# Introduction. Agents have preferences over the two goods which are determined by a utility function. Speci cally, type 1 agents utility is given by

Save this PDF as:

Size: px
Start display at page:

Download "Introduction. Agents have preferences over the two goods which are determined by a utility function. Speci cally, type 1 agents utility is given by"

## Transcription

1 Introduction General equilibrium analysis looks at how multiple markets come into equilibrium simultaneously. With many markets, equilibrium analysis must take explicit account of the fact that changes in prices (resulting from changes in supply or demand) in one market typically a ect the demand or supply in other markets, resulting in changes in prices in these markets as well. While this fact is implicitly assumed in partial equilibrium analysis, the adjustment process in other markets is ignored in this mode of market analysis. In general equilibrium analysis, we track all adjustments simultaneously. The interactions of markets lead to feedback e ects between markets which can result in signi cant non-linearities in the market. These non-linearities, in turn, can give rise to complicated equilibrium price behavior, even when underlying preferences and production processes are nicely behaved. To illustrate this, we will begin our study of general economic equilibrium by looking at an extended example of the simplest possible general equilibrium model that of a pure exchange economy having no production. An Extended Example Our extended example involves a pure exchange economy without production. There are two goods in the economy, denoted x and y; and two types of agents who are identical in terms of their basic economic characteristics (but who may, if we wish, have di erent hair color, eye color, etc.). Economic characteristics in the model are the agents preferences and endowments. Agent Characteristics Indexing agents by i = ;, we denote the endowment of an agent of type i by! i = [! x i ;!y i ] R +: The endowment is simply the goods that the agent owns at the outset, prior to trading with other agents. Agents have preferences over the two goods which are determined by a utility function. Speci cally, type agents utility is given by u (x ; y ) = Type agents utility is given by u (x ; y ) = " x + " 7 7 y # : # x + y : You may recognize these utility functions as being of the CES (constant elasticity of substitution) type. Note that our assumption that agents of each

2 type have the same economic characteristics is re ected in the assumption that each type of agent has the same preferences and endowments. Allocations An agent of type i 0 s consumption bundle is denoted (x i ; y i ) for i = ; : An equal-treatment allocation for the economy is a pair of consumption bundles [(x ; y ) ; (x ; y )] which speci es the per-capita consumption of each type of agent, together with the assumption that each agent type gets the same consumption bundle. (The reason for imposing this assumption will become clear shortly.) An allocation is said to be feasible if it uses no more than the total resources available. Thus, for the equal treatment allocation to be feasible requires that Nx + Nx N! x + N! x and Ny + Ny N! y + N!y or (since there is a factor of N on both sides) and x + x! x +! x y + y! y +!y : It should be apparent from this calculation that as long as we work with equal treatment allocations, it is su cient to work in per capita terms, which reduces the model to one in which we need only deal with two agents (one representative agent of each type). Our reason for assuming that there are N agents (and that N is large) is to justify the assumption that agents act as price takers, so that our economy s markets are competitive. We will impose one additional (and innocuous) assumption on the model by normalizing the total amount of each good to one unit, so that! x +! x =! y +!y = : This assumption is innocuous since the allocations then simply become shares of the total amount of each good allocated to each agent. Markets Since we are interested in market-mediated exchange, we assume that there are two competitive spot markets for the two goods. Agents observe the posted exchange prices on the market and make decisions as to whether they wish to sell some amount of a good they have in excess. If they choose to do so, they may use the proceeds from the sale to purchase goods they are de cient in. To

3 formalize this idea, let p x and p y denote the prices of each good. agent s budget constraint is then given by A typical p x x i + p y y i p x! x i + p y! y i : This inequality states that the value of whatever the agent purchases must be less than or equal to her income, which in this model is simply the value of her total endowment of each of the goods. One way to think of this market is that every agent brings their endowment to market and sells all of it. They receive a credit on the bank account for the value of this endowment at the going prices. They can then spend this credit on the market for each good as they wish. Note that one particular purchase is always feasible for our agent: they can always simply buy back their endowment. Whether the agent wishes to trade (i.e. buy a bundle (x i ; y i ) di erent from their endowment bundle) will depend on whether the agent s demand at the going prices is di erent from the endowment bundle (! x i ;!y i ). The demand bundle is de ned as the solution to the agent s constrained utility maximization problem max u i (x i ; y i ) (x i;y i) subject to p x x i + p y y i p x! x i + p y! y i : h i A competitive equilibrium for the economy is then an allocation ^x = (bx i ; by i ) N i= together with prices bp = [bp x ; bp y ] such that the allocation is feasible and solves each agent s budget-constrained utility maximization problem at prices bp: E ciency We say that an allocation x is e cient or Pareto optimal if it is feasible and there is no other feasible allocation y such that u i (y) u i (x) for all i, with at least one strict inequality. Hence, an e cient allocation is one in which it is impossible to make some subset of agents better o (in utility terms) without making some other agent worse o. In our simple model with two goods and two types of agents, it is easy to characterize the e cient allocations, using the concept of the Edgeworth box diagram. A typical Edgeworth box diagram is illustrated below. In the diagram, the size of the box is determined by the total amounts of each good available, which we denote as r x for the rst good, and r y for the second. By de nition, r i =! i +! i, for i = x; y. Any point in the box represents a feasible allocation, since if agent gets (x ; y ) ; then agent gets [(r x x ) ; (r y y )] : We can also graph each agent s indi erence map in the box, as shown above, taking the lower left corner as the origin for agent, and the upper right as the origin for agent. The e cient allocations occur at the tangency points of the two agents indi erence curves. To see this, start at an allocation such as x 0 in the diagram above. This allocation is not

