D) Discussing the Equilibrium in the Endowment Economy: In the endowment economy, aggregate consumption must equal aggregate endowment ineach period. Using this condition and the optimal consumption functions derived,discuss how the equilibrium real interest rate rtis determined in this economy. Explainthe role of the real interest rate in ensuring equilibrium in the lending market and why itis necessary for achieving this equilibrium(e) Analyzing the Effect of Changes in Relative Risk Aversion:Given the consumption functions, discuss how changes in the coefficient of relative riskaversion (σ) affect the household’s consumption choices and the sensitivity of these choicesto changes in the real interest rate. Provide an intuitive explanation for your findings. Equilibrium in a Two-Period Endowment Economy with CRRA UtilityThe Constant Relative Risk Aversion (CRRA) utility function is a widely used specifica-tion of preferences in economics that captures risk aversion and intertemporal consump-tion smoothing. The CRRA utility function has the desirable property that the degree ofrisk aversion is constant and independent of the level of consumption. This means thatas a household's consumption grows, its willingness to take risks remains the same. Thecoefficient of relative risk aversion (σ) measures the extent to which households are risk-averse and prefer a smooth consumption path over time. A higher value of σ indicates agreater degree of risk aversion and a stronger preference for consumption smoothing.Consider a two-period endowment economy with a large number of identical house-holds. Each household has the following lifetime utility function:Ct(j)1-01-0 - 1U(j)1-σ+ B8 (Ce+1(1)1-0 -1)where Ct (j) and C++1(j) are consumption in periods t and t + 1 for household j, re-spectively, ẞ is the discount factor, and σ > 0 is the coefficient of relative risk aversion.All households are endowed with an exogenous amount of income, Y+ in period t andY++1 in period t + 1. Households can borrow or lend at a common real interest rate, rt.intertemporal Budget ConstraintY₁ + 1+1 = C++C++11+rDeriving the Euler EquationSimplifying, we find the Euler equation:C = B(1+r)Which can also be written as:-0C++1Ct=· ß(1+r)Deriving the Optimal Consumption FunctionC++1 = (ẞ(1+r))³ C₁Final SolutionThe final expressions for optimal consumption in periods t and t + 1 are:Y++1Ye+++C+ =1+ (1+r)

The notion at discussion is the Constant Relative Risk Aversion (CRRA) utility function in a…

D) Discussing the Equilibrium in the Endowment Economy: In the endowment economy, aggregate consumption must equal aggregate endowment in each period. Using this condition and the optimal consumption functions derived, discuss how the equilibrium real interest rate rt is determined in this economy. Explain the role of the real interest rate in ensuring equilibrium in the lending market and why it is necessary for achieving this equilibrium (e) Analyzing the Effect of Changes in Relative Risk Aversion:Given the consumption functions, discuss how changes in the coefficient of relative risk aversion (σ) affect the household’s consumption choices and the sensitivity of these choices to changes in the real interest rate. Provide an intuitive explanation for your findings.

D) Discussing the Equilibrium in the Endowment Economy: In the endowment economy, aggregate consumption must equal aggregate endowment in each period. Using this condition and the optimal consumption functions derived, discuss how the equilibrium real interest rate rt is determined in this economy. Explain the role of the real interest rate in ensuring equilibrium in the lending market and why it is necessary for achieving this equilibrium (e) Analyzing the Effect of Changes in Relative Risk Aversion:Given the consumption functions, discuss how changes in the coefficient of relative risk aversion (σ) affect the household’s consumption choices and the sensitivity of these choices to changes in the real interest rate. Provide an intuitive explanation for your findings.

