Income%20and%20Substitution%20effect

Income and Subst - Department of Economics 1 Cornell University M Majumdar Economics 3 1 3 Handout 0 Income and Substitution Effects The effect of

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Unformatted text preview: Department of Economics , 1 . Cornell University - M. Majumdar Economics 3 1 3 Handout # 0 Income and Substitution Effects The effect of an increase in income with a given ratio of prices can be depicted by the income offer curve ( alternative name: income expansion curve). A good is an inferior good if its demand falls as a result of an increase in income. The effect of a fall (or rise) in the price of a good (the price of the other good and income held constant) has been analyZed by decomposing the effect as a sum of a substitution effect and an income effect. We shall see this through a numerical example first. ***NOTE: I shall assume that the consumer prefers more of each good to less; hence, the consumer selects the best point ON the budget line.*** Consider a consumer with the utility function U(x,y) =xl/2yl/3 [39y nonneg quantities] ' (*) 7 who faces prices px= 4; py = 3 and has an income m: 12. The budget set of the consumer is given by: B={(x,y): 4x + By 5 12}. But the choice problem can be described as: 'max U(x,y) subject to pxx +py y = 12. To compute the utility maximizing demands for x and y, set Ux/Uy =Px/Py Computing the relevant partial derivatives, we get: (1/2x)/ (l/3y) = 4/3 (3y)/(2x) =4/3 implies y=8X/9. ' (i) From the budget equation px +py = m we get 4x+3y= 12 (ii) From (i)and (ii) 4x+ 3 (8x19) =12 or, 20x=36 or, x: 9/5 I (iii) and [from (ii) and (iii)] y: 8/5. The utility level u~ attained by the consumer is u~=(9/5)1/2(8/5)1/3 z 1 .569 Suppose that px drops to l; py and m remain the same. The budget set is now given by B’ = {(x,y); x + 3y 5 12} Note that B is a subset of B’. It is intuitive that the consumer is “better of ”. There is a possibility of consmning more of the relatively cheap good x, and also there is an increase in the purchasing power. These intuitive ideas can be formalized in different ways. Let us first proceed numerically. ' First, what is the optimal choice for the consumer faced with the new price ratio [and the same U specified by the relation (*) and m: 12] ? It is surely NOT (9/5, 8/5) [Why not ? Think of MRS=price ratio candition! I I]. A little calculation leads to the new demands from the new exercise: maximize U [specified by (*) ] subject to x+3y = 12 : The new demands are; x~=36l5 and y~=8l5 [the property that y and y~ are the same is a special feature of the functional form chosen, do not jump to any general conclusiOnlll]. Set (Jr/Ur. = 1/3 or, y~lx~=2l9 or, y~=2x~/9. From the “new” budget line x~+3y~ = 12, we get 5x~l3= 12 or, x~=36/5; hence, y~=8/5. What we can , in principle, observe is a move from the pair (9/5, 8/5) to (36/5, 8/5). But we can “think” in terms of a “substitution move” and a “move along the income offer curve” as follows. Suppose that the consmner is asked to purchase the old optimal choice at new prices; what is the expenditure to acquire the old bundle at new prices? It is E=[9/5+ 3(8/5)] = 33/5. So we can see that the consumer has $27/5 [= 12- (33/15)] left; we can think of this as a measure of “saving” or “increased purchasing power” of the consumer, which can be used to attain a higher level of satisfaction. But we can go a bit further and think of a substitution move also. I At prices (1,3) with an “income” equal to 33/5, the old choice (although in the budget set 13’) is not optimal {Why ? It' does NOT satisfy the condition Um/Uy = 1/3}. We can show that x” = 99/25 and y”: 22/25 is the optimal choice {complete the calculations involved here}. Now we can think of the move from (99/25, 22/25)-> (36/5 , 8/5) as a move along the income-offer curve [prices remain (1,3) but income is changed from 33/5 to 12. The move from (9/5,8/5) to (99/25, 22/25) is the “substitution effect” (income adjusted appropriately and prices changed). Let us see that in the general case we can argue that the “substitution move” entails an increase in the demand for the good whose price falls. ***"‘ IN WHAT FOLLOWS I shall assume that given any (px , py ; m) [all prices and income positive], there is a UNIQUE optimal choice (x,y) with positive :1: and y. **** SuppOse that the optimal choice at (px, py; m) is (x*,y"‘). In particular, pxx*+ Py 3”“ =9“; Let the price of in: fall by A>0; i.e., the new prices are (px- A, py). If the consumer is asked to purchase the “old” choice, s/he can do it with a decrease in income by( x*)A. Let m* =m - (x*)A. We consider the budget line L ={(x,y): (px -A)x +pyy = m*} through (x*,y*) where the slope reflects the new price ratio py/[px - A]- If the consumer maximizes U subject to this budget line, then the optimal choice (x,y) must satisfy : x>x*. This is seen by noting that all points on the line L with the cocordinates (x*-fl,y’) where 320 must be in the budget set determined by the “original” (px,py;m), and , is, therefore inferior to (x*,y*) [which is supposed to be the best point in the “original” budget set]. If (x*- B, y’) is on L, px(x “‘43) + pyy’ =pxx*- pxfi + pyy’= pxx* - Pxfi'l' [m*-(px-A)(X*- fl)] = pxx“ - pxfi + [m-(x*)A- pxx* +(x*)A +p-XB -BA]= m-flASm The Hicksian substitution Effect The consumer, after-all, need not stick to the “01d” bundle (9/5, 8/5); it can choose any other bundle on the indifference curve going through that bundle and maintain the “old” level of satisfaction. Suppose that we ask the consumer to maintain the old level of satisfaction at the new prices. What will be the optimal choice ? We are really asking the consumer to solve; “minimize x +3y subject to x1/2y1/3 =u~” where u~ =l.569 Using the Lagrangian technique, from the first order conditions we get; 1.p[(1/2)x-1/2 yl/3] =0 3.“[(1/3)x1/2y-2/3] =0 xl/2yI/3 =u~; hence, from the first two equations (1/3) = (3/2)y/x, Or,x=9y/2; using the third: (9y/2)1/2y1/3=u~ - Or,y5/6 = u~[\/2/\/9] Or y= 0.6963 x=3.1333 The min expenditure is 5.2222 The income effect is a" move from (3.1333; 0.6963) to the optimal choice at new prices. A general result on the Hicksian substitution effect Suppose that at (px,py;m) the consumer chooses the bundle (x~,y~) from her budget set [draw the usual diagram]. Write u~ =u(x~,y~). With a well-behaved indifference curve we see that (x~,y~) is a commodity bundle that minimizes the cost of attaining u~ at (pmpy). Suppose that the price of It changes to px +Apx (if Apx is negative we have a fall in the price of x, if Apx is positive then we have a rise in the price of x). Let (x~ +Ax~, y+Ay~) be the commodity bundle that minimizes the cost of attaining u~ at the new prices (pX +Apx,py). Then the following inequalities hold: pxx~ + Pyy~ S px(x~ + AX~) + py(y~+Ay~) (px +Apx) (x~+ A x~) + Py (y~+ Ay~) s (px + Apar- + Py Y~ Hence, Aprxrv SO ...
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Income and Subst - Department of Economics 1 Cornell University M Majumdar Economics 3 1 3 Handout 0 Income and Substitution Effects The effect of

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