4 e cient, since moving in a northwesterly direction will make both agents better o. To see that the e cient allocations require that indi erence curves be tangent, consider reallocations that keep agent indi erent between x 0 and the reallocation. This amounts to moving along the indi erence curve through x 0 ; to the northwest. Clearly, this makes agent no worse o, while it makes agent strictly better o. This remains true until we reach an allocation at which agent two s indi erence curve is just tangent to agent s. Equivalently, this requires that the two agents utility gradients be colinear. The locus of tangencies in the Edgeworth box is called the contract curve. The contract curve contains all of the e cient allocations. We can calculate the contract curve to our example by requiring that allocations be feasible (which we do by stipulating that agent gets whatever we don t give agent ), and that the two agents marginal rates of substitutions at the allocation be the same. 4

5 The MRS s for each agent are give by MRS = MRS = 7 y x ry y : 7 r x x Taking r x = r y = and equating, we get 7 y x = y 7 x or 7 ( x ) = y x y which when solved for y gives us y = h 7 + x 7 i x : As an exercise, plot the contract curve for this model. Market-mediated Exchange We are interested in examining the competitive equilibrium for this model. To do this, we rst calculate the demand functions for each agent, add them up to get the aggregate demand function, and then equate aggregate demand to aggregate supply and solve for the competitive equilibrium price. Note that we really only need to nd the aggregate demands for one of the two goods, since Walras Law implies that once one market clears, so does the other. This is a consequence of the budget constraints, and can be demonstrated as follows. Consider the two agents budget constraints: p x x i + p y y i = p x! x i + p y! y i : Adding over the two agents (or over all the agents of each type), we have p x x + p y y + p x x + p y y = p x! x + p y! y + p x! x + p y! y ) p x (x + x ) + p y (y + y ) = p x (! x +! x ) + p y (! y +!y ) ) p x (x + x! x! x ) + p y (y + y! y! y ) = 0: Now, suppose prices are such that the market for the rst good clear, x + x! x! x = 0: Then, necessarily, p y (y + y! y! y ) = 0; which requires either that p y = 0 or y + y! y! y = 0. We can rule out the rst case by noting that with strictly monotonic preferences, if one good becomes free, the demand for it becomes in nite. Hence, we must have the second case, which tells us that the market for good clears. 5

6 We will, therefore, limit our attention to the market for the rst good (i.e.good x). To nd the individual demand functions, we start from the rstorder conditions for the budget constrained utility maximization problem, in the form that requires the marginal rate of substitution to be equal to the price, with the budget constraint satis ed. For the rst agent, these conditions require 7 y = p x x p y p x x + p y y = p x! x + p y! y : Note from these conditions that the price level is indeterminate, in the sense that if we scale the price of each good by the same amount, it leaves the price ratio unchanged, and the scaling factor appears on both sides of the budget constraint. This is just the usual result from microeconomics which states that only relative prices matter, once we take account of the fact that people earn their incomes by selling goods and services they are endowed with at going market prices. In this situation, we are free to normalize prices by taking one of the goods as the numeraire, measuring the value of the other good in terms of how many units of the numeraire good it exchanges for. We will let good y be the numeraire, and normalize prices by setting p x =p y = r; the relative price of good in terms of good. With this modi cation, the rst-order conditions become 7 y x = r rx + y = r! x +! y : Solving these two equations simultaneously, we obtain the demand functions x (r;! ) = r!x +! y r + 7 r y (r;! ) = r!x +! y + 7 r : Carrying out the parallel calculation for agent, we nd x (r;!) = r!x +! y r + 7 r y (r;! ) = r!x +! y + 7 r : As an exercise, carry out the calculations involved in solving the rst-order conditions for the demand functions. By Walras Law, we need only consider the market for good x. market, the aggregate demand is For this x (r;! ) + x (r;!) = r!x +! y r + 7 r 6 + r!x +! y r + 7 r :

7 To nd the competitive equilibrium for this market, we de ne the aggregate excess demand function z x (r;!) = x (r;! ) + x (r;! )! x! x = r!x +! y r + 7 r + r!x +! y r + 7 r! x! x : At the competitive equilibrium price ^r; we will have z x (^r;!) = 0: Hence, we need to nd the zeroes of the excess demand function. To do this, let us simplify our notation. Let = =7; and = r : Then z x (r;!) =! x +! y + +! x +! y +! x! x! x +! y h i + +! x +! y + = i! [ + ] h x +! x : In this expression, we can factor an out of each term in the denominator and cancel this against a factor of in the numerator to get z x (r;!) =! x +! y h i + [ + ] +! x +! y + h + i! x! x : Now, for the purposes of nding equilibrium, the monotonicity of preferences implies that any equilibrium price must be positive, which means that > 0: This in turn implies that the denominator of the rst term of z x (r;!) is always positive, so it is su cient to nd the zeroes of the polynomial equation! x +! y + +! x +! y = 0 (taking account of the fact that! x +! x = ): When we multiply the terms out and collect like powers of ; we get! 5 [! x +! x ] + x +!x + [! y +!y ] +!y +!y = 0: Again, using the facts that! x +! x = ; and! y +!y = this reduces to! x +!x + +!y +!y = 0: We can simplify this expression further by noting that since the initial endowment is itself an allocation, we have! x =! x ; and! y =!y : Making these substitutions, we have "! x # + + +! y = 0 7

8 or, putting the expression in canonical form, " # " # +! x! x + +! y! x = 0: As an exercise, carry through the details of this set of calculations to go from the initial expression for z x (r;!) to the equation above. Cases: Consider the case rst where! x = and! y = 0 (so that agent owns all of the rst good, and none of the second). Then the excess demand polynomial becomes + = 0: By inspection, one root of this equation is = : Dividing the polynomial by ( ) ; we get + = ( ) + ( ) + : With = =7; this is ( ) = ( ) = ( ) : Thus, the excess demand function has three real roots, = ; = =4; and = 4=: These correspond to competitive equilibrium prices of, (=4) ; and (4=) respectively. This situation is shown graphically in Figure. Now consider what happens when! x = 7=49 and! y = =49: Substituting these values in the excess demand polynomial, we get the equilibrium equation + = 0 which factors into ( ) + = 0: This equation has only one real root, so for this distribution of endowments, the competitive equilibrium is unique. To understand why we get these di erent results, consider the graph of the excess demand function for good under the rst distribution of endowments, shown in gure. 8

