Economics 101A Spring A Revised Version of the Slutsky Equation Using the Expenditure Function or, the expenditure function is our friend!

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1 Brief review... Economics 11A Spring 25 A Revised Version of the Slutsky Equation Using the Expenditure Function or, the expenditure function is our friend! e(p 1, u ) = min p 1 + p 2 x 2 s.t. U(, x 2 ) = u = p 1 x 1 (p 1, u ) + p 2 x 2 (p 1, u ) where x 1 and x 2 are the compensated demands : the choices you would make to get utility level u as cheaply as possible at prices (p 1 ). Remember that the Lagrangean for the exp-min problem is: L(, x 2, :) = p 1 + p 2 x 2! : ( U(, x 2 )! u ) The foc are: a) p 1! : U 1 (, x 2 ) = b) p 2! : U 2 (, x 2 ) = c) U(, x 2 ) = u What are the derivatives of the expenditure function w.r.t. (p 1 )? From the definition e(p 1, u ) = p 1 x 1 (p 1, u ) + p 2 x 2 (p 1, u ), and so: ( ) Me(p 1, u ) / Mp 1 = x 1 (p 1, u ) + p 1 Mx 1 (p 1, u ) / Mp 1 + p 2 Mx 2 (p 1, u ) / Mp 1. n tutorial we discussed the envelope theorem which says that the 2 nd and 3 rd terms cancel out. A quick way to prove that: Use the constraint: U( x 1 (p 1, u ), x 2 (p 1, u ) = u, and differentiate w.r.t. p 1 to get U 1 Mx 1 (p 1, u ) / Mp 1 + U 2 Mx 2 (p 1, u ) / Mp 1 = But U 1 (, x 2 ) = p 1 /: and U 2 (, x 2 ) = p 2 /: from the f.o.c. Substituting we get p 1 /: Mx 1 (p 1, u ) / Mp 1 + p 2 /: Mx 2 (p 1, u ) / Mp 1 = which means that p 1 Mx 1 (p 1, u ) / Mp 1 + p 2 Mx 2 (p 1, u ) / Mp 1 =. Thus we have:

2 Me(p 1, u ) / Mp 1 = x 1 (p 1, u ). There is a story we tell to go along with this. f you are initially minimizing expenditure, and the price of good 1 goes up, what do you do? Your first order adjustment is to simply continue buying the old bundle that will increase your spending by x 1 )p 1. That is the first term in ( ). But then you would like to re-adjust your choices of goods 1 and 2 to reflect the new prices. Those adjustments are the second and third terms in ( ). But because your initial choice was optimal (satisfying the f.o.c) when you try and re-adjust and x 2 you don t save any more. Now we are ready to analyze what happens to the uncompensated or regular demand when prices rise. Suppose we start at an intial situation with prices (p 1 ) and income. The initial choices are = (p 1, ), x 2 = x 2 (p 1, ) where the ( ) and x 2 ( ) functions without superscripts are the regular demand functions We decompose the effect of a change in price )p 1 = p 1 N! p 1 as follows: a) starting from, x 2, think of the adjustment you would make if you could keep utility constant (remain on the old indifference curve). This gets you to a new position, x 2. Since prices have risen this position costs more than you were initially spending. This move is called the substitution effect of the price increase. b) then from, x 2 think of the adjustment you make to get back to spending only the amount of income that you actually have. This is a movement inward along an income expansion path (EP). You end up at N, x 2 N. This move is called the income effect of the price increase. Note that the total change in is ) = N!

3 = ( N! ) + (! ) = ) + ) S. How big are these two parts? To begin, notice that (, x 2 ) and (, x 2 ) are on the u indifference curve. = (p 1, ) But it is also true that = (p 1, u ). Also, = (p 1 N, u ) So ) S =! = (p 1 N, u )! (p 1, u ). Mx 1 (p 1, u ) / Mp 1 )p 1 The substitution effect depends on the rate at which compensated demands change: this is purely a function of the curvature of the indifference curve. How about the income effect? ) = N! First note that N = (p 1 N, ) : its just your regular demand choice at (p 1 N, ). But what is? t is the choice when you have enough income to get to the old indifference curve at the new prices. How much money do you need to get there? That s just e(p 1 N, u )! So Thus = (p 1 N, e(p 1 N, u ) ) Make sure this makes sense to you! ) = (p 1 N, )! (p 1 N, e(p 1 N, u ) ). M (p 1, ) / M (! e(p 1 N, u ) ) So the income effect depends on the income derivative of demand times the size of the income change ) =! e(p 1 N, u ). Note that ) <, since you need more than to get to the u indifference curve when prices are (p 1 N ).

4 But how big is )? We have to use one last trick. We know that = e(p 1, p 2, u ). So we can write ) =! e(p 1 N, p 2, u ). = e(p 1, p 2, u )! e(p 1 N, p 2, u ). Me(p 1, p 2, u ) / Mp 1 ( p 1 - p 1 N) = Me(p 1, p 2, u ) / Mp 1 (!)p 1 ) =! Me(p 1, u ) / Mp 1 )p 1 Note that this is negative for a rise in the price of good 1. Finally (almost done) we have Me(p 1, u ) / Mp 1 = x 1 (p 1, u ) from the first page of this lecture note = because is also the compensated demand choice All this together means that ).! )p 1. Note that the size of the income effect depends on how much you were buying. Pulling it all together, ) = M (p 1, ) / M ) =! M (p 1, ) / M )p 1 Thus ) = ) + ) S =! M (p 1, )/M )p 1 + Mx 1 (p 1, u ) / Mp 1 )p 1 or ) / )p 1 =! M (p 1, )/M + Mx 1 (p 1, u ) / Mp 1. Now take the limit as )p 1 gets small and the ratio ) / )p 1 tells us the derivative of the regular demand function. We have established: M (p 1, ) / Mp 1 =! M (p 1, )/M + Mx 1 (p 1, u ) / Mp 1

5 This is called Slutsky s equation, after the Russian economist who first proved it about 1 years ago. Slutsky s equation says that the derivative of the regular demand with respect to p 1 is a combination of the income and substitution effects. The income effect depends on the derivative of demand w.r.t income, times the amount of you initially consume. The substitution effect depends on the derivative of the compensated demand for good 1. A neat thing about the Slutsky equation is that it gives us a way to recover information about indifference curves from the derivatives of demand w.r.t. prices and incomes. n principle, we can observe M (p 1, ) / Mp 1 and M (p 1, )/M. So we can infer: Mx 1 (p 1, u ) / Mp 1 = M (p 1, ) / Mp 1 + M (p 1, )/M Suppose we get an estimate of Mx 1 (p 1, u ) / Mp 1 that is close to. That means indifference curves must be almost like right angles.