1. OLG model with and without a pension scheme. In an overlapping generationseconomy a representative consumer lives for two periods denoted by t = 1,2. Shereceives w units of endowment in period 1 and no further endowment in period 2,while she leaves no bequests. Her consumption in the two periods is c, and c2, andher utility function is given byc - 1+B1- 0ct-o - 1U =-σ> 0, βε(0,1).1- 0Population grows at a constant rate, n > 0, and every young agent has access to astorage technology with gross rate of return (1 + r).(a) Write down the problem for a young agent and, given factor prices, solve for theoptimal allocation of her endowment between consumption in the two periodsand savings.(b) Now set o = 1 (i.e., logarithmic utility) and assume there exists a social securitysystem in the economy such that young agents pay lump-sum taxes in theamount t each period, and when they get old, they receive a pension in theamount p. Assume that the system is self-financed in each period so that p =(1 + n)t. Solve again for the individual's optimal levels of consumption andsavings.(c) Continuing with o = 1, describe the single condition (between n and r) underwhich the social security system improves the optimal lifetime utility for theindividual (Hint: first solve for the optimal consumption allocations in the absenceof the social security system, and then compare the levels of optimal utility withand without the social security system). Explain.(d) Continuing with o = 1, and in the absence of a social security system, supposethe representative consumer now also has a second period endowment equal tow(1 + y) units, where y > 0. Illustrate how y affects the individual's optimalsavings. Explain.

Given , Utility function : U = c11-σ-11-σ + βc21-σ-11 -σ In period t1 Income in period t1 = w…

economy a representative consumer lives for two periods denoted by t = 1,2. She receives w units of endowment in period 1 and no further endowment in period 2 while she leaves no bequests. Her consumption in the two periods is c, and c2, and her utility function is given by ct-o - 1 U + B 1-0 c - 1 o > 0, Be(0,1). 1-0 Population grows at a constant rate, n > 0, and every young agent has access to a storage technology with gross rate of return (1 + r). (a) Write down the problem for a young agent and, given factor prices, solve for the optimal allocation of her endowment between consumption in the two periods

economy a representative consumer lives for two periods denoted by t = 1,2. She receives w units of endowment in period 1 and no further endowment in period 2 while she leaves no bequests. Her consumption in the two periods is c, and c2, and her utility function is given by ct-o - 1 U + B 1-0 c - 1 o > 0, Be(0,1). 1-0 Population grows at a constant rate, n > 0, and every young agent has access to a storage technology with gross rate of return (1 + r). (a) Write down the problem for a young agent and, given factor prices, solve for the optimal allocation of her endowment between consumption in the two periods

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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An agent lives for 2 periods and gains utility only by consuming. Future consumption is discounted at rate θ where 0<θ<1. Agent consumes c, saves s at interest rate r in first period of life from income y. In second period agent consumes from savings plus interest earned but must pay a flat fixed tax. Find the Euler equation for this agent and what it shows.
1. If we observe a consumer choosing (x1, x2) when (y1, Y2) is available one time, are we justified in concluding that (x1,x2) > (y1, Y2)? 2. Consider a group of people A, B, C and the relation "at least as tall as," as in “A is at least as tall as B." Is this relation transitive? Is it complete? 3. Take the same group of people and consider the relation “strictly taller than." Is this relation transitive? Is it reflexive? Is it complete?
I JUST WANT THE DIAGRAM FOR EACH PART. PLEASE DRAW THE DIAGRAMS, DONT TYPE IT!!!!!! WRONG ANSWERS WILL BE REPORTED. Consider the representative consumer who decides consumption and leisure. The preference is given by U (C,L) = αln C + (1 −α) ln L. Assume h = 1, i.e., the time endowment is one day. Suppose the non-wage income π −T increases while the wage rate w falls at the same time. The size of the changes can be different. Determine the effects on consumption demand and labour supply (i.e., leisure demand). Use the indifference map to explain your results in terms of income and substitution effects for the following cases: (i) The increase in π −T exactly cancels out the drop in w, i.e., |∆ (π −T)|= |∆w|. (ii) The increase in π −T is greater than the drop in w, i.e., |∆ (π −T)|> |∆w|. (iii) The increase in π −T is smaller than the drop in w, i.e., |∆ (π −T)|< |∆w|.
2. Constructing an Equilibrium Households live two periods and have preferences U(c) + BU(c2), where 0 0. The type 1 agent faces budget constraints Y1 c+s' rs' where consumption for the type i agent in period j is denoted c. The type 2 agent faces budget constraints G + s² Y2 +rs? = The resource constraints are Nịc + N2c? Nịc+ N2c (a) State the maximization problem solved by each type of agent and derive the first- order and second-order conditions. Derive the solution using the implicit function theorem. (b) Determine the equilibrium conditions for the three markets using the resource constraints and the budget constraints. Provide a statement of the equilibrium. (c) Assume logarithmic utility U(c) = In(c) and derive a closed form solution for consumption in both periods and savings for both types of agents. (d) Sel
2. Constructing an Equilibrium Households live two periods and have preferences U(c) + BU(c2), where 0 0. The type 1 agent faces budget constraints Y1 c+s' rs' where consumption for the type i agent in period j is denoted c. The type 2 agent faces budget constraints G + s² Y2 +rs? = The resource constraints are N1c + N2c Nịc+ N2c (a) State the maximization problem solved by each type of agent and derive the first- order and second-order conditions. Derive the solution using the implicit function theorem.
Q7: An individual lives for only two periods and has preferences given by the follow- ing intertemporal utility function: U = lng + aln(1-h₁) + B[lno₂+ a ln(1 — h₂)] (1) where, c₁, c₂ denotes consumption in period 1 and period 2 respectively h₁, h₂ denote the labour supply in period 1 and 2 respectively. Therefore 1-h is the amount of leisure time in period 1. The term 3 is the discount factor. The problem of the individual at period 1 is to choose consumption in both periods and labour supply in both periods subject to the following budget constraints: +8= why and 0₂= w₂h₂ + (1+r)s where s denote the saving, w denotes the wates and r denotes the real interest rate. (a) Provide an economic interpretation of the two budget constraints written above. (b) Combine the two budget constraints written above and prove the economic interpretation of of life time budget constraint. (c) Set the Lagrangean function and find the first order conditions with respect to C₁ C₂, hi and h₂. (d) Find the…
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A decision maker allocates an endowment of W > 0 dollars across two periodst = 1, 2. He discounts the future by β ∈ (0, 1) while facing a gross interest rateof R > 1. His utility is the same as studied in class. Solve for the intertemporalchoice problem. Show that the optimal consumption is decreasing over time ifβR < 1, constant over time if βR = 1, and increasing over time if βR > 1.
(a) Will dynastic preferences, such as those discussed in Section 5.2, lead to infinite-horizon maximization if the instantaneous utility functions of future generations are different (ut(.) potentially different for each generation t)? (b) How would the results be different if an individual cares about the continuation utility of his offspring with discount factor β, but also cares about the continuation utility of the offspring of his offspring with a smaller discount factor δ?
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