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter3: Preferences And Utility

Section: Chapter Questions

Problem 3.9P

See similar textbooks

Similar questions

Betty is looking for a job. She considers job opportunities intwo cities. Bettyís utility is given by y- x, where y is the lifetime income andx is the amount spent on buying a house. The income from City 1 fluctuatesalthough the house price is stable. On the contrary, the income from City2 is stable while the house price fluctuates. If she moves to City 1, Bettycan earn a lifetime income y1 with probability alpha and 1 + y1 with probability1-alpha . The house price in City 1 is x1. Moving to City 2 means that Bettycan earn an income of y2. However, the house price is x2 with probabilitygamma and 1 + x2 with probability 1-gamma . Do the following: (a) Write down theexpected utilities associated with living in the two respective cities, i.e., V1and V2. (b) Derive the condition under which Betty chooses City 1.
The consumer choice is not restricted to the choice of consumptiongoods. In fact, it can apply to all our decisions that involve trade-offs. Suppose Mary has awage per hour of 10 euros. With her earned income she consumes. That isC=wH per day.She also decides how many hours to work of take leisure time each day.H=24-N, whereHis work and N is leisure. Her utility is given by （picture） Solve for the optimal decision of labor/leisure. Plot the budget constraint and the indif-ferent curve. What is the labor supply function?
Zach's preferences are representable by the utility function u = 90.3927, where 9₁ and 92 denote his consumption of goods 1 and 2. (Answers to each of these questions are rounded, where required, to two decimal places.) Still assuming endowments of e₁ = 5 and e₂ = 9 and market prices p₁ = 20 and p2 = 30, what is the maximised value of Zach's utility? O Au= 6.74 O B. u = 5.39 O Cu = 5.65 O D.u = 7.56 O E. None of the above
You and a coworker are assigned a team project on which your likelihood or a promotion will be decidedon. It is now the night before the project is due and neither has yet to start it. You both want toreceive a promotion next year, but you both also want to go to your company’s holiday party that night.Each of you wants to maximize his or her own happiness (likelihood of a promotion and mingling withyour colleagues “on the company’s dime”). If you both work, you deliver an outstanding presentation.If you both go to the party, your presentation is mediocre. If one parties and the other works, yourpresentation is above average. Partying increases happiness by 25 units. Working on the project addszero units to happiness. Happiness is also affected by your chance of a promotion, which is depends on howgood your project is. An outstanding presentation gives 40 units of happiness to each of you; an aboveaverage presentation gives 30 units of happiness; a mediocre presentation gives 10 units…
Anna has endowment 1500 now and 500 later. Internet rate is 2.0%. She prefers smooth consumption to time (i.e., u0=u1=u). a. Assume utility function, u(c)= log c. What are the optimal consumption c0and c1if Anna's beta=1, and she wants to maximize her utility? b. Now assume that the utility function, u(c)=c0.5. If everything else remains the same as Problem 1(a), what are the optimal consumption c0and c1if Anna wants to maximize her utility?
In class discussions about uncertainty we assumed that the utility levels in each state of nature depends on c, which we might interpret as some aggregate con- sumption and we expressed utility as U(c). Now, let's extend this to a case where the utility level depends on consumption of two goods (this was the type of utility we used mainly in this course). Ben is a farmer who grows wheat and barley. However, his harvest is uncertain. If weather is good, he gets 200 lbs of wheat and 200 lbs of barley. If weather is bad, he gets only 100 lbs of wheat and 100 lbs of barley. His utility in each state of nature is U(w, b) = w¹/4b³/4, where w and b represent his consumption of wheat and barley, respectively. Prices of wheat and barley are $1 in both state of nature. The probability of good weather is T. Question 3 Part a Express Ben's expected utility function. (Hint: find Ben's optimal consumption in each state of nature first) Question 3 Part b Let's assume = 0.5. Knowing that bad weather…
