PROPERTY, POWER AND EXCHANGE MUTUAL GAINS AND CONFLICTS LESSON 1 Advanced Macro and Micro, ECO3020F Andrew Kerr, UCT School of Economics These slides are based on slides prepared by Lindo Njozela. All errors are my own. LEARNING OBJECTIVES • People have exchanged goods for mutual benefit for thousands of years. • We wish to understand mutual gains through trade • and the conflict over the distribution of these gains. • Understand how an allocation of goods can be evaluated on grounds of Pareto efficiency and fairness. • Understand how property rights, the exercise of power, and other aspects of the rules of the game will affect the extent of mutual gains and the inequality of their distribution. • Use mathematical tools (equations and graphs) to illustrate the issues. READINGS FOR THE WEEK • Bowles and Halliday Chapter 4. • Excluding: • Section 14.3 • M-Note 4.6 • M-Note 4.7 • Legislation in Section 14.10 • There are also revision notes and maths notes which may be helpful. • The first two weeks are tough as you will be revising content from 2003 and learning new content at the same time. CLASSICAL CONSTITUTIONAL CONUNDRUM “The problem of finding a set of laws, policies and social norms (“rules of the game”) so that people are free to choose their actions, whilst avoiding outcomes that none of them would have chosen had they been able to co-ordinate.” PLAN FOR THE WEEK • This week we look at these interdependencies between individuals’ choices. • We are going to think through this as conflict and cooperation • mutual gains from cooperation (trade) exist alongside conflict over the distribution. • Similar to the content from ECO2003F • Our main goal is to see how different institutions/rules of the game can affect the equity and efficiency of the outcomes of interactions between two people. PLAN FOR THE WEEK • The institutions we will focus on are: • A social planner (called Impartial Spectator in the textbook) • Market institutions with symmetric exchange • Market institutions with asymmetric exchange (power) • Take-It-Or-Leave-It-Power • Price-setting power ASSUMED KNOWLEDGE • We assume you are comfortable with: • Pareto efficiency, pareto criteria, pareto improvements • Utility maximization (graphical and mathematical analysis of marginal rates of substitution, marginal rates of transformation, indifference curves, feasible frontiers). • Nash equilibria • Cobb-Douglas and quasi-linear utility functions. • If you aren’t then you must go over the revision slides. • We will do some brief revision in class. SETTING THE SCENE • We are going to think about a society as populated by two people, with only two goods. • In our society we have citizens Ayanda and Biko, and goods “data” and coffee. • The total amount of each good in the society is fixed and does not change. • The point of the exercise is to find a socially optimal allocation of the two goods and then see what the outcomes will be under different “rules of the game” that govern how the two citizens interact. • We examine bartering of goods without money even though in the modern world we use indirect exchange and money• we exchange our work hours for money and then buy goods with our money SETTING THE SCENE: KEY ASPECTS • Initial endowments or endowments or endowment bundle are the goods each person has before any exchanges • we are going to assume they just picked their goods up off the floor • i.e., they were randomly allocated those goods. • Final endowments or post-exchange bundle are the goods each person has after all exchanges. • We will use these terms interchangeably, although I prefer initial endowment and final endowments. INDIFFERENCE CURVES: REFRESHER • Indifference curves (IC): all combinations of goods which give the same utility i.e., the combinations for which the person is indifferent between other combinations. • Marginal rate of substitution (MRS): • The willingness to pay for a 1 unit increase in the amount of good x expressed as the maximum amount of good y the person would be willing to give up for this. • It is also the negative of the slope of an indifference curve. • It is also the minimum amount of units of good Y that the person is willing to receive if required to give up a small (marginal) amount of good X. FEASIBLE FRONTIERS: REFRESHER • Feasible frontier: The border of the feasible set, showing for any value of good x the maximum value of good y that is feasible, meaning that the decision-maker can obtain. • Marginal rate of transformation (MRT): the negative of the slope of the feasible frontier. • It measures the sacrifice of the good y necessary in order to get more of the good x. • It is therefore the opportunity cost of the x good in terms of the y good. CONSTRAINED UTILITY-MAXIMIZATION • An outcome bundle (x, y) is constrained utility maximising if there is no point in the feasible set that is on a higher indifference curve. • This occurs where marginal rate of substitution (slope of indifference curve) is equal to the marginal rate of transformation (slope of feasible frontier). • Graphically this is the point of tangency between the IC and the feasible frontier. THE EDGEWORTH BOX: BASICS • We have two goods: coffee and data. • Total good x (coffee) = π₯π₯ = 10 kgs • Total good y (data) = π¦π¦ = 15 gigs • Any feasible allocation between A and B must be: • π₯π₯ π΄π΄ + π₯π₯ π΅π΅ ≤ π₯π₯ • π¦π¦ π΄π΄ + π¦π¦ π΅π΅ ≤ π¦π¦ • We will only consider the case in which π₯π₯ π΄π΄ + π₯π₯ π΅π΅ = π₯π₯ and π¦π¦ π΄π΄ + π¦π¦ π΅π΅ = π¦π¦ • This means that all goods are allocated to either A or B with nothing leftover. THE EDGEWORTH BOX: BASICS • The Edgeworth Box is a way of representing and visualizing the distribution and allocative efficiency of goods in a society/economy. • On the left and right axes we represent the number of gigs of data (good y) A and B have. • The left axis is for A and the right axis is for B. THE EDGEWORTH BOX: BASICS • The number of gigs of data A has increases upwards. • The number of gigs of data B has increases downwards. • Why? Because we have a finite amount of data and all of it must be allocated between the two players. So how ever much data A has, B has 15 gigs minus A’s data. THE EDGEWORTH BOX: BASICS • The left and right axes range from 0 to 15 which means that at most a player can have all the data in society (15 gigs) and the other player none of the data in society (zero gigs). • We first consider a possible initial endowment at point z. • Point z shows B has 14 gigs and A has 1 gig. THE EDGEWORTH BOX: BASICS • The top and bottom axes show the amount of coffee (good x) A and B have. • The top axis is B’s coffee and the bottom axis is A’s coffee. • A’s coffee increases from left to right and B’s coffee increases from right to left. • Why? How ever much coffee A has, B must have 10kg minus A’s holdings of coffee. THE EDGEWORTH BOX: BASICS • At the initial endowment of point z, A has 9kgs of coffee and B has 1kg of coffee. THE EDGEWORTH BOX: UTILITY • We want to introduce utility to the box. • Before that, remember that more of each good always results in higher utility. THE EDGEWORTH BOX: UTILITY • A’s utility increases as his allocation of goods moves away from the bottom-left corner because he is getting more of both goods. • B’s utility increases as his allocation of goods moves away from the top-right corner for the same reason. THE EDGEWORTH BOX: UTILITY • There is obviously conflict. • For