I are exogenous. (a) What is the demand function for X? (b) What is the demand function for Y?If f(k) = 6k0.5, s = 0.1, n = 0.1, and d = 0.2, what is the value of f(k) at equilibrium? A. 6 B. 12 C. 18 D. 24