9 For this case, consider the standard tatonnement procedure for nding the competitive equilibrium prices. Pick an initial price r 0 : If the excess demand at that price is positive, increase the price. If it is negative, decrease the price. Continue adjusting prices in this way until you arrive at an equilibrium. If we apply this procedure to any initial price with r 0 less than, we converge to the equilibrium at (=4) : If we start with r 0 > ; we go to the equilibrium at (4=) : For this tatonnement procedure, the equilibrium with r = is unstable; starting in any neighborhood of r = ; the tatonnement diverge from that neighborhood. Note the curious fact here: if r is close to but not equal to, then increasing the price increases excess demand, while decreasing the price decreases excess demand. To see why this happens, we need to look at the relationship between income and substitution e ects in the model. Recall that for this case, agent makes all of her income from the sale of the rst good, while agent makes her income from the sale of the second. To analyze this, we will consider the e ects of changes in the price of good in various cases, which are illustrated in Figure 4. Consider situation A, where the price of good one is low relative to that of good two, and we increase the price of good one slightly. Increasing the price 9

10 of good one has two e ects. First, it makes good one relatively more expensive than good. Second, it increases agent s income since the value of what she sells is now greater. The increase in relative prices will generate a standard negative substitution e ect, while the increase in value of agent s endowment will lead to a positive income e ect (since these are normal goods). Whether agent s overall demand for good increases or decreases depends on which of these two e ects dominates. In case A, it should be clear from the diagram that the substitution e ect dominates, and increasing the price of good decreases agent s demand for good. Since agent experiences no income e ects due to the increase in the value of good, but does experience the substitution e ect, her demand for good one also falls. Hence, aggregate demand for good one falls when the price rises. In situation B, the income e ect dominates and an increase in the price of good one generates an increase in agent s demand for good one. While agent s demand for the good continues to decrease when the relative price increases (i.e. substitution e ect dominates), agent s increase in demand is larger, so aggregate demand increases. This is a simple example of how the distribution of wealth can a ect the equilibrium of the economy. You should contrast this to the case where! x = 7=49 and! y = =49: If you check, you will nd that this endowment distribution (with agent having! x = =49 and! y = 7=49) lies on the contract curve and is Pareto optimal to start with. For this distribution of endowments, agents will not nd trade pro table. This situation is frequently referred to as a no-trade 0

11 equilibrium. We saw above that in this case, the equilibrium is unique; in fact, it is easy to show that any no-trade equilibrium must be unique. One of things that economists are frequently interested in is how equilibrium changes when we change the underlying fundamentals of the economy. For the simple model we are examining here, the fundamentals include the distribution of endowments, and the preferences of the agents. We will focus for the remainder of this section on analyzing how changes in the distribution of endowments a ect the equilibrium of the economy. Looking at the e ects of changing preferences can be done using similar techniques. To begin our analysis, let us introduce the set E (r;!) = (r;! x ;! y ) R ++ j Z x (r;!) = 0 : The set E (r;!) is called the equilibrium set of the economy when resources are xed but the distribution of endowments can be varied. By studying the properties of this set, we can provide information about the feasibility of comparative

12 static analysis of equilibrium when we change the distribution of wealth. We ve seen from our analysis above that changing the wealth distribution (even continuously) can lead to changes in the number of (discrete) equilibria, which implies that any relationship between equilibrium prices and endowments must include some discontinuities. Can we determine how prevalent these discontinuities are? If they occur only rarely, then we can be con dent that for the most part, small changes in the distribution of endowments lead to small changes in the equilibrium price (or prices). On the other hand, if there are many discontinuities, then small changes in the distribution of wealth can lead to large, radical changes in the nature of the economic equilibrium. Our jumping o point for this analysis is the equilibrium polynomial " # ^Z (;! x ;! y ) = +! x " #! x + +! y! x = 0: If we let P = " #! x we can write the polynomial as Q = +! y! x This equation can be put into the form by making the substitution + P P + Q = 0: w + Aw + B = 0 w = + P where and A = P [ + P ] B = P + 9P + 7Q : 7 Note that we have implicitly de ned a mapping from our original parameter space to the space of parameters (w; A; B) : To see that this mapping is locally

13 one-to-one, calculate the derivative of the mapping (; P; Q) 7! (w; A; B) : Since 4 w + p A 5 = 4 P ( + P ) B 7 P + 9P + 7Q 5 the derivative matrix is J = P P + P This matrix will be non-singular as long as P 6= : You can check directly that when P = ; this corresponds to (and any value of!y ), so that along the vertical line in the Edgeworth box over! x : = 0:88 we will lose the invertibility of the mapping. But this set has measure zero in the two-dimensional Edgeworth box, so that the two invertible mappings on either side of the line of singularity can be "glued" together by continuity. Let h (r;! x ;! x ) = (w; A; B) be the map de ned above, and consider (subject to the caveat on inverting the invertibility of the mapping) the set n ^E = (w; A; B) R j ^Z o h (; P; Q) = 0 : In canonical coordinates, then, the set ^E is de ned by 5 : ^E = (w; A; B) R j w + Aw + B = 0 : Figure 5 shows a Matlab plot of this set looking down at the surface ^E from above in the direction of the w axis. Note how for values of (A; B) toward the right side of the diagram, the value of w for which w + Aw + B = 0 is unique, while toward the left side of the diagram, there are multiple roots to this equation. We can characterize the parameter values of (A; B) for which the number of equilibria change by de ning the so-called natural projection : ^E! (A; B) R which associates the zeros of the equilibrium polynomial with parameters A and B: The mapping is given explicitly by (w; A; B) = w + Aw + B; A; B : It should be obvious that for any (w; A; B) ^E; this expression is precisely [0; A; B] which is just the projection of ^E on the parameter space. Now, from the diagram, it should be clear that the number of zeroes of the equilibrium polynomial change precisely when we cross one of the folds of the surface. This is equivalent to the statement that the natural projection fails to map onto the