B to be happier, his allocation of goods must move closer to the bottom-left corner. • But as B’s allocation moves to the bottom-left corner, A is losing goods and her utility is decreasing. THE EDGEWORTH BOX: UTILITY • For A to be happier, her allocation of goods must move closer to the top-right corner. • But as A’s allocation moves to the top-right corner, B is losing goods and his utility is decreasing. THE EDGEWORTH BOX: UTILITY • The basics are done. Now we need to make it more interesting. • We are going to give A and B standard Cobb-Douglas utility functions. • Don’t confuse superscripts and exponents! • α is the exponent and the capital letters A and B are the superscripts. • Superscripts indicate the person (A or B) we are talking about. COBB-DOUGLAS: REFRESHER • The key characteristics of a Cobb-Douglas utility function are: • ππ = πππππΆπΆ πππ·π· • π’π’π₯π₯ > 0, π’π’π¦π¦ > 0 Marginal utility is positive • π’π’π₯π₯π₯π₯ < 0, π’π’π¦π¦π¦π¦ < 0 • ππ > 0, πΌπΌ, π½π½ > 0 Diminishing marginal utility • πΌπΌ ππππππ π½π½ represent intensity of preference for x and y. COBB-DOUGLAS: REFRESHER • ππ ππ, ππ = πππππΆπΆ πππ·π· • π΄π΄π΄π΄π΄π΄ ππ, ππ = πΆπΆ ππ ππππ = π·π· ππ ππππ • Note that MRS depends both on the relative intensity of preferences for the two goods (exponents), as well as the relative quantity of two goods. • Put differently, it is the ratio of marginal utilities for good x and good y. THE EDGEWORTH BOX: UTILITY • Linking this back to the box. • It is VERY important to understand and remember the direction in which each person’s utility is increasing and decreasing! • This seems obvious now, but it can be harder to see as the boxes become more complicated. EDGEWORTH BOX: INDIFFERENCE CURVES • With the CD utility functions, we get familiar looking indifference curves. • A’s utility increases the further away A’s indifference curve is from the origin. • Notice point z: A has 1 gig data and 9 kgs of coffee. • We write this as: (π₯π₯π§π§π΄π΄ , π¦π¦π§π§π΄π΄ ) = (9,1) EDGEWORTH BOX: INDIFFERENCE CURVES • Standard indifference curves for B as well. • B’s utility increases the further away B’s indifference curve is from the origin. • Notice point z: B has 14 gigs data and 1 kg of coffee • We write this as: (π₯π₯π§π§π΅π΅ , π¦π¦π§π§π΅π΅ ) = (1,14) EDGEWORTH BOX: BIKO’S IC’S • Let’s take a step back and remember what the box looks like. • Look at where B’s origin is. • Without A, B’s origin is at the bottom left (on the previous slide). But with A, B’s origin is in the top right. • So what will B’s ICs look like in the box? EDGEWORTH BOX: AYANDA’S IC’S • We need to combine our ICs with the Edgeworth Box. • Firstly, we simply include A’s ICs into the box. • This representation is exactly the same as A’s ICs without the box. • Now let’s see what happens with B’s ICs in the box. EDGEWORTH BOX: BIKO’S IC’S • Putting B’s ICs into the box is slightly more complicated. • Notice B’s origin in the orange circle. • B’s utility increases as he moves away from the origin. EDGEWORTH BOX: BIKO’S ICs • Unlike A, B’s ICs are rotated. • Notice B’s origin. • His utility is still increasing as he moves away from the origin. • It’s just that now we’ve changed where the origin is. EDGEWORTH BOX: INDIFFERENCE CURVES • To construct the Edgeworth Box, all we have done is rotate B’s indifference curve maps. • But this smart move comes at a cost: things start looking messy. PUTTING IT ALL TOGETHER • How did we get here? • All we’ve done is rotate B’s indifference curves. • MAKE SURE YOU CAN SEE THIS! • It looks complicated, but it really is not. You’ll get the hang of it. • NOTE: Every possible allocation in the box has indifference curves for A & B that pass through that allocation point. PUTTING IT ALL TOGETHER • In rotating B’s ICs we’ve had to add a lot more information to the box. • Let’s see what it tells us: • Point z is still the same initial endowment as before. • The lines labelled “u” represent A and B’s utility at that indifference curve. PUTTING IT ALL TOGETHER • We have modelled a situation in which A’s and B’s indifference curves are related. • This represents interdependencies in economic interactions. • A’s welfare is directly linked to B’s welfare. • No man (person) is an island anymore. PARETO COMPARISONS & EFFICIENCY • We need some more ideas to make progress. • Pareto criteria: No one must be made worse-off to make anyone better-off. • An outcome is Pareto superior to another if it allows at least one of those involved to be better off without anyone being worse off. • A Pareto-superior outcome is also called a Pareto improvement over the outcome it was compared to. • An outcome is Pareto efficient if no other feasible outcome is Pareto superior to it. • If we can rank two outcomes such that one is Pareto superior to the other, then we say that these two outcomes can be Pareto compared. PARETO CRITERION: STRENGTHS AND WEAKNESS • Pareto efficiency cannot decide on a best outcome – it only helps us avoid worse ones. • It cannot answer which among many Pareto efficient outcomes are “good” and which is “bad” • Too strong? Using this criterion may result in status quo bias (unwillingness to change current policies/institutions) • Too weak? An outcome may be efficient but may not be desirable for society. • (Pareto) efficiency and equity are two different goals. PARETO Efficient Allocations • There are many possible allocations of the two goods between A and B. • A and B might decide to limit the set of possible allocations to those that are Pareto efficient. • Pareto efficient outcomes for an allocation between two players occur when the MRS of A and the MRS of B are equal. • This is called the MRSA=MRSB rule • Why are such allocations Pareto efficient? • When the MRS of A and B are equal this means the indifference curves are tangent, that A and B have the same willingness to pay for the two goods and thus there are no feasible Pareto improving allocations. PARETO Efficient Allocations • Note that this is another tangency rule. • For individual constrained optimization problems, the utilitymaximising solution was to set MRS=MRT • Now we have a societal allocation problem where the solution that generates a Pareto efficient allocation is one in which MRSA=MRSB. EDGEWORTH BOX: PARETO EFFICIENCY • Let’s think through an example (see M-Note 4.2) • We are going to use gigs of data per kg of coffee because MRS represents the willingness to forego good x (coffee) in terms of y (gigs) • Let’s say ππππππ π΄π΄ = 0.22 and ππππππ π΅π΅ = 7 • This means A is willing to sell 1 kg of coffee for 0.22 gigs of data, while B is willing to pay 7 gigs for a kg of coffee. • i.e., ππππππ π΄π΄ < ππππππ π΅π΅ • Since B is willing to pay more than A is willing to accept, there is room for a Pareto improving trade to occur. EDGEWORTH BOX: PARETO EFFICIENCY • If ππππππ π΄π΄ = ππππππ π΅π΅ then we are at a Pareto-Efficient allocation. Neither person can be made better-off without making the other else worse-off. • The intersection between A’s ICs and B’s ICs at point z (a few slides back) means that we can immediately tell that point z is not Pareto efficient • the slopes of the ICs (MRSs) aren’t equal. • NB: We are talking about intersecting ICs between A and B and NOT intersection of A’s or B’s own indifference curves. • Your own individual ICs CAN NEVER INTERSECT. EDGEWORTH BOX: PARETO EFFICIENCY • Any and all intersections between A’s ICs and