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- 1) Consider observable effort. Assume that if the Agent does not accept the wage the Principal offers his outside option gives him a net utility of v =2. The probability of high profit under e=1 is %, the probability of high profit under low effort is 1/4. Calculate the minimum wage that the agent will accept to work and supply the asked effort when the Principal asks him to supply e=0, and e=1. Let us call these wages wo and wi. Now, assume that instead of offering him a flat wage, the Principal is offering the agent a wage schedule (, w) where the agent receives when the (gross) profit is High and w when the (gross) profit is low. Calculate all the lowest cost wage schedules the agent will accept to supply e=1, and e=D0. Does the principal's expected net profit change when he pays the minimum cost wage schedule instead of the flat wage minimum cost wage? Explain the intuition for your answer.typed please 3.12 CES utility with weights a. Show that the CES function 46 + ото 8 is homothetic. How does the MRS depend on the ratio y/x? b. Show that your results from part (a) agree with our discussion of the cases 8 = 1 (perfect substitutes) and 80 (Cobb-Douglas). = c. Show that the MRS is strictly diminishing for all values of 8 < 1. d. Show that if x = y, the MRS for this function depends only on the relative sizes of a and B. e. Calculate the MRS for this function when y/x = 0.9 and y/x 1.1 for the two cases 8 = 0.5 and 8 = -1. What do you conclude about the extent to which the MRS changes in the vicinity of x y? How would you interpret this geometrically? =the agent has a maximum of 12 hrs in a day that she can work, and her aspiration is not a utility level, but a target of making at least $500 in a work day. Her utility from working q hours at an hourly rate w is given by wq - q2(so she likes total income but dislikes working). If her target is not achievable, then she just max imizes her utility. If the target level of income is achievable, then she chooses any feasible number of hours that yield at least her target income. Formulate the model by doing the following (warning: (i) and (ii) require care): (i) Specify the choice domain. (ii) Write the agent's menu when the hourly rate is w. (iii) Determine the agent's choice from menus corresponding to wages $22/hr and $50/hr respectively.2 (iv) Point out how the choices relate to the finding in the study.
- Consider a competitive economy that has four different jobs that vary by their wage and risk level. The table below describes each of the four jobs.Job Risk ( r ) Wage ( w)A 1/5 $ 3B 1/4 $12C 1/3 $23D 1/2 $25All workers are equally productive, but workers vary in their preferences. Consider a worker who values his wage and the risk level according to the following utility function:u(w, r) = w + 1/r2Where does the worker choose to work? Suppose the government regulated the workplace and required all jobs to have a risk factor of 1/5 (that is, all jobs must become A jobs). What wage would the worker now need to earn in the A job to be equally happy following the regulation?Problem 4 Consider the leisure demand/labor supply model studied in class, and let the consumer have a spccific utility function U(N, Y) = N2/3y!/3. As in lecture, let the price of consumption be normalized to 1, and let w denote the wage. (a) Say that w = 10. What are the optimal N* and Y*? How many hours does the agent work? Draw a sketch to illustrate this situation. (b) Solve for the general demand functions N(w) and Y(w) as a function of the wage, as well as the labor supply function H(w). Calculate the elasticity of labor supply with respect to the wage, w. Do you notice anything special about this particular example? (c) Say the wage rises to some w' > w. What is the change in leisure demand N(w)? Carefully draw a sketch that decomposes this into an income effect and a substitution effect. (d) Sketch the labor supply function. (e) Let H(w) be the labor supply as a function of the (take-home) wage w. Say now that the government imposes an income tax of a. Let T(a) denote the…Consider the Edgeworth Box illustrated in the figure to the right. I and I, are individual j's indifference curves and I, and are individual d's indifference curves. Bundle e represents the initial endowment. Explain why point e in the figure to the right is not on the contract curve. Bundle e is not on the contract curve because Contract curve O A. the consumers' marginal rates of substitution are equal at bundle e. O B. both consumers can be made better off if consumerj trades wood for candy with consumer d. O C. the consumers' indifference curves are tangent at bundle e. O D. one party cannot be made better off without harming the other from trade. O E. both consumers can be made better off if consumer i trades candy for wood with consumer d. e Candy
- Consider a worker who earns $20/hour in the labor market and receives $50 per week innon-labor income. Assume the total number of hours available for work (h) and leisure(L) is 168 hours per week (i.e., ? = 168 = ℎ + ?). a. Draw the budget constraint for this individual. Label the endowment point. b. What is the maximum value of consumption that this individual could achieve ina week? c. In the United States, the Fair Labor Standards Act requires workers to be paid 1.5times their usual hourly wage for “overtime” work, defined as work in excess of40 hours in a week. If this law applies to the worker described above, how does itchange the budget constraint? d. Now suppose Congress passes an income tax. The income tax applies to labor andnon-labor income. The Fair Labor Standard Act is still in effect. The first $1,000in weekly income is exempt from the tax. However, every dollar above $1,000 istaxed at a rate of 10%. (To be clear: the tax on $1,000 in income is 0, the tax on$1,001 is…In the figure below, which represents how the labor/leisure choice of a consumer changes when the wage rate increases to w' > w, the substitution effect is zero and the overall income effect is positive (in the figure, x represent the amount of leisure). A. True B. False motivate it with a short comment (max 4 lines). 24w' Py (w' > w) 24w C Py A = B BLa L 24Which of the following preference doesn't satisfy monotonicity? O Quasilinear preference O Perfect substitutes preference Bads O Cobb-Douglas Preference
- The most general form of a social welfare function (SWF ) can be written as W = W(U1; ... ; UH). a. Explain the following properties that a SWF may satisfy: non paternalism, Pareto principle, anonymity (the names of the agents do not matter), and concavity (aversion to inequality). b. Consider two agents h = 1; 2 with utilities U1 and U2. Depict the social indifference curve of the utilitarian SWF in (U1; U2)-space. Which of the properties in part a does it satisfy? c. Depict the social indifference curves of the maxim in or Rawlins SWF. Contrast to the utilitarian SWF with respect to the aversion to inequality. Which properties does the Rawlins SWF satisfy? d. The Bernoulli-Nash social welfare function is given by the product of individual utilities. Discuss the distributional properties of the Bernoulli-Nash SWF.Question 6 Unsaved DD 24 D B E BL₁ с BL₂ GOOD 1 If bundle A is chosen on BL₁ and bundle C is chosen on BL2 then: We can say that bundle C is revealed preferred to bundle A. We cannot make a comparison between bundles A and C. We can say that this consumer is not satisfying WARP. We can say that bundle A is revealed preferred to bundle C.Prove that if the Walrasian demand function is generated by a rational preference relation, then it must satisfy the weak axiom. Is the converse true? If yes, prove it, or else give a counterexample?