in the second, whereN = N1 + Ng.Households have access to a credit market where they can borTow (s < 0) or saves>0. The type 1 agent faces budget constraintsY = c+srs!where consumption for the type i agent in period j is denoted e. The type 2 agentfaces budget costraints2 +rsThe resource constraints areNye+ NạGNic+ NạNa2 =(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 functiontheorem.(b) Determine the equilibrium conditions for the three markets using the resourceconstraints and the budget constraints. Provide a statement of the equilibrium.(c) Assume logarithmic utility U(e) = In(c) and derive a closed form solution forconsumption in both periods and savings for both types of agents.(d) Solve the social planning problem. Compare the solution of the social planningproblem with the competitive equilibrium. Demonstrate that the decentralizedsolution solves the social planning problem for a particular set of Pareto weights.Explain how this is an example of the First Welfare Theorem. 2. Constructing an EquilibriumHouseholds live two periods and have preferencesU(c) + BU(c2),where 0 < B < 1 and U is the utility function and satisfies our usual assumptions.There are Ñ households in the economy. N¡ of these households have endowment y,in the first period and no endowment in the second - these agents are called “Type 1".The remaining N2 have no endowment in the first period and y2 in the second period- these agents are called “Type 2." Hence the resources of the economy arein the first period and

Equilibrium is an economic concept, where two market forces, i.e. demand and supply equate each…

(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. (e) Assume logarithmic utility U(e) - In(e) and derive a closed form solution for consumption in both periods and savings for both types of agents. (d) Solve the social pla

(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. (e) Assume logarithmic utility U(e) - In(e) and derive a closed form solution for consumption in both periods and savings for both types of agents. (d) Solve the social pla

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter3: Preferences And Utility