B’s ICs will always be a Pareto-inefficient allocation since the MRS’s at those intersections are not equal. • Since z is a point of intersection and therefore not Pareto-Efficient, there are gains from trade to be made • i.e., another allocation can be found which would at least improve the welfare of A or B without making the other person worse-off. • The existence of rents (gains from trade) from finding a ParetoEfficient point from point z is why A and B would be willing to trade. • We’re going to look at this graphically. EDGEWORTH BOX: PARETO EFFICIENCY • But first, take a moment to properly understand what’s happening in the box. • Spend some time talking yourself through it and seeing the conflict and the Paretoinefficiency. • It’s a good idea to take a moment and draw it. • Also notice that in this figure, none of the ICs are tangent to each other. EDGEWORTH BOX: PARETO EFFICIENCY • We have now re-drawn our Edgeworth to include ICs that are tangent and ICs that are intersecting. • We’ve also changed the utility functions: 2 3 • π’π’ π΄π΄ = (π₯π₯ π΄π΄ ) (π¦π¦ π΄π΄ ) 1 3 • π’π’π΅π΅ = (π₯π₯ π΅π΅ ) (π¦π¦ π΅π΅ ) 1 3 2 3 EDGEWORTH BOX: PARETO EFFICIENCY • We have included points d and h which are points of intersection and therefore Pareto-inefficient. • But point h is Pareto-superior to point z and point d • B is made better off while A remains with the same utility • Gains from trade can be made from points d, z, or h (all three are pareto inefficient) EDGEWORTH BOX: PARETO EFFICIENCY • Now let’s focus on points π‘π‘ π΄π΄ and π‘π‘ π΅π΅ . • At π‘π‘ π΄π΄ , we can see that A’s IC (π’π’3π΄π΄ ) is tangent to B’s IC (π’π’π§π§π΅π΅ ) • Similarly, at point π‘π‘ π΅π΅ , π’π’π§π§π΄π΄ is tangent to π’π’4π΅π΅ . • NOTE: π’π’π§π§π΄π΄ and π’π’π§π§π΅π΅ represent the ICs at the initial endowment of point z. EDGEWORTH BOX: PARETO EFFICIENCY • Because of the tangency at point π‘π‘ π΄π΄ and point π‘π‘ π΅π΅ , we know that point π‘π‘ π΄π΄ and point π‘π‘ π΅π΅ are Paretoefficient: • ππππππ π΄π΄ = ππππππ π΅π΅ . EDGEWORTH BOX: PARETO EFFICIENCY • Notice the curve that goes through both points π‘π‘ π΄π΄ and π‘π‘ π΅π΅ . We can use this curve to say things about point i, even if we cannot see the ICs at point i. • The pink curve is called the ParetoEfficiency Curve (PEC). It is a locus/collection of all Pareto efficient points in the box. • That is, allocations along the PEC are pareto-efficient allocations. EDGEWORTH BOX: PARETO EFFICIENCY • NOTE: The PEC doesn’t always have the same shape. It can be straight, concave, convex etc. • The shape of the curve will depend on the two people’s utility functions and goods available • since the PEC depends on the two MRS’s. EDGEWORTH BOX: PARETO EFFICIENCY • Since point i is on the PEC, we know that point i must be a pareto-efficient allocation. • Note that the distribution of goods along the PEC differs substantially. • A and B face two issues- • which Pareto efficient outcome to choose and • A conflict of interest SOLVING FOR THE PEC MATHEMATICALLY • Having done the graphical analysis, we’ll do some mathematical analysis to find the PEC. • We will use the following utility functions: 2 3 • π’π’ π΄π΄ = (π₯π₯ π΄π΄ ) (π¦π¦ π΄π΄ ) 1 3 1 3 2 3 • π’π’π΅π΅ = (π₯π₯ π΅π΅ ) (π¦π¦ π΅π΅ ) • To find the Pareto Efficienct curve, we need to find the MRSs, since the PEC is defined as where the MRSs are equal. SOLVING FOR THE PEC • Remember (and be able to show! See maths notes) that MRS is the π΄π΄ π’π’π₯π₯ π΄π΄ ratio of the marginal utilities: for person A ππππππ = π΄π΄ (1) π’π’π¦π¦ • From the utility functions: 2 1 1 1 π΄π΄ πππ’π’ 2 2 ( −1) − • π’π’π₯π₯π΄π΄ = π΄π΄ = (π₯π₯ π΄π΄ ) 3 (π¦π¦ π΄π΄ )3 = (π₯π₯ π΄π΄ ) 3 (π¦π¦ π΄π΄ )3 (2) 3 3 πππ₯π₯ 2 1 2 2 π΄π΄ πππ’π’ 1 π΄π΄ π΄π΄ 3 1 π΄π΄ (3 −1) π΄π΄ 3 π΄π΄ −3 • π’π’π¦π¦ = π΄π΄ = (π₯π₯ ) (π¦π¦ ) = (π₯π₯ ) (π¦π¦ ) (3) 3 3 πππ¦π¦ 1 1 2 π΄π΄ − 3 (π¦π¦ π΄π΄ )3 (π₯π₯ ) • Sub (2) and (3) into (1): ππππππ π΄π΄ = 31 2 2 − π΄π΄ )3 (π¦π¦ π΄π΄ ) 3 (π₯π₯ 3 SOLVING FOR THE PEC • ππππππ π΄π΄ = 1 1 2 π΄π΄ − π΄π΄ (π₯π₯ ) 3 (π¦π¦ )3 3 2 2 1 π΄π΄ − 3 (π¦π¦ π΄π΄ ) 3 (π₯π₯ ) 3 π¦π¦ π΄π΄ • ππππππ π΄π΄ = 2 π΄π΄ π₯π₯ • By the same process as person A: ππππππ π΅π΅ π΅π΅ π’π’π₯π₯ 1 π¦π¦ π΅π΅ = π΅π΅ = 2 π₯π₯ π΅π΅ π’π’π¦π¦ SOLVING FOR THE PEC • We can find the PEC in terms of A’s allocation or B’s allocation. • We first find the PEC in terms of A’s allocation. • Recall that in our example: • π₯π₯ π΄π΄ + π₯π₯ π΅π΅ = π₯π₯ = 10 ππππππ • π¦π¦ π΄π΄ + π¦π¦ π΅π΅ = π¦π¦ = 15 ππππππππ • Rearranging to isolate π₯π₯ π΅π΅ and π¦π¦ π΅π΅ : • π₯π₯ π΅π΅ = π₯π₯ − π₯π₯ π΄π΄ = 10 − π₯π₯ π΄π΄ and π¦π¦ π΅π΅ = π¦π¦ − π¦π¦ π΄π΄ = 15 − π¦π¦ π΄π΄ (1) • And weπ΄π΄ set πππππππ΅π΅ π΄π΄ = ππππππ π΅π΅ since this defines the PEC. • 2 π¦π¦ π₯π₯ π΄π΄ = 1 π¦π¦ 2 π₯π₯ π΅π΅ • Sub (1) into (2): • 2 π¦π¦ π΄π΄ π₯π₯ π΄π΄ (2) 1 15 − π¦π¦ π΄π΄ = 2 10 − π₯π₯ π΄π΄ SOLVING FOR THE PEC • Simplifying, we have 2 π¦π¦ π΄π΄ π₯π₯ π΄π΄ 1 15 − π¦π¦ π΄π΄ = 2 10 − π₯π₯ π΄π΄ π΄π΄ (1) • With some rearranging to isolate π¦π¦ we obtain the Pareto efficient curve in terms of A’s allocation: • π¦π¦ π΄π΄ 15π₯π₯ π΄π΄ = (PEC) 40 −3π₯π₯ π΄π΄ SOLVING FOR THE PEC • We can do the same exercise in terms of B’s allocation by substituting in π₯π₯ π΄π΄ and π¦π¦ π΄π΄ instead of π₯π₯ π΅π΅ and π¦π¦ π΅π΅ like we did before. •2 15 − π¦π¦ π΅π΅ 10 − π₯π₯ π΅π΅ 1 π¦π¦ π΅π΅ = 2 π₯π₯ π΅π΅ • With some rearranging to isolate π¦π¦ π΄π΄ , we find the Pareto efficiency curve in terms of B’s allocation: π΅π΅ 60π₯π₯ • π¦π¦ π΅π΅ = (PEC) π΅π΅ 10 −3π₯π₯ SOLVING FOR THE PEC • In a simpler example with identical utility functions for A and B, with πΌπΌ = 0.5 1 1 1 1 • π’π’ π΄π΄ = (π₯π₯ π΄π΄ )2 (π¦π¦ π΄π΄ )2 and π’π’π΅π΅ = (π₯π₯ π΅π΅ )2 (π¦π¦ π΅π΅ )2 • In the general case: π΄π΄ π’π’ • ππππππ π΄π΄ = π₯π₯π΄π΄ = π’π’π¦π¦ 1 1 1 π΄π΄ − π΄π΄ (π₯π₯ ) 2 (π¦π¦ )2 2 1 1 1 π΄π΄ − (π₯π₯ )2 (π¦π¦ π΄π΄ ) 2 2 • Since A and B are identical: • ππππππ π΅π΅ π¦π¦ π΅π΅ = π΅π΅ π₯π₯ π¦π¦ π΄π΄ = π΄π΄ π₯π₯ SOLVING FOR THE PEC π΄π΄ π΅π΅ π¦π¦ π¦π¦ • ππππππ π΄π΄ = ππππππ π΅π΅ ⇒ π΄π΄ = π΅π΅ π₯π₯ π₯π₯ • From the same process of substitution and rearranging as before: 3 2 3 π΅π΅ π΅π΅ • π¦π¦ = ( )π₯π₯ in terms of B’s allocation. 2 • π¦π¦ π΄π΄ = ( )π₯π₯ π΄π΄ in terms of A’s allocation. REACHING PARETO EFFICIENCY • In our example from the previous two slides A and B have identical utility functions with identical weights (exponents) placed on data and coffee. • π’π’ π΄π΄ = π’π’π΅π΅ = π₯π₯ 0.5 π¦π¦ 0.5 • The PEC is now a straight line – nothing else meaningful changes. REACHING PARETO EFFICIENCY • We know that z is inefficient. • So, 1) how do we get to a Pareto efficient allocation, and 2) which one do we get to? • This is where institutions and power will matter. • We’ll focus on an impartial social planner (government); take-it-orleave it power, and price-setting power. IMPARTIAL SOCIAL PLANNER • The hypothetical “impartial social planner” is assumed to be a third party who does not get allocated any goods. • This third-party acts like a government who makes and enforces a distribution of goods between A and B. • We noted above that A and B could not agree on which Pareto Optimal point would be best. • The question we answer now is “What allocations would the impartial social planner choose for A and B”? IMPARTIAL SOCIAL PLANNER • We are going to assume that the social planner cares about procedural and substantive fairness. • Procedural fairness: Are the procedures that determined the allocation fair? E.G. random