EquivalentVariation
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Hale Waihona Puke No we can calculate a simple example. Assume u (x; y ) = x 2 y 2 : The problem is
fx;y g
1
1
max x 2 y 2 s:t: p x x + py y
1
1
m x
0; y
0
The Lagrangian to the problem is $ (x; y; ) = x 2 y 2
When she is compensated with the amount CV, her new income is
1 0 0 0 E p1 = E p0 + CV; x ; py ; u x ; py ; u 1 hence she will be able to attain u0 at prices p1 x ; py : Lets now analyze the Equivalent Variation (EV). This measures how much the consumer is willing to pay (or accept, depending on wether the price is 1 1 0 higher or lower after the change) to avoid the price change from p0 x ; py to px ; py , 1 but attain the new utility level u : Following a similar argument as above, the 0 consumer has income m in the initial situation p0 x ; py and attains utility level 0 1 1 u . If prices change to px ; py and her income does not change, she will attain u1 : The Equivalent variation is how much of her income the consumer is willing to give up (to pay) in order not to su¤er the price change but attain u1 : From the expenditure function, we know that if a consumer wants to attain 0 u1 when prices are p0 x ; py (this is the situation explained above), the minimum 0 1 amount of income needed is E p0 : Hence the consumer will now be x ; py ; u 0 1 willing to pay an amount such that her new income is E p0 : Since x ; py ; u 1 1 1 originally the consumer had income m E px ; py ; u ; the amount that she has to pay is 1 1 0 1 EV = E p1 E p0 ; x ; py ; u x ; py ; u
This function is telling us what is the minimum amount of income that 0 0 the consumer needs, if prices are p0 x ; py ; and we want to attain u : It is very important to note that from the UMP we know that this amount is m: We will take this last statement as true although it has not been proved in class (but we have already used it when we derived the Slutzky Equation). 0 1 1 Suppose that now there is a price change from p0 x ; py to px ; py : The solu1 1 tion to the utility maximization problem now is x p1 p1 x ; py ; m ; y x ; py ; m . Again we can plug this number in the utility function to …nd the utility level, u x
when she pays EV her income becomes
0 1 1 1 E p0 = E p1 x ; py ; u x ; py ; u
EV;
which is exactly the income needed to attain utility level u1 at the original 0 prices p0 x ; py :
fx;y g 0 min p0 x x + py y
s:t: u (x; y )
u0
0 0 0 0 : The ; y c p0 The solution to this problem will be xc p0 x ; py ; u x ; py ; u expenditure function will be 0 0 c 0 0 0 c 0 0 E p0 = p0 + p0 p0 ; x ; py ; u x x px ; p y ; u yy x ; py ; u
A Note on Compensating Variation and Equivalent Variation
Marcos Dinerstein
In this notes we will see more explicitly where the Compensating Variation and the Equivalent Variation take part in the utility maximization problem. Let u (x; y ) be the utility function that represents an individual´s preferences. Assume that preferences are such that Axiom 3 (monotonicity) holds and that u (x; y ) is a continuos function (do not worry about this last assumption, you do not need to know thism). The Utility Maximization (UM) problem is
fx;y g
max u (x; y ) s:t: p x x + py y
m; x
0; y
0
Where px is the price of good x; py is the price of good y and m is income. If this problem is well de…ned (do not worry about what this means now) the solution is x (px ; py ; m) ; y (px ; py ; m) : Suppose that prices are initially 0 0 0 p0 x ; py :Note that since px ; py ; m are just numbers, this is just a vector of two numbers. If we plug in those numbers in the utility function, the utility level is 0 0 u x p0 p0 = u0 : x ; py ; m ; y x ; py ; m The Expenditure Minimization problem is
1 p1 x ; py ; m ; y 1 p1 x ; py ; m
= u1 :
1
1 Note that now the minimum amount of income needed at the new prices p1 x ; py 1 1 is E p1 = m: x ; py ; u Now that we have set up our problem we can think about what is the Equivalent Variation and Compensating Variation. Recall from lecture that Compensating Variation (CV) is how much the 0 1 1 consumer has to be compensated if prices change from p0 x ; py to px ; py in order to keep attaining the original utility level u0 : We can think of this in the following 0 way. The consumer starts with an income m; and prices are p0 x ; py , hence she 1 attains utility level u0 :If prices change to p1 ; p but her income is still m; she will x y attain utility level u1 : Compensating Variation is the extra amount of income 1 needed in order to be able to attain u0 when prices are p1 x ; py : From the expenditure function, we know that if the consumer wants to 1 attain utility u0 at prices p1 x ; py ; the minimum amount of income needed is 1 0 E p1 : Hence the consumer has to be compensated in such a way that x ; py ; u 0 1 her income is E p1 x ; py ; u . Since before the price change she had income m ; then she only needs the di¤erence 1 0 CV = E p1 x ; py ; u 0 0 E p0 : x ; py ; u