14 parameter space at these points. This can perhaps be best seen from looking at a cross-section of the surface, as in Figure 6. At the two fold points, maps the tangent line at the fold point into a single point. Anywhere else along the curve, it maps the tangent line onto the B axis. From advanced calculus, we know that for a mapping or transformation to exhibit this kind of singular behavior, it must be that the Jacobian matrix D (w;a;b) has less than maximal rank. For the mapping above, this matrix takes the form w + A w D (w;a;b) = : 0 0 This fails the have rank precisely when w + A = 0: Putting this together with the requirement that (w; A; B) ^E; the locus of points at which singularity occurs will be the solutions to q Since w = w + Aw + B = 0 w + A = 0: A ; substituting this into the cubic and solving for A yields A = B : 4

15 This equation is plotted in Figure 7. In the diagram, values of B are plotted along the horizontal axis, with values of A plotted along the vertical axis. Positive values of A correspond to the case where the cubic has only one real root. Negative values of A lying inside the cusp gure above correspond to the case where there are real roots. Negative A 0 s outside the cusp correspond to unique roots, while combinations of A and B lying on the cusp correspond to repeated roots. We call the cusp gure itself the critical set, and denote it by : You can also view the cusp as the projection on the A B plane of the fold region of the surface ^E; as indicated in Figure 8, where the fold region is highlighted. One interesting thing to note here is that the fold region is thin, that is, it has measure zero relative to the natural measure or area on the plane or on a -dimensional surface. This tells us that if we pick an endowment distribution at random, then the probability that we will pick one in is zero. Hence, for regular economies (i.e. those not lying in ), we can be sure that su ciently small variations in the underlying distribution of endowments will generate continuous variations in the equilibrium prices, so that standard comparative static analysis will be valid, with probability one. 5

16

### ECON 301: General Equilibrium I (Exchange) 1. Intermediate Microeconomics II, ECON 301. General Equilibrium I: Exchange

ECON 301: General Equilibrium I (Exchange) 1 Intermediate Microeconomics II, ECON 301 General Equilibrium I: Exchange The equilibrium concepts you have used till now in your Introduction to Economics,

### 2. Efficiency and Perfect Competition

General Equilibrium Theory 1 Overview 1. General Equilibrium Analysis I Partial Equilibrium Bias 2. Efficiency and Perfect Competition 3. General Equilibrium Analysis II The Efficiency if Competition The

### Monotonic transformations: Cardinal Versus Ordinal Utility

Natalia Lazzati Mathematics for Economics (Part I) Note 1: Quasiconcave and Pseudoconcave Functions Note 1 is based on Madden (1986, Ch. 13, 14) and Simon and Blume (1994, Ch. 21). Monotonic transformations:

### Increasing Returns to Scale

Increasing Returns to Scale 1 New Trade Theory According to traditional trade theories (Ricardian, speci c factors and HOS models), trade occurs due to existing comparative advantage between countries

University of British Columbia Department of Economics, Macroeconomics (Econ 502) Prof. Amartya Lahiri Handout 7: Business Cycles We now use the methods that we have introduced to study modern business

### Oligopoly. Chapter 10. 10.1 Overview

Chapter 10 Oligopoly 10.1 Overview Oligopoly is the study of interactions between multiple rms. Because the actions of any one rm may depend on the actions of others, oligopoly is the rst topic which requires

### CAPM, Arbitrage, and Linear Factor Models

CAPM, Arbitrage, and Linear Factor Models CAPM, Arbitrage, Linear Factor Models 1/ 41 Introduction We now assume all investors actually choose mean-variance e cient portfolios. By equating these investors

### Chapter 2 TWO-PERIOD INTERTEMPORAL DECISIONS

Chapter 2 TWO-PERIOD INTERTEMPORAL DECISIONS The decisions on consumption and savings are at the heart of modern macroeconomics. This decision is about the trade-o between current consumption and future

### Representation of functions as power series

Representation of functions as power series Dr. Philippe B. Laval Kennesaw State University November 9, 008 Abstract This document is a summary of the theory and techniques used to represent functions

### Price Discrimination: Exercises Part 1

Price Discrimination: Exercises Part 1 Sotiris Georganas Royal Holloway University of London January 2010 Problem 1 A monopolist sells in two markets. The inverse demand curve in market 1 is p 1 = 200

### c 2008 Je rey A. Miron We have described the constraints that a consumer faces, i.e., discussed the budget constraint.

