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Consider a population of consumers. When a consumer is a member of a club providing a level of provision G and having n members, they obtain utility
Where k is a positive constant and G n is the charge for club membership.
a. Derive the optimal membership for the club if it maximizes the utility of each member.
b. Assuming the club chooses G optimally given its member ship; calculate the loss due to membership of a club with suboptimal size.
c. Assume that the total population is of size m, with k
d. What club size maximizes total utility produced by the club? Contrast to the answer for part a.
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- 3. A firm that is located in country H, where price levels are p = (1,1), needs to send one of its two employees to its branch in country F. However, in country F price levels are p′ ≫ p, so the firm will have to pay additional salary to ensure that its employee is equally well-off in country F as she was in country H. Suppose the utility functions of the two employees are u1(x1, x2) = x1 + x2 and u2(x1, x2) = min {x1, x2}. The two employees are otherwise identical, including current salary. If the firm wants to minimize the additional salary it needs to pay, which employee should it send? Explain.Consider two consumers (1; 2), each with income M to allocate between two goods. Good 1 provides 1 unit of consumption to its purchaser and units of consumption to the other consumer. Each consumer i, i = 1; 2, has the utility function is consumption of good 1 and is consumption of good 2. a. Provide an interpretation of α. b. Suppose that good 2 is a private good. Find the Nash equilibrium levels of consumption when both goods have a price of 1. c. By maximizing the sum of utilities, show that the equilibrium is Pareto-ancient if α = 0 but incident for all other values of α. d. Now suppose that good 2 also provides 1 unit of consumption to its purchaser and a, 0 ≤ α ≤ 1, units of consumption to the other consumer. For the same preferences, find the Nash equilibrium and show that it is ancient for all values of α. e. Explain the conclusion in part d.Beans and doughnuts: The consumer receives positive benefits from the consumption of beans (B) and donuts (K). Utility function of the consumer is the following: U(B,K) = 100∙B^0.25 · K^0.75 The price of beans (can) is ISK 2,000. but the price of a donut (box) is ISK 4,000. Consumption restrictions are placed on the consumer since his income is ISK 400,000. Put on all form donuts on the x-axis and beans on the y-axis. a) Show an equation for the bean's success rate for a single donut in light of the utility function. Draw the equivalence curve on a picture and explain what the equation is performance ratio is stated at each point on the equivalence curve. Explain with the concept of the efficiency ratio of the curvature of the equivalent curve. b) Find the most efficient consumption combination and draw on the diagram. c) The government decides to support the consumption of beans so its price drops to 1,000. Who is the most economical consumption combination based on the changed price…
- Finn is in charge of decorations for an upcoming festival, and he is planning to decorate withclovers (C) and flags (F). Suppose his preferences over decorations can be represented by theutility function U(C, F) = C^(3/4)F^(1/4) For this problem, assume C and F are infinitely divisible so you don’t need to worry aboutrestricting to whole-number answers.(a) Write Finn’s budget constraint as a function of the prices PC, PF , and his budget I.(b) Write Finn’s constrained optimization problem in Lagrangian form and derive the threefirst order conditions.(c) Use two of the first order conditions to show that Finn’s marginal rate of substitution(MRS) equals the marginal rate of transformation (MRT) at the optimum. (Note: Youdo not need to solve the constrained optimization any more than this.)Consider an economy with 2 goods and 2 identical agents, each of whom has the following utility function, u (x1; x2) = ln x1 + 2 ln x2. The aggregate endowments of the 2 goods are given by (1; 2). Suppose there is a social planner who cares about agents equally.(a) Set up the plannerís problem. Calculate the first-best outcomeEvery month, a family of three spends $2,000 on food (F) and other items (O). The family’s preferences are represented by the utility function U(F,O) = F1/5O4/5. The unit price of food and the unit price of other items are both Suppose the club did NOT charge a membership fee: how much money would the family spend on food? How much food would the family buy? (