luck vs hard work/earned endowments. • Substantive fairness: Is the outcome itself fair? Which is the “best” outcome? • We are only going to focus on substantive fairness. • For clarity, the social planner has “(social) welfare” while the actual people have “utility”. • They are conceptually identical- each wants the most of this thing, subject to constraints. IMPARTIAL SOCIAL PLANNER • We are also going to assume that the social planner has diminishing marginal welfare/value on increases in utility for B or A. • That is, the greater the utility of A, the less welfare the social planner derives from an addition unit of utility for A. The same thing applies to B. • In addition, we assume our social planner cares equally about A & B“impartial”. • These are assumptions. • A social planner can prefer one person over another and there is no need for them to have diminishing marginal welfare/value. WELFARE FUNCTION • We assume the social planner has a Cobb-Douglas welfare function: • We assume our social planner cares equally about A as they do B, meaning the weights / exponents must be equal • We choose to set: • So, we have the social welfare function as: ISO-WELFARE FUNCTIONS • Just like we had indifference curves which map every combination of goods which result in the same level of utility, we can have iso-welfare functions for the social planner. • Iso-welfare function: combination of utilities for A and B which result in the same level of welfare for the social planner (“iso” literally means “same”) • Everything about the welfare functions and the iso-welfare functions is conceptually identical to utility functions and indifference curves. • The slope of the iso-welfare functions is the MRS and a function of A and B’s utility levels. • You can hopefully see we are heading to another type of constrained optimization problem… UTILITY POSSIBILITY FRONTIER • There are a variety of ways in which the social planner can determine a “fair” outcome. • An equal share of the goods between A and B. • Maximising total utility regardless of who has more of the goods. • We need to shift gears from coffee and data to B’s utility and A’s utility i.e. utility space. • The distribution of data and coffee still matters since that determines the utility of A & B • but it is easier to represent the social planner’s problem directly as utilities, so we use the utility possibility frontier: • The boundary of the set of feasible utility pairs of A & B. • Similar to feasible frontier from second year. OPTIMAL SOCIAL WELFARE • This graph shows: • “Utility space” i.e. A and B’s utility on the axes. • All possibility combinations of A & B’s utility from available goods -the shaded region • The SP’s iso-welfare functions downward sloping blue lines. • The Utility Possibilities Frontier -the straight line connecting the two axes & boundary of shaded region. • This particular UPF comes from the example above with identical utility functions where we found the PEC was a straight line. OPTIMAL SOCIAL WELFARE • To draw the UPF, we map the utilities for all pareto-efficient allocations of goods. • The frontier itself shows all combinations of A’s and B’s utilities associated with allocations on the Pareto-efficient curve • i.e. it is the limit that defines which combinations of utility are feasible and infeasible. • You can see that it looks like the feasible frontiers in the case of one person choosing how much to consume of two goods. OPTIMAL SOCIAL WELFARE • The UPF is downward sloping because it represents conflict between A and B about the distribution of goods and thus the distribution of utility. • The slope of the UPF is again called the MRT. OPTIMAL SOCIAL WELFARE • The shaded area represents Paretoinefficient points because utility for A and/or B can be increased by Paretoimprovements. • Our initial endowment of point z lies in the shaded area and points along the PEC lie on the UPF itself (π‘π‘ π΄π΄ and π‘π‘ π΅π΅ ). OPTIMAL SOCIAL WELFARE • The intersection with either of the axes represents cases in which all the goods are allocated to one person and thus all the utility goes to one person. OPTIMAL SOCIAL WELFARE • For the SP to maximise their welfare, they need to find a point on the UPF which is tangent to their highest isowelfare curve i.e. MRT = MRS. SOCIAL PLANNER: MAXIMISATION • The point of tangency is point i. So the SP will choose point i. • The final allocation of point i allocates equal utility between A and B (roughly 6 utils for both people) • Equal utility doesn’t have to be the outcome – it’s just because of how we have constructed the problem. SOCIAL PLANNER: MAXIMISATION • Since the final allocation has equal utility (approximately 6), • And A and B are identical, • This means that the final allocation of coffee and data must be an equal split of the goods between A and B. SOCIAL PLANNER: MAXIMISATION • The final allocation of goods doesn’t always have to be an equal split. • It is equal here because of the way we have constructed the problem. SOCIAL PLANNER: MAXIMISATION • We can see the link between the Edgeworth Box and the allocation of goods, and the SP’s maximization problem by looking at the graphs next to each other. SOCIAL PLANNER: MAXIMISATION • Point i on the UPF corresponds roughly to 7.5gigs for both A and B, and 5kgs for both A and B. THOUGHTS ON THE SOCIAL PLANNER • Thinking about the hypothetical social planner allows to place policy issues in our (simple) framework. • The weights we choose for the two different people could be the weights we place on two groups of people, for example. LIMITS OF THE SOCIAL PLANNER IDEA • In reality, things are never this simple. • Democracy requires debate and the government/SP is rarely (never?) impartial, nor does it treat people equally. • Even without democracy, there might be a rule that says the social planner is an absolute monarch/chief who again isn’t impartial. • In addition, people have different social standings. An older person in some societies might have more power than a younger person in determining social welfare/choosing/lobbying the SP. Same thing with gender – men might have more power than women. • The SP in our example isn’t very realistic. MARKET INSTITUTIONS • We now consider A and B to be interacting in markets, rather than sitting in a digs as friends. • We ask how things are different. • We assume that a market “game” includes • the rule of law, private property, fallback options. • Players may have differing bargaining power, partly determined by the rules of the game. • We assume an initial endowment as the starting point and that it is an inefficient allocation (point z). MARKET INSTITUTIONS • Each person has a fallback position/participation constraint which is their initial endowment (the next best alternative to potential exchanges). • That is, if a person refuses an exchange, then they will always at least have their initial endowment. • The goods are private property – each person can completely exclude the other person from consuming the goods they own. • Improvements in utility after an exchange are called rents • the difference in utility between the post-exchange allocation and the initial allocation. PARTICIPATION CONSTRAINTS/FALLBACK • Unlike the SP case, the introduction of private property/ownership gives A and B some control over the outcome. • The ability to refuse a trade has an important consequence: the initial