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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 incomeoffer 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*(pxA)(X* ﬂ)] = pxx“  pxﬁ + [m(x*)A pxx* +(x*)A +pXB BA]= mﬂASm The Hicksian substitution Effect The consumer, afterall, 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)x1/2 yl/3] =0
3.“[(1/3)x1/2y2/3] =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 wellbehaved 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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 '06
 MASSON
 Microeconomics, optimal choice

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