De…ning the compensating variation, cv, and the equiv-alent variation, ev.de…ncvandev.tex and .pdf October 21, 2014More complicated stu¤, but important stu¤ for you to understandI mislead you; the area under the inverse demand function (marginal willingness-to-pay function) only approximates consumer’s surplus. I lied. It turn outthat CS cannot be de…ned as an area under Marshallian demand function.Consider the inverse demand curves for Diet Coke and Diet Pepsi (assumingeach has a price and then the price of Coke rises. The problem is that whenthe price of Coke changes this causes the demand curve for Pepsi to shift, andthis causes the price of Pepsi to change, which then causes the demand curvefor Coke to shift. And vice-versa. It turn out that the CS as an area underdemand functions is not well de…ned.So a new money measure of an individual’s utility change needed to be found.What properties would one like this measure to have?1. Postive for improvements (policies that cause the individual’s utility toincrease)2. Negative for deteriorations (policies that cause the individual’s utility todecrease)3. The money measure should be an adjustment to your income such thatafter the adjustment the individual is indi¤erent between the original state(without the policy in place) and the new state (with the policy in place)4. The money measure is well de…ned for both improvements and deteriora-tions.There are two CS measures that …t the bill, the compensating variation, cv,and the equivalent variation, ev.They di¤er in terms of whether the income adjustment occurs in the originalstate or the new state.Simple de…nitions of the compensating and equivalent variation11 The cv and ev are both measures of consumer’ssurplus.The cviand eviare calculated for individual i for a change in the state of theworld.1.1 States of the worldA state of the world is all that in‡uences your life that is exogenous to you.What do we mean by a change in the state of the world? We will de…nea state of the world from an individual’s perspective. Let S0idenote the ini-tial state of the world as experienced by individual i, and let S1idenote thenew/proposed state. This is just useful notation.We want to determine individual i’s monetary value for a change from S0ito S1i.The question is how to describe/characterize a state of the world for indi-vidual i?A state of the world is de…ned in terms of that which is exogenous to theindividual (prices, levels of public goods, environmental quality, etc.)2For example, Lady Gaga is one of the characteristics of the state you livein. So, is the prices of coke and Coke, and the fact that the FlatIrons areplaced where they are placed, and that, at the moment, your signi…cant otheris probably not named Wanda Sue.3Now let’s introduce some notation for states of the worldLooking at things simply, the initial state for individual i is completelydescribed by the following vectorS0i (y0i; P0; C0)where y0iis individual i’s income in the initial state,P0p01; p02; :::; p0M is a the price vector for market goods in the initial stateC0c01; c02; :::; c0J is a vector of the levels of the nonmarket commoditiesin the initial state (including the existence of Ms. Gaga)For example, p04is the price of market good 4 in the initial state and c07isthe level of non-market commodity 7 in the initial state.A state of the world is de…ned in terms of all those things that a¤ect thequality of your life, but whose levels you cannot control.The proposed state for individual i is completely described by the levels of ex-ogenous variables that would exist in the proposed state; e.g. S1i fy1i; P1; C1g.For example, if all that changed between the two states is the levels ofsome of the non-market commodities the change would be from fy0i; P0; C0g tofy0i; P0; C1g.We are interested determining the value an individual places on a change inthe state of the world. This value can be positive or negative.For example, if we inact policies to reduce global warming, the rate of globablwarming will be reduced, the price of carbon energy will increase, the price ofproducts that are carbon intensive will increase, your income might go up ordown, etc. How would you as individual i value such a change in the state ofthe world.Do you prefer the initial state or this new state with a reduction in GW?4The introduction of the City of Boulder’s Open-space Program many yearsago a¤ected Boulder in many ways (land preservations, housing prices, incomes,congestion, growth patterns in neighboring communities, etc.). Note that theprogram has a¤ected many people over the years, many of whom have neverlived in Boulder. How would one value such a change in the state of the world?52 One measure of the value of a state changeis the compensating variation, another is theequivalent variationThe cv an the ev are the money measures of welfare changes (the consumer’ssurplus measures) that we want to estimate and want to use, not the area underthe inverse demand function of a particular commodity.Individual i’s cv for a change from S0to S1, cvi, is how much money wouldhave to be subtracted from her income in in the new state to make her in-di¤erent between the initial state and the new state with the subtraction fromincome. Formally the cv is de…ned asfy0i; P0; C0g s f(y1i cvi); P1; C1g:where s denotes indi¤erence.cv > 0 for improvements and cv < 0 for deteriorations.For an improvement, cv is willingness to pay, W T P , for the new state (howmuch she would pay in the new state to be in the new state). For a deterioration,cv is, in absolute terms, willingness to accept, W T A, the new state (how muchshe would have to be paid in the new state to accept it).Individual i’s ev for a change from S0to S1, evi, is how mu ch money wouldhave to be added to income in the initial state to make her indi¤erent betweenthe new state and the initial state with the income additionf(y0i+ evi); P0; C0g s fy1i; P1; C1g:ev > 0 for improvements and ev < 0 for deteriorations. For an improvement,ev is W T A the initial state (how much she would have to be paid in the initialstate to forego the improvement). For a deterioration, ev is, in absolute terms,W T P to not switch to the new state (how much she would pay in the initialstate to not switch).One can show that cv  ev for both improvements and deteriorations. Notethat cv and ev are negative for deteriorations.6For a price or quality