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# Optimalxy x thatbasket a ratio is py 1 p 4 m rsxy

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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(Cobb-Douglas utility2.5) optimal?XY = X . thatbasket (A), ratio is Py = 1 . P 4 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 Cobb-Douglas 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. of !"#\$%&'( )*+',-.,/0.*12\$',/3%+-\$2\$ ! 4+/45/+6'/-*7-8-72#&/-1/9-&&-*:/+./ +,#7'/.*'/2*-+/.;/</;.,/.*'/2*-+/ .;/=>//);/[email protected]/ABC/#*7/?< @/ADCC5/ +6'/-*7-8-72#&/E#*/F'/\$#7'/F'++',/ .;;/FG/+,#7-*:/.*'/2*-+/.;/< ;.,/H/2*-+1/.;/=>/I./\$#"-\$-J' 2+-&-+G5/+6'/-*7-8-72#&/9-&&/ E.*+-*2'/+./'"E6#*:'/</;.,/=/ 2*+-&/KLM/-1/'N2#&/+./+6'/,#+' #+/96-E6/:..71/E#*/F'/+,#7'7 ;.,/.*'/#*.+6',/-*/+6'/\$#,O'+/P#+ QR>/ "#\$%&%'##!%(%)### *(%)+%(%\$! . ! " # # , '-" . *%(%'"%(%\$! \$ )# !" Figure 5: An interior optimum example. Figure 4: An Interior Optimum for a Cobb-Douglas 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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