Unformatted text preview: ( x, y ) = (10, function, M RS Note For the price
x
4
of the two baskets ( A) (we y) = (4, 4) ora(CobbDouglas utility2.5) optimal?XY = X . thatbasket (A), ratio is Py = 1 .
P
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M RSXY = 1. For basket (B), M RSXY = 2 .5 = 1/4. Therefore (B)y is the optimal basket. 10
Since we are dealing with a CobbDouglas utility function, MRSxy = x . For basket ( A), MRSxy = 1. For
Notice that at (A), the market says the consumer must give up one unit of Y to get four
1
basket (B), MRSxy = 2.5, but she Therefore give up one unit of Xto be the unit of Y . She can therefore that at ( A), the
unit of 10 = 4 . is willing to ( B) is supposed to get one optimal basket. Notice
X
market says the consumer must by trading one unit of yon the market. unit ofa x, but understanding to give up one
be made better o↵ give up some Y for X to get four To get deeper she is willing
unit of x to get onethis, study .Figure can therefore be made better off by trading some y for x on the market.
unit of y She 5.
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$ )# !" Figure 5: An interior optimum example.
Figure 4: An Interior Optimum for a CobbDouglas Utility Function U ( x, y) = xy
As an exercise, pretend you don’t know that (X, Y ) = (10, 2.5) is the optimal bundle and
use equation (1) and
Example 2: Perfect Complementsthe budget constraint to derive it. Sometimes the tangency condition will not hold at the optimal bundle because the indifference curve is
2.1 the marginal rate
not differentiable and Corner Points of substitution does not exist for the optimal bundle. Recall that
the utility functioncorner solutioncomplements, U ( xbundle in which at By}, one good undeﬁned marginal rate of
A for perfect describes an optimal , y ) = min{ Ax, least has an is not being
substitution at Ax = By. The tangency condition (1) may not hold at that the optimal bundle must lie on the point
consumed. With a little thought, one can see a corner solution. The usual method
Ax = By. At anyof solving above this point, the consumer is consuming too much (1:) she can trade y for x and
bundle the consumer’s problem is to check whether the tangency condition y holds for
hop onto a higher indifference curve. The opposite applies for any bundle 0, use the budget Thus, to solve
any positive X and Y . If no such point exists, check the corners: set X = below the line.
the consumer’s problem when she is choosing between function; complements, use Ax =see and the budget
line to solve for Y an...
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 Summer '13
 Microeconomics, py, Corner Solutions

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