endowment REALLY matters. • Different starting points lead to different outcomes. • Any initial allocation results from prior games, including ones in which the market institutions we have described were not enforced • Eg some people may have had their property taken away from them by force. PARTICIPATION CONSTRAINTS/FALLBACK • There is a history to the initial allocations we consider. • The history that brought us to current initial allocations limits the possible results of exchanges taking place after the initial allocations. • If we start at one inefficient point and move to another inefficient point (which is pareto-improving), this move will determine the set of possible allocations after that. PARTICIPATION CONSTRAINTS/FALLBACK • Let’s go back to the box. • Remember that π’π’π§π§π΅π΅ and π’π’π§π§π΄π΄ are the indifference curves and utility at our initial endowment of point z. • Because no one will agree to an exchange which makes them worse-off, this means that any exchange from point z must fall between the indifference curves π’π’π§π§π΅π΅ and π’π’π§π§π΄π΄ . PARTICIPATION CONSTRAINTS/FALLBACK • Allocations outside of this area will necessarily make one of the players worse-off compared to the starting point. • Neither of the players will voluntarily agree to this. • So π’π’π§π§π΅π΅ and π’π’π§π§π΄π΄ are our participation constraints/fallback position. PARTICIPATION CONSTRAINTS/FALLBACK • With these constraints, the best B can achieve is π’π’4π΅π΅ which is tangent to A’s participation constraint • A won’t be made worse-off by moving to point π‘π‘ π΅π΅ so she could agree to this trade if the other option was her fallback position. • Similarly, the best A can achieve is moving to π’π’3π΄π΄ . PARTICIPATION CONSTRAINTS/FALLBACK • Let’s start with a different history - initial endowment is at point h. • Now π’π’3π΅π΅ is B’s participation constraint but π’π’π§π§π΄π΄ is still A’s participation constraint. PARTICIPATION CONSTRAINTS/FALLBACK • The possible allocations will not lie between π’π’π§π§π΅π΅ and π’π’π§π§π΄π΄ . • The best B can do from point h is π’π’4π΅π΅ (which is the same as before when we started from point z). • BUT the best A can do is now close to point i (the indifference curves aren’t drawn but we know that this will be pareto-efficient). • Point i for A is at a lower utility than point π‘π‘ π΄π΄ . PARTICIPATION CONSTRAINTS/FALLBACK • With the exact same institutions, point h as a history is potentially worse for A than point z as the history. PARETO IMPROVING LENS • Let’s again assume that point z is the initial endowment. • We show the possible potential gains from trade (pareto improvements) with the yellow shaded area in the box. • We call this area the Pareto improving lens. • Every allocation within the lens represents a pareto-improvement relative to point z. PARETO IMPROVING LENS • All the points in the yellow area could potentially be reached by voluntary changes in the endowment allocation through exchange by A and B. BARGAINING SET • Just like before, we can map the Pareto-improving lens in utility-space. • This is the exact same thing we did with the social planner. • To make the language clear, we will call pareto-improvements in utility-space the bargaining set. BARGAINING SET • The main difference between the social planner and market exchanges are the participation constraints. • The SP didn’t have any restrictions because the players had no control. BARGAINING SET • Now, the players have some control. • We map the participation constraints as the vertical and horizontal dashed line that go through point z. BARGAINING SET • A will refuse any exchange that puts him to the left of the vertical dashed line. • Similarly, B will refuse any exchange that puts him below the vertical dashed line. • Combining these constraints means that any exchanges must result in allocations to the right and upwards from point z • but could include the dotted lines above z or to the right of z. BARGAINING SET • Everything in the yellow region represents an increase in utility for either A, B, or both, compared to point z. PARETO IMPROVING LENS • Why does the lens include areas above the PEC? • All of these points are pareto improving so both could agree to trade to get there. PARETO IMPROVING LENS • A first trade from point z to point h means that the post-trade pareto-improving lens reduces at the expense of A. • This is the same thing as having started from h as the initial endowment in the first place. Summary so far • Thus far we have set up an interaction with two players • We have explained how market institutions like ownership, private property and fall-back options lead to participation constraints and narrow down the set of possible outcomes to the bargaining set. • We have noted that rules of the game determine bargaining power but we have not yet provided any specific rules and so we only have very broad restrictions on the outcome of bargaining • (eg the outcome of the game must be weakly better than fallback options). • Bowles and Halliday then give different possibilities for specific rules of the game, and explain how these lead to different sets of outcomes • We work through some of them Rules of the Game possibility 1- with Symmetric Bargaining Power • We assume the players agree to the following rules: 1. At any allocation point where the two MRSs are not equal, choose a price for the exchange that is mid-way between the two MRSs. • One player will be giving up good X and receiving good Y at this price and the other will be giving up Y and receiving X at this price. 2. Find out how much each player is willing to trade at this price. 3. Since trades must be voluntary, the amount of trade that must occur is the amount for the player who wants to exchange the least amount. 4. Check whether the new post-trade allocation indifference curves intersect (ie MRSs are equal) • If they do intersect, repeat steps 1-4 from the new post-trade allocation. • If they do not intersect, ie they are tangent, end the game with this allocation. Rules of the game possibility 1 -outcome • The outcome of the game with the rules specified on the previous slide will be on the Pareto-efficient curve. • This is because 1. Each trade must be Pareto improving (otherwise it wouldn’t have been undertaken) and so the players move in the PEC direction. 