- Problem 4 A person has been mugged in the street by a thief, and there are n witnesses. All of the witnesses prefer someone else to chase the thief but none of them want to be the person trying to catch her. Each person gets 0 utility if the crime goes unreported, v if the crime is reported by someone else, and v −c if they report the crime. Moreover, we assume that the cost of chasing the thief is less than the utility of catching the thief: c < v. a. Define the strategic game described above. b. Find players’ best responses. c. Find the Nash equilibria of the game.Rosario has to finish her dissertation within 10 days, that is, at time t = 1, t = 2, ..., or t = 10. It takes one day to finish the dissertation, and on the day Rosario does so, she incurs an instantaneous disutility cost equivalent to $10. Rosario is a hyperbolic discounter with β = 0.85 and δ = 1. Her (instantaneous) utility function is u(x) = x.(a) Suppose the university has a system in which it charges Rosario $1 in fees for every day she does not finish her dissertation (paid each day that it is incurred). E.g., finishing on day 2 incurs a cost of $1 paid on day 1. When does Rosario finish if she is naive? How much does she pay in penalties? (Hint, past penalties are sunk, e.g., from the perspective of t = 2 self, any penalties paid in t = 1 are sunk, and do not factor into decisions or utilities going forward.)(b) Still in the $1/day system, when does Rosario finish if she is sophisticated?(c) Now suppose that the university has a deadline system: Rosario incurs a penalty of…Problem 1: Closed CityAssumptions of the model:The city is a circle with a radius that extends from the point 0 to the bound- ary of the city x ̄. There is a fixed population N, which we for now consider exogenous or fixed. All workers receive and income y and face a commuting cost t per mile that they drive. Workers have preferences over a compos- ite consumption good c with their utility function given by U(c) = c. All households demand a fixed quantity of land q, which is exogenously deter- mined and carries the rent r(x) per unit of land at location x. The rent at the boundary of the city is equivalent to the agricultural rent ra, that is r(x ̄) = ra All rental income is assumed to paid to absentee landlords who live outside the city. The household budget constraint is then given by y − tx = ri(x)q + c a) Make a list of your exogenous and endogenous variables. b) Solve for the urban bid-rent function.c) Solve for the remaining endogenous variables
- A seller sells a good of quality q at a price t. The cost of producing at quality level q is given by q2/2. There is a buyer who receives a utility of Xq − t by consuming the unit of quality q at price t. If he decides not to buy, he gets a utility of zero. X can take two values X1 = 1 and X2 = 4. (a) Suppose the seller can observe X. Derive the profit maximizing price-quality pairs offered when the type is X1 = 1 and when the type is X2 = 4. (b) Show that the full information price-quality pairs are not incentive compatible if the seller cannot observe X.Let vij be bidder i's valuation for object j, where i in {1,2,3} and j in {1,2}. Bidder i knows its valuation vi; but other bidders only know that vi; is drawn uniformly from [0, 100]. If bidder i wins object 1 at price p1 and object 2 at price p2, bidder i's payoff is v;1 If bidder i wins only object j at price p;, his payoff is vij – Pj. If bidder i does not win any object, his payoff is 0. The auction proceeds as follows. The initial prices are zero for both objects. All bidders sit in front of their computers and observe the prices for both items in real-time. Initially, all bidders are invited to enter the bidding race for both items. At any moment in time, each bidder has the option to withdraw from the bidding race for either object or both. If a bidder withdraws from the bidding for one object, he can no longer get back to the bidding for that object, but he can stay in the bidding race for the other object if he hasn't withdrawn from it previously. The price for an object…Let vij be bidder i's valuation for object j, where i in {1,2,3} and j in {1,2}. Bidder i knows its valuation vi; but other bidders only know that vi; is drawn uniformly from [0, 100]. If bidder i wins object 1 at price p1 and object 2 at price p2, bidder i's payoff is v;1 If bidder i wins only object j at price p;, his payoff is vij – Pj. If bidder i does not win any object, his payoff is 0. The auction proceeds as follows. The initial prices are zero for both objects. All bidders sit in front of their computers and observe the prices for both items in real-time. Initially, all bidders are invited to enter the bidding race for both items. At any moment in time, each bidder has the option to withdraw from the bidding race for either object or both. If a bidder withdraws from the bidding for one object, he can no longer get back to the bidding for that object, but he can stay in the bidding race for the other object if he hasn't withdrawn from it previously. The price for an object…