Section: Chapter Questions

Problem 3.9P

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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.
2. Constructing an Equilibrium Households live two periods and have preferences U(ci) + BU(c2), where 0 0. The type 1 agent faces budget constraints Y1 c+s' rs' %3D where consumption for the type i agent in period j is denoted c. The type 2 agent faces budget cOnstraints Y2 +rs2 = 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. (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) Solve the social planning problem. Compare the solution of the social planning problem with the competitive equilibrium. Demonstrate that the decentralized solution solves the…
1. Suppose there are two consumers, A and B. The utility functions of each consumer are given by: UA(X,Y) = X*Y UB(X,Y) = X*Y3 Therefore: • For consumer A: MUX = Y; MUY = X • For consumer B: MUX = Y3; MUY = 3XY2 The initial endowments are: A: X = 10; Y = 6 B: X = 14; Y = 19 show all work a) Suppose the price PY = 1. Calculate the price of X, PX that will lead to a competitive equilibrium. b) How much of each good does each consumer demand in equilibrium? Consumer A’s Demand for X: Consumer A’s Demand for Y: Consumer B’s demand for X: Consumer B’s demand for Y: c)What is the marginal rate of substitution for consumer A at the competitive equilibrium?
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
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…
Exercise 4 Consider an economy with two consumers, Alexia and Bart, who live two periods, t = 0 and t = 1. In each period they can consume one type of good and their preferences for consumption are given by U (co, c²) = c(c²)² _i = A, B. Alexia and Bart have the following endowment of good in each period M=1, M₁ = 1, MB = 2, MB = 2. In t = 0, Alexia and Bart can exchange a financial contract for the delivery of one unit of consumption good in t = 1 (a bond). Name p the price of the bond and b² the amount bought by agent i = =A, B. (a) Write down each agent's utility maximization and budget constraints assuming that he/she can trade the bond without restrictions. (b) Find each agent's optimal quantity b² as a function of the bond net return r. (c) Find the equilibrium value of r and the equilibrium demand/supply of each agent.
2. General Equilibrium. consumers, each with the same Cobb-Douglas preferences except with differ- ent parameters. Consumer 1 has utility function u(x, x)= (x})"(x})!-«, while Consumer 2 has utility function u(x, x;) = (xP(x)-P. The endowment of good j owned by consumer i is denoted w. The price of good 1 is p, and the price of good 2 is 1. In the superscript, we denoted the consumer i = 1,2; in the subscript, we denote the good j= 1,2. Consider an exchange economy with two Write the maximisation problem faced by each consumer i = (a) 1,2, taking care to define the objective function and the budget constraint. Set up the Lagrangian and find the first order conditions. (b) For each consumer i = 1,2 , use the first-order conditions to determine the demand functions for each consumer i = 1,2 and for each good j = 1,2, in terms of the price p. (c) Find the aggregate demand for each good j = 1,2 and clear the markets for each good. Hence, show that the equilibrium price pi is given by the…
1. [This question gives sum kind of foundation for "adding up utilities" as a mea- sure of efficiency.] Consider an exchange economy with finitely many consumers I and n > 1 goods. Let Z := R¹ and X = Z × R, and suppose each consumer i E I has a utility u : X → R and an endowment w E X. For the purposes of this question, a bundle is any member of X (in particular, consumers are allowed to consume negative quantities of good n). As usual, and allocation is any profile of consumption bundles such that the sum of all consumption is equal to the sum of all the endowments (so what people consume as a group is the same as what they start with as a group). Suppose every agent has preferences that are quasilinear in the nth good, and so has some v¹: R-¹ such that any z' – (z', r'′) has u¹(x¹) = v¹(zª) +. Say an allocation = (z¹, ™½)ie, is surplus maximizing if every other allocation I=(2, 2)ier has Eier (2) 2 Σier (2¹) (a) Show that, if the allocation à is surplus maximizing, then it is Pareto…
Suppose there are two consumers. A and B. The utility functions of each consumer are given by: U₁XX)-x²-y U₂XY)-x-² Therefore For consumer A: MUX-2XY: MU-X² For consumer B: MUx-Y²; MUy-2XY The initial endowments are: A:X-60; Y-150 B.X-45: Y-75 al Suppose the price of Y, Py-1. Calculate the price of X, P, that will lead to a competitive equilibrium. How much of each good does each consumer demand in equilibrium? Consumer A's demand for X: Consumer A's demand for Y Consumer B's demand for X: Consumer B's demand for Y d ts) What is the marginal rate of substitution for consumer A at the competitive equilibrium?
4. Consider an exchange economy of two goods and two individuals. Consumer A has an endowment of 100 units of good 1 and 12 units of good 2 wA = (100,12), while consumer B has an endowment of 100 units of good 1 and 3 units of good 2 wB = (100, 3). The consumers' utility functions are given by: UA х,4 + Inx,A and UB x,B + 2lnx,B %3D Which of the following allocations is not efficient? a. (x,4,x24) = (100,5), (x,³, x2³) = (100,10) b. (x,4, x24) = (50,5), (x,®, x,³) = (150,10) c. (x,4, x24) = (0,4), (x,³, x,") = (200,11) d. (x,4, x24) = (100,0), (x1",x,") = (100,15) %3D
Exercise 4 Consider an economy with two consumers, Alexia and Bart, who live two periods, t = 0 and t = 1. In each period they can consume one type of good and their preferences for consumption are given by U (co, ci) = c(ci)² _i = A, B. Alexia and Bart have the following endowment of good in each period M=1, M₁ = 1, MB = 2, MB = 2. In t = 0, Alexia and Bart can exchange a financial contract for the delivery of one unit of consumption good in t = 1 (a bond). Name p the price of the bond and b² the amount bought by agent i = =A, B. (a) Write down each agent's utility maximization and budget constraints assuming that he/she can trade the bond without restrictions. (b) Find each agent's optimal quantity b² as a function of the bond net return r. (c) Find the equilibrium value of r and the equilibrium demand/supply of each agent.