Lecture 2b: Utility c 2008 Je rey A. Miron Outline: 1. Introduction 2. Utility: A De nition 3. Monotonic Transformations 4. Cardinal Utility 5. Constructing a Utility Function 6. Examples of Utility Functions

### Economics 140A Identification in Simultaneous Equation Models Simultaneous Equation Models

Economics 140A Identification in Simultaneous Equation Models Simultaneous Equation Models Our second extension of the classic regression model, to which we devote two lectures, is to a system (or model)

### Public Goods : (b) Efficient Provision of Public Goods. Efficiency and Private Goods

Public Goods : (b) Efficient Provision of Public Goods Efficiency and Private Goods Suppose that there are only two goods consumed in an economy, and that they are both pure private goods. Suppose as well

### Mathematics for Economics (Part I) Note 5: Convex Sets and Concave Functions

Natalia Lazzati Mathematics for Economics (Part I) Note 5: Convex Sets and Concave Functions Note 5 is based on Madden (1986, Ch. 1, 2, 4 and 7) and Simon and Blume (1994, Ch. 13 and 21). Concave functions

### The Fisher Model. and the. Foundations of the Net Present Value Rule

The Fisher Model and the Foundations of the Net Present Value Rule Dr. Richard MacMinn mailto:macminn@mail.utexas.edu Finance 374c Introduction Financial markets perform of role of allowing individuals

### Midterm March 2015. (a) Consumer i s budget constraint is. c i 0 12 + b i c i H 12 (1 + r)b i c i L 12 (1 + r)b i ;

Masters in Economics-UC3M Microeconomics II Midterm March 015 Exercise 1. In an economy that extends over two periods, today and tomorrow, there are two consumers, A and B; and a single perishable good,

### Collusion: Exercises Part 1

Collusion: Exercises Part 1 Sotiris Georganas Royal Holloway University of London January 010 Problem 1 (Collusion in a nitely repeated game) There are two players, i 1;. There are also two time periods,

### The Market-Clearing Model

Chapter 5 The Market-Clearing Model Most of the models that we use in this book build on two common assumptions. First, we assume that there exist markets for all goods present in the economy, and that

### Lecture Notes 5: General Equilibrium. mrkts

Lecture Notes 5: General Equilibrium We are now ready to analyze the equilibrium in all markets, that is why this type of analysis is called general equilibrium analysis. Recall our graphical overview

### PART IV INFORMATION, MARKET FAILURE, AND THE ROLE OF GOVERNMENT CHAPTER 16 GENERAL EQUILIBRIUM AND ECONOMIC EFFICIENCY

PART IV INFORMATION, MARKET FAILURE, AND THE ROLE OF GOVERNMENT CHAPTER 6 GENERAL EQUILIBRIUM AND ECONOMIC EFFICIENCY QUESTIONS FOR REVIEW. Why can feedback effects make a general equilibrium analysis

### Common sense, and the model that we have used, suggest that an increase in p means a decrease in demand, but this is not the only possibility.

Lecture 6: Income and Substitution E ects c 2009 Je rey A. Miron Outline 1. Introduction 2. The Substitution E ect 3. The Income E ect 4. The Sign of the Substitution E ect 5. The Total Change in Demand

### Utility Maximization

Utility Maimization Given the consumer's income, M, and prices, p and p y, the consumer's problem is to choose the a ordable bundle that maimizes her utility. The feasible set (budget set): total ependiture

### Answer Key: Problem Set 2

Answer Key: Problem Set February 8, 016 Problem 1 See also chapter in the textbook. a. Under leasing, the monopolist chooses monopoly pricing each period. The profit of the monopolist in each period is:

### Lecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization

Lecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization 2.1. Introduction Suppose that an economic relationship can be described by a real-valued

### ME128 Computer-Aided Mechanical Design Course Notes Introduction to Design Optimization

ME128 Computer-ided Mechanical Design Course Notes Introduction to Design Optimization 2. OPTIMIZTION Design optimization is rooted as a basic problem for design engineers. It is, of course, a rare situation

### Unraveling versus Unraveling: A Memo on Competitive Equilibriums and Trade in Insurance Markets

Unraveling versus Unraveling: A Memo on Competitive Equilibriums and Trade in Insurance Markets Nathaniel Hendren January, 2014 Abstract Both Akerlof (1970) and Rothschild and Stiglitz (1976) show that

### 1. Briefly explain what an indifference curve is and how it can be graphically derived.

Chapter 2: Consumer Choice Short Answer Questions 1. Briefly explain what an indifference curve is and how it can be graphically derived. Answer: An indifference curve shows the set of consumption bundles

### Demand. Lecture 3. August 2015. Reading: Perlo Chapter 4 1 / 58

Demand Lecture 3 Reading: Perlo Chapter 4 August 2015 1 / 58 Introduction We saw the demand curve in chapter 2. We learned about consumer decision making in chapter 3. Now we bridge the gap between the

### Economics 326: Duality and the Slutsky Decomposition. Ethan Kaplan

Economics 326: Duality and the Slutsky Decomposition Ethan Kaplan September 19, 2011 Outline 1. Convexity and Declining MRS 2. Duality and Hicksian Demand 3. Slutsky Decomposition 4. Net and Gross Substitutes

### Economics 165 Winter 2002 Problem Set #2

Economics 165 Winter 2002 Problem Set #2 Problem 1: Consider the monopolistic competition model. Say we are looking at sailboat producers. Each producer has fixed costs of 10 million and marginal costs

### MATH MODULE 11. Maximizing Total Net Benefit. 1. Discussion M11-1

MATH MODULE 11 Maximizing Total Net Benefit 1. Discussion In one sense, this Module is the culminating module of this Basic Mathematics Review. In another sense, it is the starting point for all of the

### 14.451 Lecture Notes 10

14.451 Lecture Notes 1 Guido Lorenzoni Fall 29 1 Continuous time: nite horizon Time goes from to T. Instantaneous payo : f (t; x (t) ; y (t)) ; (the time dependence includes discounting), where x (t) 2

### Consider a consumer living for two periods, and trying to decide how much money to spend today, and how much to save for tomorrow.

Consider a consumer living for two periods, and trying to decide how much money to spend today, and how much to save for tomorrow.. Suppose that the individual has income of >0 today, and will receive

### Import tari s, export subsidies and the theory of trade agreements

Import tari s, export subsidies and the theory of trade agreements Giovanni Maggi Princeton University and NBER Andrés Rodríguez-Clare IADB July 2005 Abstract Virtually all the existing models of trade

### Partial Derivatives. @x f (x; y) = @ x f (x; y) @x x2 y + @ @x y2 and then we evaluate the derivative as if y is a constant.