In class discussions about uncertainty we assumed that the utility levels in each state of nature depends on c, which we might interpret as some aggregate con- sumption and we expressed utility as U(c). Now, let's extend this to a case where the utility level depends on consumption of two goods (this was the type of utility we used mainly in this course). Ben is a farmer who grows wheat and barley. However, his harvest is uncertain. If weather is good, he gets 200 lbs of wheat and 200 lbs of barley. If weather is bad, he gets only 100 lbs of wheat and 100 lbs of barley. His utility in each state of nature is U(w, b) = w¹/46³/4, where w and b represent his consumption of wheat and barley, respectively. Prices of wheat and barley are $1 in both state of nature. The probability of good weather is π. Question 3 Part a Express Ben's expected utility function. (Hint: find Ben's optimal consumption in each state of nature first) Question 3 Part b Let's assume π = 0.5. Knowing that bad weather…
2. Mr. A has the following utility function and budget constraints: Max 0.1Ln(C1) + 0.7Ln(C2) Subject to S1 + C1 = 100 C2 + S2 = (1 + r)S1 where C1 and C2 are consumption level at young and that at old respectively. Likewise, S1 and S2 are saving at young and saving at old respectively. a) Find out Mr. A’s optimal consumption levels (i.e. C1*, C2*) and optimal savings (i.e. S1*, S2*) in terms of interest rate r. b) Show clearly the results in part a) in a suitable diagram (with C1 as x-axis and C2 as y-axis). c) Is Mr. A a saver ? or a borrower ? d) If r is equal to 0 (i.e. saving gives no returns), will Mr. A still choose to save when he is young (i.e. is S1 still bigger than 0) ? Why ? e) Suppose that Mr. A is not allowed to save (i.e. S1 = 0). What are his optimal consumption levels ? Show his optimal consumption levels in the same diagram you prepare for part a) (with a suitable indifference curve). f) If r increases,…
1. Suppose that the representative consumer has a utility function defined over consumption over two dates of the form U(C₁, C₂) = c₁²c₂². The general form of the slope of the indifference curve for the representative consumer is - C₂/C₁. Moreover, remember that c₁ = y₁ − S and C₂ = y₂ + s(1 + r). a. Assume that the representative consumer has an endowment of consumption goods in the two periods of y₁ = 20 and y₂ = 10. Assuming an interest rate r = 1, compute the equilibrium allocation and the implied savings. b. Suppose that, because of an attack of pessimism, the representative consumer assumes that future income will drop so that y₂ = 0. What happens to the savings s in the first period? c. In the previous part, the interest rate remained at 1. Now, consider the savings function, that is, the relationship between the real rate of interest and the amount saved. The equilibrium interest rate is then determined as a market price in the Saving-Investment diagram. Given the typical shape…
Suppose the administration of a University that is located a high-crime urban area mustdecide on a policy to increase policing on its campus. The student population is 10,500 students but iscould increase if crime is reduced. The cost of added policing in terms of additional salary and equipmentis estimated to be $2.4 million per year. Suppose the administration plans to make their decision with thehelp of a benefit cost analysis of the policy. VSL is $11.8 million a. Explain how the concept of Value of a Statistical Life might factor into a benefit cost analysis of thispolicy. b. How many lives would need to be save to justify the added policing costs if only prevented deathswere considered as the policy benefit? c. Suppose it is believed that student enrollment will increase in the future if crime on campus is reduced.Write a formula to calculate the present value of benefits of policing if student enrollment increases by4% each year.
Consider a town with a single street of 1 km long with 3,000 people spread uniformly along it. Two stores, 1 and 2, are located at the opposite ends of the street and sell the same product (store 1 is locatedattheleftend).Thecostofwalkingist1 =$6perkmtostore1andt2 =$9perkmtostore2for each consumer. The net utility of a consumer located at point x from buying a product at store 1 is U1(x) = 100 – p1 – t1x, where pi is a price of the product at store i = 1,2. The net utility from buying at store 2 is U2(x) = 100 – p2 – t2(1 – x). The average cost of the product for each store is c = 4. (a) Assume that all consumers buy product from the sellers. Find the demand functions Di(p1,p2) and the profit functions πi(p1,p2) for each store i = 1,2 as functions of prices p1,p2.(b) Find the equilibrium prices.
Suppose the consumer's utility function is given by U(x,y) = X3/5y2/5, and the exogenous variables are given by M = 500, Px = 5, and Py = 4. Price of y changes to Py = 2. What is the income effect ony?O A.-14.53O B. +14.53O C. +24.21O D. +25.79O E. None of the above