Unformatted text preview: onsumer a
Px . These non-negativity into cases where the be in force, but like to not written
P y run
2. the non-negativity constraints2 ; x, y
good to zero. negative amount of a good. Whenever this happens, we simply set her consumption of that 3.1 Interior Optimum 0 Most of the time we will have an interior optimum : the consumer’s optimal bundle
is positive in all goods. Figure 4 shows an interior optimum. At an interior optimum,
an indi↵erence curve will be tangent to the budget line. This means that, at the optimal
bundle, the slope of the budget line ( PX ) will be equal to the slope of the indi↵erence curve
( M RSXY ):
= M RSXY .
PY Most of the time we will have an interior optimum: the consumer’s optimal bundle is positive in all goods.
This expression has very strong economic intuition. The ratio of prices is the rate at which
If the indifference curve is differentiable, it will befor good Y in theto the budget line at an interior optimum. Figure
the consumer can exchange good X tangent market place. The marginal rate
and Y is
the consumer would
3 illustrates an interior optimum. substitution between Xthe optimal the rateisat whichfor which the consumerbeis willing to
exchange X for Y . Thus,
bundle the one
exchange X for Y at precisely the rate the market is willing to exchange X for Y . ! "#$%$#$&'$()*#$'+*,& 7 -.+*/01('2,*'$(3*&+$#*,#(4,15+*,&6
!" Figure 4: An interior optimum. Figure 3: An Interior Optimum
It is important that you understand why this must be the case. Consider the following 1 We assume that the goods market is competitive so that the consumer takes the prices as given.
non-negativity constraints will always be in force, but usually not written explicitly. We will seldom run into cases where
the consumer would like to consumer a negative amount of a good.
2 These 3 x
At the interior optimum, the slope of the budget line Py will be equal to the slope of the indifference
curve, the negative marginal rate of substitution M RSxy : this means Px
Py (6) This has a strong economic intuition. The ratio of prices is the rate at which the consumer can exchange
good x for good y in the market place. The marginal rate of substitution between x and y is the “subjective
rate” at which the consumer would be willing to exchange x for y. The optimal bundle is the one for which
the consumer is willing to exchange x for y at precisely the rate the market is willing to exchange x for y.
Example 1: Cobb-Douglas
example. Let U be Y ) case, I = consider 50 following example with a Cobb-Douglas utility
To understand why this must (X, the= XY , let’s 1000, PX =the and PY = 200. Are either of the two
baskets optimal: (A) (X, Y ) = (4, 4) or (B) (X, P) I (10, ( 5)? Note that ). price ratio 4
function U ( x, y) = xy and the exogenous variables ( Px , Y y ,= ) = 2.50, 200, 1000theIn Figure is below, are either
. Since x, are dealing with B)...
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