2. Trade concludes at a Pareto-efficient outcome, • ie the MRSs are equal, • ie this is on the PEC by definition. • These rules imply the outcome is somewhere on the PEC • Exactly where will depend on the players’ initial endowments and their utility functions. • The more unequal the initial allocation the more unequal the final outcome will be. • We don’t attempt to work out where the final allocation is (unlike in later possible rules of the game) Rules of the game option 2 -outcome • Bowles and Halliday mention a second set of rules that also involves symmetric bargaining. • It is the same as option 1 but instead of picking a rate of exchange/price midway between the two prices, • the players flip a coin and the exchange happens at either • just below A’s willingness to pay for data in terms of coffee (if the coin lands on heads) • or just above B’s willingness to pay for coffee in terms of data (if the coin lands on tails). • Again the final outcome will be on the PEC, although where depends on where the coins land. Rules of the game with ASYMMETRIC EXCHANGE • One person usually has more control over the outcome of interactions than the other person. • This results in asymmetric exchange or asymmetric power. • Asymmetric power leads to one person being able to gain more of the rents from the exchange than the other person. • We will consider two options for rules of the game with asymmetric exchange/power: • Price-setting power/first mover advantage • Take-it-or-leave-it-power (TIOLI) PRICE-SETTING POWER (PSP) • We are going to start with Price-setting power (PSP). • This is sometimes also called first-mover advantage. • To start we need to revise budget constraints and prices. • This should be familiar to you from other ecos courses. BUDGET CONSTRAINT • Your budget is given by ππ = πππ¦π¦ π¦π¦ + πππ₯π₯ π₯π₯ • The total amount of money you spend on good x and good y, at the price of good π¦π¦ (πππ¦π¦ ) and the price of good π₯π₯, (πππ₯π₯ ), will equal the total amount of money you have (ππ). • If we assume ππ is fixed and that you spend all your money, every combination of prices and quantities must still equal ππ. • We will assume that πππ¦π¦ = 1. • The implication is that πππ₯π₯ is the price of coffee in terms of data, ie the relative price of coffee, or the amount of coffee one can buy for 1 unit of data. • The slightly new idea is that we take concepts from constrained optimization decisionmaking for one person, purchasing two goods, to maximise their utility • and then apply these to the types of interactions between two players that we have been considering this week. • The budget constraint is now a line giving feasible combinations of data and coffee available to each player through exchange at some given price. BUDGET CONSTRAINT • Mathematically, we are going to represent the current holdings of π₯π₯ and π¦π¦ for player B as π¦π¦π§π§π΅π΅ and π₯π₯π§π§π΅π΅ • z is like the initial allocation point. • The budget constraint for B is: ππ = π¦π¦π§π§π΅π΅ + πππ₯π₯ π₯π₯π§π§π΅π΅ • The final allocation is represented as π¦π¦ π΅π΅ and π₯π₯ π΅π΅ (i.e. no z subscript). • In our setup with two players, the value of the final endowment must be equal to the value of the initial endowment. • Therefore, π¦π¦ π΅π΅ + πππ₯π₯ π₯π₯ π΅π΅ = π¦π¦π§π§π΅π΅ + πππ₯π₯ π₯π₯π§π§π΅π΅ (1) BUDGET CONSTRAINT • It is useful to isolate the price when we consider price setting power. • Rearranging (1) to isolate the prices: • πππ₯π₯ π₯π₯ π΅π΅ − π₯π₯π§π§π΅π΅ = (π¦π¦π§π§π΅π΅ −π¦π¦ π΅π΅ ) (2) π¦π¦π§π§π΅π΅ −π¦π¦ π΅π΅ • πππ₯π₯ = π΅π΅ π΅π΅ (2) π₯π₯ −π₯π₯π§π§ • This is the slope of the budget constraint (change in y over change in x). • The negative of the slope of a feasible frontier in general is called the Marginal Rate of Transformation (MRT). BUDGET CONSTRAINT • What (2) is saying is that the value of the extra coffee you get – price of coffee times change in quantity of coffee, ππ π₯π₯ π΅π΅ − π₯π₯π§π§π΅π΅ , – is equal to the value of data given up, π¦π¦π§π§π΅π΅ − π¦π¦ π΅π΅ , – price of data times change in quantity of data. PRICE-SETTING POWER (PSP) • We will discuss price-setting power (PSP) which, as the name suggests, one person has the power to set the price but not the quantity. • PSP is sometimes called first-mover-advantage. • The person with PSP will offer/set prices and the person without PSP will decide on the quantities. • In real life an example is a firm selling to a customer. PRICE-SETTING POWER (PSP) • In our A and B examples the person with PSP can say: • “I will give you 1 kg of coffee for every 3 gigs of data you give me. You can decide how much data you would like to exchange for coffee at that ratio (the price), but the ratio is not going to change. Of course you are free to buy nothing.” • The person without PSP will then say: “At that ratio (price), I will exchange 10 gigs of data.” • Let’s formalise this logic. WHAT IS PRICE-SETTING POWER? • Let’s assume player A has PSP. • A will stipulate a price (but not quantity). • At a given price, B can decide what quantities he’s prepared to exchange. • A will need to figure out how B will respond (in terms of quantity exchanged) to the prices A offers. • A must satisfy B’s participation constraint (defined as above) AND now also B’s incentive compatibility constraint. • B’s best response to every potential price that A offers. HOW DOES PRICE-SETTING POWER WORK? • The goal for A is to get B to agree voluntarily to exchange an amount of data and coffee that gives A the highest possible utility at the price A proposes. • Stage 1: A needs to think about what B will do when A offers different prices. • This is B’s best-response function. • Stage 2: What should A do, given what A thinks B will do in response to the prices offered? • This is an example of A using “backward induction”. STAGE 1 • Stage 1 is about A figuring out the many solutions to B’s constrained optimization problems at all the different prices A could offer. • In real life this could come from prior interactions with B or with another customer, perhaps similar to B. • B’s constrained optimization problem is the same as from second year micro, • just that now there are many optimization problems, each one having a different price. • We can show all the optimal responses on the best response function. PRICE-SETTING POWER: GRAPH STAGE 1 • A proposes ππ4 to B. • This is a standard constrained maximization problem like in second year micro • Although it is “the wrong way around” on the figure. PRICE-SETTING POWER: GRAPHS STAGE 1 • Given ππ4 , at the initial endowment of point z, B’s IC falls within his budget constraint, so it is not the utility maximising point: • This wouldn’t be B’s best response since he could do better by moving from π’π’π§π§π΅π΅ to π’π’2π΅π΅ . • Point ππ4 will be B’s optimal bundle at ππ4 . PRICE-SETTING POWER: GRAPHS STAGE 1 • Notice that the ππ4 budget constraint goes through point z. • This is because we said the value of post-exchange endowment must equal value of initial endowment (in this case, initial endowment is point z). • This shows graphically what our budget constraint equation showed. PRICE-SETTING POWER: GRAPHS STAGE 1 • This method of figuring out B’s best response can be applied to every potential price. • The figure shows 3 possible prices that A offers B. • At each of the 3 prices, notice that B has a point of tangency between his ICs and the price. • The best response function is ALL the points of best response to allπ΅π΅ the possible prices, where ππππππ = ππ. • B’s best-response function is called his incentive compatibility constraint. STAGE 2 • A now thinks about B’s responses to all the prices A might have offered-B’s best response function (BRF). • New idea- From A’s point of view the slope of B’s BRF is A’s MRT• how much data A can get for coffee, given how B responds to the different prices. • B’s BRF is A’s feasible frontier. • In simple 1 person constrained optimization problems MRT, the constraint, is constant and equals “the” price. • Here MRT varies. • From this point, A still does constrained optimization, setting MRS=MRT. • We look at this graphically. GRAPHS STAGE 2 • A’s constraint is B’s BRF. • To max U, A must find tangency between his ICs (MRS) and B’s BRF (MRT). • Tangency is at point n. • Everything below B’s BRF is a feasible allocation for A. • But choosing a point below B’s BRF wouldn’t be the max utility A could get. GRAPHS STAGE 2 • Point n is called the Nash Equilibrium. • Why? Because at this point, given