Partial Derivatives Partial Derivatives Just as derivatives can be used to eplore the properties of functions of 1 variable, so also derivatives can be used to eplore functions of 2 variables. In this

### Problem Set #5-Key. Economics 305-Intermediate Microeconomic Theory

Problem Set #5-Key Sonoma State University Economics 305-Intermediate Microeconomic Theory Dr Cuellar (1) Suppose that you are paying your for your own education and that your college tuition is \$200 per

### Chapter 4 Online Appendix: The Mathematics of Utility Functions

Chapter 4 Online Appendix: The Mathematics of Utility Functions We saw in the text that utility functions and indifference curves are different ways to represent a consumer s preferences. Calculus can

### Chapter 16 General Equilibrium and Economic Efficiency

Chapter 16 General Equilibrium and Economic Efficiency Questions for Review 1. Why can feedback effects make a general equilibrium analysis substantially different from a partial equilibrium analysis?

### General Equilibrium. The Producers

General Equilibrium In this section we will combine production possibilities frontiers and community indifference curves in order to create a model of the economy as a whole. This brings together producers,

### Exact Nonparametric Tests for Comparing Means - A Personal Summary

Exact Nonparametric Tests for Comparing Means - A Personal Summary Karl H. Schlag European University Institute 1 December 14, 2006 1 Economics Department, European University Institute. Via della Piazzuola

Advanced Microeconomics General equilibrium theory I: the main results Harald Wiese University of Leipzig Harald Wiese (University of Leipzig) Advanced Microeconomics 1 / 52 Part F. Perfect competition

### Chapter 27: Taxation. 27.1: Introduction. 27.2: The Two Prices with a Tax. 27.2: The Pre-Tax Position

Chapter 27: Taxation 27.1: Introduction We consider the effect of taxation on some good on the market for that good. We ask the questions: who pays the tax? what effect does it have on the equilibrium

### Risk Aversion. Expected value as a criterion for making decisions makes sense provided that C H A P T E R 2. 2.1 Risk Attitude

C H A P T E R 2 Risk Aversion Expected value as a criterion for making decisions makes sense provided that the stakes at risk in the decision are small enough to \play the long run averages." The range

### I. Utility and Indifference Curve

Problem Set #1 Econ 203, Intermediate Microeconomics (Due on Sep. 26 th (Thursday) by class time) I. Utility and Indifference Curve Definition: Utility Function, Indifference Curve, Marginal Rate of Substitution,

### 1 Another method of estimation: least squares

1 Another method of estimation: least squares erm: -estim.tex, Dec8, 009: 6 p.m. (draft - typos/writos likely exist) Corrections, comments, suggestions welcome. 1.1 Least squares in general Assume Y i

### An example of externalities - the multiplicative case

An example of externalities - the multiplicative case Yossi Spiegel Consider an exchange economy with two agents, A and B, who consume two goods, x and y. This economy however, differs from the usual exchange

### We refer to player 1 as the sender and to player 2 as the receiver.

7 Signaling Games 7.1 What is Signaling? The concept of signaling refers to strategic models where one or more informed agents take some observable actions before one or more uninformed agents make their

### A Closed Economy One-Period Macroeconomic Model

A Closed Economy One-Period Macroeconomic Model Chapter 5 Topics in Macroeconomics 2 Economics Division University of Southampton February 2010 Chapter 5 1/40 Topics in Macroeconomics Closing the Model

### Topic 7 General equilibrium and welfare economics

Topic 7 General equilibrium and welfare economics. The production possibilities frontier is generated using a production Edgeworth box diagram with the input goods on the axes. The following diagram illustrates

### Our development of economic theory has two main parts, consumers and producers. We will start with the consumers.

Lecture 1: Budget Constraints c 2008 Je rey A. Miron Outline 1. Introduction 2. Two Goods are Often Enough 3. Properties of the Budget Set 4. How the Budget Line Changes 5. The Numeraire 6. Taxes, Subsidies,

Advanced Microeconomics Ordinal preference theory Harald Wiese University of Leipzig Harald Wiese (University of Leipzig) Advanced Microeconomics 1 / 68 Part A. Basic decision and preference theory 1 Decisions

### EconS Bargaining Games and an Introduction to Repeated Games

EconS 424 - Bargaining Games and an Introduction to Repeated Games Félix Muñoz-García Washington State University fmunoz@wsu.edu March 25, 2014 Félix Muñoz-García (WSU) EconS 424 - Recitation 6 March 25,

### Chapter 14: Production Possibility Frontiers

Chapter 14: Production Possibility Frontiers 14.1: Introduction In chapter 8 we considered the allocation of a given amount of goods in society. We saw that the final allocation depends upon the initial

### ARE211, Fall2012. Contents. 2. Linear Algebra Preliminary: Level Sets, upper and lower contour sets and Gradient vectors 1

ARE11, Fall1 LINALGEBRA1: THU, SEP 13, 1 PRINTED: SEPTEMBER 19, 1 (LEC# 7) Contents. Linear Algebra 1.1. Preliminary: Level Sets, upper and lower contour sets and Gradient vectors 1.. Vectors as arrows.

### Corporate Income Taxation

Corporate Income Taxation We have stressed that tax incidence must be traced to people, since corporations cannot bear the burden of a tax. Why then tax corporations at all? There are several possible

### Lecture 1 Newton s method.

Lecture 1 Newton s method. 1 Square roots 2 Newton s method. The guts of the method. A vector version. Implementation. The existence theorem. Basins of attraction. The Babylonian algorithm for finding

### IDENTIFICATION IN A CLASS OF NONPARAMETRIC SIMULTANEOUS EQUATIONS MODELS. Steven T. Berry and Philip A. Haile. March 2011 Revised April 2011

IDENTIFICATION IN A CLASS OF NONPARAMETRIC SIMULTANEOUS EQUATIONS MODELS By Steven T. Berry and Philip A. Haile March 2011 Revised April 2011 COWLES FOUNDATION DISCUSSION PAPER NO. 1787R COWLES FOUNDATION

### Of the nature of Cournot competition

Of the nature of Cournot competition Alex Dickson and Roger Hartley y March 30, 2009 Abstract Bilateral oligopoly generalizes Cournot competition by allowing strategic behavior on both sides of the market,

### Prot Maximization and Cost Minimization

Simon Fraser University Prof. Karaivanov Department of Economics Econ 0 COST MINIMIZATION Prot Maximization and Cost Minimization Remember that the rm's problem is maximizing prots by choosing the optimal

### Chapter 15 Introduction to Linear Programming

Chapter 15 Introduction to Linear Programming An Introduction to Optimization Spring, 2014 Wei-Ta Chu 1 Brief History of Linear Programming The goal of linear programming is to determine the values of

### Lecture 3: Growth with Overlapping Generations (Acemoglu 2009, Chapter 9, adapted from Zilibotti)

Lecture 3: Growth with Overlapping Generations (Acemoglu 2009, Chapter 9, adapted from Zilibotti) Kjetil Storesletten September 10, 2013 Kjetil Storesletten () Lecture 3 September 10, 2013 1 / 44 Growth

### Chapter 3 Consumer Behavior

Chapter 3 Consumer Behavior Read Pindyck and Rubinfeld (2013), Chapter 3 Microeconomics, 8 h Edition by R.S. Pindyck and D.L. Rubinfeld Adapted by Chairat Aemkulwat for Econ I: 2900111 1/29/2015 CHAPTER

### Chapter 6: Pure Exchange

Chapter 6: Pure Exchange Pure Exchange Pareto-Efficient Allocation Competitive Price System Equitable Endowments Fair Social Welfare Allocation Outline and Conceptual Inquiries There are Gains from Trade

### CONSUMER PREFERENCES THE THEORY OF THE CONSUMER

CONSUMER PREFERENCES The underlying foundation of demand, therefore, is a model of how consumers behave. The individual consumer has a set of preferences and values whose determination are outside the

### Rational inequality. Sunil Kumar Singh. 1 Sign scheme or diagram for rational function

OpenStax-CNX module: m15464 1 Rational inequality Sunil Kumar Singh This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 2.0 Rational inequality is an inequality

### Increasing for all. Convex for all. ( ) Increasing for all (remember that the log function is only defined for ). ( ) Concave for all.

1. Differentiation The first derivative of a function measures by how much changes in reaction to an infinitesimal shift in its argument. The largest the derivative (in absolute value), the faster is evolving.

### Case 1 - Economic profits Case 2 - Zero economic profit Case 3 - Losses while continuing to operate Case 4 - Firm closes down

TOPIC VII: PERFECT COMPETITION I. Characteristics of a Perfectly Competitive Industry II. III. The Firm as a Price Taker Short Run Decisions A. Possible profit (loss) cases for the firm Case 1 - Economic

### Partial Fractions Decomposition

Partial Fractions Decomposition Dr. Philippe B. Laval Kennesaw State University August 6, 008 Abstract This handout describes partial fractions decomposition and how it can be used when integrating rational

### Time Preference and the Distributions of Wealth and. Income

Time Preference and the Distributions of Wealth and Income Richard M. H. Suen This Version: February 2010 Abstract This paper presents a dynamic competitive equilibrium model with heterogeneous time preferences

### Figure 4.1 Average Hours Worked per Person in the United States

The Supply of Labor Figure 4.1 Average Hours Worked per Person in the United States 1 Table 4.1 Change in Hours Worked by Age: 1950 2000 4.1: Preferences 4.2: The Constraints 4.3: Optimal Choice I: Determination

### Demand and Supply Analysis: Introduction

READING 13 Demand and Supply Analysis: Introduction by Richard V. Eastin, PhD, and Gary L. Arbogast, CFA Richard V. Eastin, PhD, is at the University of Southern California (USA). Gary L. Arbogast, CFA

### In economics, the amount of a good x demanded is a function of a person s wealth and the price of that good. In other words,

LABOR NOTES, PART TWO: REVIEW OF MATH 2.1 Univariate calculus Given two sets X andy, a function is a rule that associates each member of X with exactly one member ofy. Intuitively, y is a function of x

### 1 Present and Future Value

Lecture 8: Asset Markets c 2009 Je rey A. Miron Outline:. Present and Future Value 2. Bonds 3. Taxes 4. Applications Present and Future Value In the discussion of the two-period model with borrowing and

### Newton s Method on a System of Nonlinear Equations

Newton s Method on a System of Nonlinear Equations Nicolle Eagan, University at Bu alo George Hauser, Brown University Research Advisor: Dr. Timothy Flaherty, Carnegie Mellon University Abstract Newton

### 1 Cobb-Douglas Functions

1 Cobb-Douglas Functions Cobb-Douglas functions are used for both production functions Q = K β L (1 β) where Q is output, and K is capital and L is labor. The same functional form is also used for the

### Economics 326: Marshallian Demand and Comparative Statics. Ethan Kaplan

Economics 326: Marshallian Demand and Comparative Statics Ethan Kaplan September 17, 2012 Outline 1. Utility Maximization: General Formulation 2. Marshallian Demand 3. Homogeneity of Degree Zero of Marshallian

### U = x 1 2. 1 x 1 4. 2 x 1 4. What are the equilibrium relative prices of the three goods? traders has members who are best off?