what the other player is doing, neither player has an incentive to move to any other point. PSP-PARETO EFFICIENCY • Here we include B’s IC at point n. π΅π΅ . • You can see that π’π’πππ΄π΄ intersects with π’π’ππ • This means n is not a Pareto efficient point, since that requires the players’ ICs to be tangent. • The area between the intersecting ICs represents a Pareto-improving lens. • The outcome also means B gains some rents, since the final allocation puts him on a higher IC. • This is not true in the game we study next… ASYMMETRIC EXCHANGE: TIOLI • We now study a fourth possible set of rules of the game. • With PSP, the person with power only controlled the prices. • What happens if the person with power controls both the prices and the quantities? • Power over prices and quantities is called Take-It-Or-Leave-It (TIOLI) power. • We are going to limit TIOLI so that the other player still has the right to refuse a transaction • i.e., their participation constraint must be reached. • Put differently, they have property rights over their current endowment. TIOLI VS PSP • Differences between TIOLI and PSP: • TIOLI: • Power over prices and quantities. • Only the participation constraint must be respected. • Example- a firm advertising a job with wages (price) and hours of work (quantity) stipulated. • PSP: • Power over only prices but not quantities. • Participation AND incentive compatibility constraints must be respected. • Example- a firm setting a price for its product. • PSP is a “weaker” form of power. TIOLI AS AN ULTIMATUM • We can think of TIOLI as an ultimatum game. • In the ultimatum game, the proposer offers an allocation, and the responder can only reject or accept the offer. • The proposer is saying to the responder: “take it or leave it”. • You are probably familiar with the idea of “TIOLI” from other ECOs courses. • The player with TIOLI power can often capture most or all rents from an interaction with another player. HOW DOES TIOLI WORK? • Let’s assume A has TIOLI power. • A says to B, “I’ll give you 2 kilograms of coffee and you give me 9 gigabytes of data. If you refuse, I will not agree to any other trade you might propose.” • In other words, “either accept the allocation I impose, or we both stay at our endowment, z” • For A’s TIOLI power to work, his threat that he will not offer any other deal must be credible • B must believe that this is the only offer possible- a credible threat. • If B doesn’t believe A, then he will bargain for other offers more in his favour. • We will assume B does believe A’s threat is credible- similar to many labour market interactions • Will you/have you bargained with your employer?! HOW DOES TIOLI WORK? • As with PSP, A uses backward induction to help her decide what to do. • She asks what B will do in response to various options, and then given those responses, her utility maximising response. • So again this is a constrained maximisation problem for A. • Note that B has much less agency with TIOLI- he can’t choose quantities anymore- only whether to accept or reject A’s offer. • B will reject any offer that makes him worse off than before. • So B’s participation constraint is what limits what A can offer. HOW DOES TIOLI WORK? • A will not make an offer that isn’t the best she can do. • B will not accept an offer that makes him worse-off. • Where these two conditions are satisfied, there are no more mutual gains from trade • This will mean ππππππ π΄π΄ = ππππππ π΅π΅ and indifference curves of the two players are tangent • This implies Pareto efficiency has been achieved. • Not because A cares about B or about Pareto efficiency itself, but • Because she wants to do the best she can for herself, while taking into account B’s participation constraint. HOW DOES TIOLI WORK? • In reality, A might offer an allocation just a tiny bit better than B’s participation constraint • because A might think that B would reject an offer that doesn’t improve B’s utility at all. • We are going to keep things simple and assume B accepts an offer that simply meets B’s participation constraint. TIOLI: UTILITY FUNCTIONS • Before we explain TIOLI further, we are going to make two changes: • A and B’s utility functions will be quasi-linear instead of Cobb-Douglas. • We will talk about money and time instead of coffee and data. • Quasi-linear utility functions are linear in one good (money in our case) and non-linear in the other (time). • Very general form: π’π’ =y +h(x) where h is a non-linear function. • These changes are just to simplify things and don’t have meaningful consequences except these two technical points: • The MRS will now depend only on the non-linear good. • The ICs will be parallel. TIOLI: MRS • With quasi-linear utility functions, the MRS depends only on the nonlinear good. • This is because the MRS is the ratio of the marginal utilities. • The marginal utility of the linear good is constant (why!?), so MRS only depends on the non-linear good. • We are going to work through an example next. TIOLI: MRS • One example of a quasi-linear utility function is: π’π’π΅π΅ = π¦π¦ π΅π΅ + 32π₯π₯ π΅π΅ − (π₯π₯ π΅π΅ )2 • π¦π¦ π΅π΅ is B’s money (the linear good) measure in rands. • π₯π₯ π΅π΅ is B’s time (the non-linear good) measured in hours. • We can now find the marginal utilities: πππ’π’π΅π΅ • π΅π΅ = 1 = π’π’π¦π¦ πππ¦π¦ • πππ’π’π΅π΅ π΅π΅ = 32 − 2π₯π₯ = π’π’π₯π₯ π΅π΅ πππ₯π₯ TIOLI: MRS π’π’π₯π₯ 32 −2π₯π₯ π΅π΅ • ππππππ = = = 32 − 2π₯π₯ π΅π΅ π’π’π¦π¦ 1 • From the MRS above, we can see that changes in money/the linear good (y) do not affect MRS. • This means that the indifference curves will now be parallel: • for a given x, the distance between the indifference curves will be constant. • Let’s look at this now. TIOLI: MRS • Notice that for a given amount of time (hours of living), say 8 hours, point f on π’π’1π΅π΅ is parallel to point g on π’π’2π΅π΅ , and parallel to point h on π’π’3π΅π΅ . • This is true for all ICs and all given values of time. TIOLI: MRS • Also notice that for a given level of money, the ICs flatten out with more time. • To see this, compare point e and point g. • Point e is steeper than point g. • This is because of diminishing marginal utility in time. TIOLI: MRS • We will come back to quasi-linear preferences throughout the course. • They aren’t very realistic descriptions of preferences, but they do make our lives simpler without much loss to the broader and main conceptual analysis. • One of the key benefits is that we can now measure utility in the same units as the linear good (y) i.e., money in rands. • A one unit increase in y leads to an upward/vertical shift of the IC by one utility unit. TIOLI: EMPLOYMENT • To illustrate TIOLI, we are going to use an employer and employee example. • A is the employer and B is the potential employee. • A and B will be bargaining over wages and labour hours. • Employers have the power to set the price (wage) and quantity (hours worked) of the exchange. • This means that they have TIOLI. • Employees only have the power to refuse an employment offer. • A and B have the same quasi-linear utility functions – shown 5 slides back. TIOLI: EMPLOYMENT • We are going to continue thinking about the two goods as time and money. • Pay careful attention to how we frame time here: A and B are bargaining over B’s time. • As the employer, A wants B’s hours of work. • As the employee, B prefers free time to working, all else equal, but