Chapter 7 General Equilibrium Exercise 7. Suppose there are 00 traders in a market all of whom behave as price takers. Suppose there are three goods and the traders own initially the following quantities:

### First Welfare Theorem

First Welfare Theorem Econ 2100 Fall 2015 Lecture 17, November 2 Outline 1 First Welfare Theorem 2 Preliminaries to Second Welfare Theorem Last Class Definitions A feasible allocation (x, y) is Pareto

### Optimal insurance contracts with adverse selection and comonotonic background risk

Optimal insurance contracts with adverse selection and comonotonic background risk Alary D. Bien F. TSE (LERNA) University Paris Dauphine Abstract In this note, we consider an adverse selection problem

### Chapter 22: Exchange in Capital Markets

Chapter 22: Exchange in Capital Markets 22.1: Introduction We are now in a position to examine trade in capital markets. We know that some people borrow and some people save. After a moment s reflection

### The Graphical Method: An Example

The Graphical Method: An Example Consider the following linear program: Maximize 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2 0, where, for ease of reference,

### 14.452 Economic Growth: Lectures 2 and 3: The Solow Growth Model

14.452 Economic Growth: Lectures 2 and 3: The Solow Growth Model Daron Acemoglu MIT November 1 and 3, 2011. Daron Acemoglu (MIT) Economic Growth Lectures 2 and 3 November 1 and 3, 2011. 1 / 96 Solow Growth

### Chapter 12 Modal Decomposition of State-Space Models 12.1 Introduction The solutions obtained in previous chapters, whether in time domain or transfor

Lectures on Dynamic Systems and Control Mohammed Dahleh Munther A. Dahleh George Verghese Department of Electrical Engineering and Computer Science Massachuasetts Institute of Technology 1 1 c Chapter

C H A P T E R 3 Simulation of Business Processes The stock and ow diagram which have been reviewed in the preceding two chapters show more about the process structure than the causal loop diagrams studied

### The Binomial Distribution

The Binomial Distribution James H. Steiger November 10, 00 1 Topics for this Module 1. The Binomial Process. The Binomial Random Variable. The Binomial Distribution (a) Computing the Binomial pdf (b) Computing

### Formula Sheet. 1 The Three Defining Properties of Real Numbers

Formula Sheet 1 The Three Defining Properties of Real Numbers For all real numbers a, b and c, the following properties hold true. 1. The commutative property: a + b = b + a ab = ba. The associative property:

### Markov Chains, part I

Markov Chains, part I December 8, 2010 1 Introduction A Markov Chain is a sequence of random variables X 0, X 1,, where each X i S, such that P(X i+1 = s i+1 X i = s i, X i 1 = s i 1,, X 0 = s 0 ) = P(X

### The Not-Formula Book for C1

Not The Not-Formula Book for C1 Everything you need to know for Core 1 that won t be in the formula book Examination Board: AQA Brief This document is intended as an aid for revision. Although it includes

### Economics 200B Part 1 UCSD Winter 2015 Prof. R. Starr, Mr. John Rehbeck Final Exam 1

Economics 200B Part 1 UCSD Winter 2015 Prof. R. Starr, Mr. John Rehbeck Final Exam 1 Your Name: SUGGESTED ANSWERS Please answer all questions. Each of the six questions marked with a big number counts

### Chapter 3 Linearity vs. Local Linearity

Chapter 3 Linearity vs. Local Linearity This chapter helps you understand the main approximation of di erential calculus: Small changes in smooth functions are approximately linear. Calculus lets us use

### 36 CHAPTER 1. LIMITS AND CONTINUITY. Figure 1.17: At which points is f not continuous?

36 CHAPTER 1. LIMITS AND CONTINUITY 1.3 Continuity Before Calculus became clearly de ned, continuity meant that one could draw the graph of a function without having to lift the pen and pencil. While this

### ECONOMICS 101 WINTER 2004 TA LECTURE 5: GENERAL EQUILIBRIUM

ECONOMICS 0 WINTER 2004 TA LECTURE 5: GENERAL EQUILIBRIUM KENNETH R. AHERN. Production Possibility Frontier We now move from partial equilibrium, where we examined only one market at a time, to general

### Managerial Economics Prof. Trupti Mishra S.J.M. School of Management Indian Institute of Technology, Bombay. Lecture - 13 Consumer Behaviour (Contd )

(Refer Slide Time: 00:28) Managerial Economics Prof. Trupti Mishra S.J.M. School of Management Indian Institute of Technology, Bombay Lecture - 13 Consumer Behaviour (Contd ) We will continue our discussion

### Review of Utility Theory, Expected Utility

Review of Utility Theory, Expected Utility Utility theory is the foundation of neoclassical economic demand theory. According to this theory, consumption of goods and services provides satisfaction, or

### Accident Law and Ambiguity

Accident Law and Ambiguity Surajeet Chakravarty and David Kelsey Department of Economics University of Exeter January 2012 (University of Exeter) Tort and Ambiguity January 2012 1 / 26 Introduction This

### Chapter 4. Simulated Method of Moments and its siblings

Chapter 4. Simulated Method of Moments and its siblings Contents 1 Two motivating examples 1 1.1 Du e and Singleton (1993)......................... 1 1.2 Model for nancial returns with stochastic volatility...........

### Tastes and Indifference Curves

C H A P T E R 4 Tastes and Indifference Curves This chapter begins a 2-chapter treatment of tastes or what we also call preferences. In the first of these chapters, we simply investigate the basic logic

### Econ306 Intermediate Microeconomics Fall 2007 Midterm exam Solutions

Econ306 Intermediate Microeconomics Fall 007 Midterm exam Solutions Question ( points) Perry lives on avocado and beans. The price of avocados is \$0, the price of beans is \$, and his income is \$0. (i)