also likes money. • So in terms of B’s time, what is “good” for A is the opposite of what is “good” for B. TIOLI: EMPLOYMENT • We have a total of 16 hours of possible work and a total of R400 to allocate. • As the employer, A will start with an initial endowment of all the money in society and none of B’s time. • As the employee, B will start with an initial endowment of all of his own time (called “hours of living” in the textbook) and none of the money. • Again using point z to indicate our initial endowment, this means: • (π₯π₯π§π§π΄π΄ , π¦π¦π§π§π΄π΄ ) = (0, 400) • (π₯π₯π§π§π΅π΅ , π¦π¦π§π§π΅π΅ ) = (16, 0) TIOLI: EMPLOYMENT • Because we have changed from Cobb-Douglas to quasilinear utility functions this Edgeworth Box looks a bit different to those we have looked at so far. • But nothing changes conceptually. TIOLI: EMPLOYMENT • Firstly, we have money on the left and right axes, and B’s time on the top and bottom axes. • On the top axis, π₯π₯ π΅π΅ , B’s utility increases with more hours of living. • On the bottom axis, π₯π₯ π΄π΄ , A’s utility increases as B works more hours for A. • Our initial endowment is point z. TIOLI: EMPLOYMENT • You will also see that the Pareto Efficient Curve is a vertical line. • This is because we have used quasi-linear preferences. • The meaning and implications of the Pareto Efficient Curve don’t change at all. TIOLI: EMPLOYMENT • π’π’π§π§π΅π΅ and π’π’π§π§π΄π΄ are B and A’s participation constraints. • Recall that our initial endowment is point z: • A has R400, and 0 hours hired of B’s time. • B has R0 and all 16 hours of his own hours of living. • Also recall that because of the quasi-linearity, we can measure utility in rands. TIOLI: EMPLOYMENT • Remember that the utility functions are: • π’π’ π΄π΄ = π¦π¦ π΄π΄ + 32π₯π₯ π΄π΄ − (π₯π₯ π΄π΄ )2 • π’π’π΅π΅ = π¦π¦ π΅π΅ + 32π₯π₯ π΅π΅ − (π₯π₯ π΅π΅ )2 • Starting with π’π’π§π§π΄π΄ , with zero π₯π₯ π΄π΄ , A’s utility is simply π¦π¦ π΄π΄ = 400 = π’π’π§π§π΄π΄ • For B, plugging (π₯π₯π§π§π΅π΅ , π¦π¦π§π§π΅π΅ ) = (16, 0), π’π’π§π§π΅π΅ = 256. TIOLI: EMPLOYMENT • Lastly, in terms of increasing utility, • π’π’3π΅π΅ > π’π’2π΅π΅ > π’π’π§π§π΅π΅ • π’π’3π΄π΄ > π’π’2π΄π΄ > π’π’π§π§π΄π΄ TIOLI: EMPLOYMENT • Hopefully, you can see that while things look different, nothing fundamental about the Edgeworth Box analysis has changed. • So what was the point of all of this? • We can now move on to understand how differences in who has TIOLI power may affect outcomes. TIOLI: EMPLOYMENT • Since we are talking about employment, we will focus on two common rules of the game (i.e., institutional arrangements): • The employer (A) makes a TIOLI offer. • Employee trade union (represented by B) makes a TIOLI offer. • So either the employer says: “I will pay you π¦π¦ rands for π₯π₯ hours”, or the employee says: “I want π¦π¦ rands for π₯π₯ hours”. • In both cases, the person with TIOLI is setting hours (quantity) and price (wage). EMPLOYER’S TIOLI • Assume the employer (A) makes a TIOLI offer: • A must respect B’s participation constraint. • In maximizing her utility, A will offer an allocation that will not make B worse-off. • The offer will be Pareto efficient otherwise A could do better if she made another offer. EMPLOYER’S TIOLI • Compare A and B’s utility at π‘π‘ π΄π΄ versus the initial endowment: • π’π’3π΄π΄ − π’π’π§π§π΄π΄ = 652 − 400 = 252 • π’π’π§π§π΅π΅ − π’π’π§π§π΅π΅ = 256 − 256 = 0 • With TIOLI, A would capture all the rents at point π‘π‘ π΄π΄ . EMPLOYEE’S TIOLI • Now assume the employee trade union (B) can make a TIOLI offer: • B must respect A’s participation constraint. • In maximizing his utility, B will offer an allocation that will not make A worse-off. • The offer will be Pareto efficient otherwise B could do better if he made another offer. EMPLOYEE’S TIOLI • Now that B has TIOLI, relative to the initial endowment, he will capture all the rents at π‘π‘ π΅π΅ : • π’π’π§π§π΄π΄ − π’π’π§π§π΄π΄ = 400 − 400 = 0 • π’π’3π΅π΅ − π’π’π§π§π΅π΅ = 508 − 256 = 252 • Clearly, who has TIOLI matters for the distribution of utility. EMPLOYEE/EMPLOYER TIOLI • It is crucial to note that in both cases, regardless of who had TIOLI, the outcomes (π‘π‘ π΄π΄ or π‘π‘ π΅π΅ ) were Pareto efficient: • We have tangency (ππππππ π΄π΄ = ππππππ π΅π΅ ) • In both cases, A and B were utility maximizing subject to the other’s IC. • Both lie on the Pareto efficient curve CALCULATING A’S TIOLI OFFER • To calculate A’s TIOLI offer, we need know two things: • The UPF equation, because we know the outcome must be pareto efficient. • B’s participation constraint (his utility at the initial endowment) because this will be the most A will offer B. • The PEC is simpler in the above example than in the earlier case we worked through finding the PEC. • The graph makes it clear the PEC is the straight line π₯π₯ π΅π΅ = 8 for player B and π₯π₯ π΄π΄ = 8 for player A. • How do we get that mathematically?... Obtaining the PEC • We can express the PEC in terms of A or B. • We do it for B here. • As in our previous calculations to obtain the PEC there are two steps. • The first is to use the total goods to express π₯π₯ π΅π΅ in terms of the total amount of x available in the economy/this game: • π₯π₯ π΄π΄ + π₯π₯ π΅π΅ = 16 so π₯π₯ π΄π΄ = 16 − π₯π₯ π΅π΅ (1) Obtaining the PEC • The second step is to set the MRS of A and B equal to each other, since that is the definition of the PEC. • Slide 135 has the MRS for B. • Since A’s utility function is the same as B’s, their MRS is the same: • ππππππ π΅π΅ = 32 − 2π₯π₯ π΅π΅ and ππππππ π΄π΄ = 32 − 2π₯π₯ π΄π΄ • Equating these implies that π₯π₯ π΄π΄ = π₯π₯ π΅π΅ • We then substitute in π₯π₯ π΄π΄ from equation (1) to find that π₯π₯ π΅π΅ = 16 − π₯π₯ π΅π΅ • This implies π₯π₯ π΅π΅ = 8 CALCULATING A’S TIOLI OFFER to B • Recall we need B’s PC and the PEC to find A’s TIOLI offer. • π₯π₯ π΅π΅ = 8 (PEC) • π’π’π§π§π΅π΅ = π¦π¦π§π§π΅π΅ + 32π₯π₯π§π§π΅π΅ − (π₯π₯π§π§π΅π΅ )2 = 0 + 32 16 − 16 2 = 256 (B’s PC) • So then we ask “what amount of money for 8 hours work will give B utility of 256?” • Answering that means equating the PC and the utility function with π₯π₯ π΅π΅ =8 substituted in: • 256=π¦π¦ π΅π΅ + 32(8) − (8)2 • This implies that π¦π¦ π΅π΅ = 64 and we know that π₯π₯ π΅π΅ = 8 so the TIOLI offer for B is (8,64), which we assume B accepts. • Since we know the total money and hours it is easy to figure out A’s allocation when B accepts A’s TIOLI offer. • This is the point π‘π‘ π΄π΄ in the figures above. TIOLI VS PSP Outcomes TIOLI: • The outcome is Pareto efficient & highly unequal in favour of the party with TIOLI. • Player with TIOLI has power over price and quantity. • Player with TILOI is constrained only by other player’s participation constraint. Price-setting power: • The outcome is Pareto inefficient & favours player with PSP but not by as much as TIOLI. • Player with PSP has power to set prices. • Player without PSP has power to set quantities. • Power of Player with PSP is constrained by other player’s best response/incentive compatibility constraint & participation constraint. WHAT HAVE WE LEARNT? • There have always been exchanges of goods between people, which result in gains for at least one party. • The rules of the game/institutions governing the exchanges matter for the efficiency and equity of the outcomes. • Pareto-efficiency can be achieved with equal gain (the social planner) or with gain only to one party (TIOLI).
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