B 1 at the optimum we 6 4 are supposed to have x y

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Unformatted text preview: d plug this into the utility perfect do the same with Y = 0 and By line to solve for the two utility is greater. which is goods. For example, let U ( x, y) = min{ x, y}, ( Px , Py , I ) = (50, 200, 1000), and A = B = 1. At the optimum, we 6 4 are supposed to have x = y. We can plug this into the budget line: 50x + 200y = 1000 ) ) ) ) 50x + 200x = 1000 250x = 1000 x=4 y=4 So the optimal bundle is ( x, y) = (4, 4). See Figure 5 to see this graphically. !"#$%&'( )*+',-.,/0.&1+-.*/2-+3/4',5'6+ 7.$%&'$'*+8 9:; 9 )7< =%+-$#& >1*?&' )7@ )7B )7A C < ; < BD Figure 5: An Example with Perfect Complements. !" Figure 7: An example with perfect complements. 3.2 Corner Solutions A corner solution describes an optimal bundle in which at least one good is not being consumed. The tangency condition (6) may not hold at a corner solution. The usual method of solving the consumer’s problem is to check whether the tangency condition (6) holds for any positive x and y. If no such point exists, check the corners: set x = 0, use the budget line to solve for y and plug this into the utility function; do the same with y = 0 and see which is utility is greater. Example: Perfect Substitutes Let U ( x, y) = x + y and ( Px , Py , I ) = (50, 200, 1000). The price ratio is bundles ( x, y). It is clear that the tangency condition cannot hold. 9 Px Py = 1 4 and the MRSxy = 1 for all • Lets check the corner solution for x = 0. The consumer spends all her money on y so y = 5 so she gets utility U (0, 5) = 5. • If y = 0, then x = 1000 50 I Py = 1000 200 = = 20 and U (20, 0) = 20. Therefore the consumer’s optimal bundle is (20, 0). This method is a brute force method of solving for the optimal bundle. Using a bit of economic intuition can sometimes help you skip to the end. You may have notice that both x and y have marginal utility of 1 but y costs 4 times as much as x. No rational consumer would buy any y. To see this, imagine two gas stations right next to each other, one offers gas at $1 per litre and one offers gas at $4 per litre. Which station would you choose? More formally, good x gives 1/50 units of marginal utility per dollar while good y gives only 1/200 units of marginal utility per dollar. This indicates that good x is the better buy. See P. 112 in your text to see how this measurement (marginal utility per dollar) relates to the tangency condition. A third method of looking for corner solutions is to draw the budget line and indifference curve. See Figure 6 for this. Recall that as the indifference curves move out from the origin they represent higher and higher utility. 5 1/200 units of marginal utility per dollar. This indicates that good X is the better buy. See page 112 in your text to see how this measurement (marginal utility per dollar) relates to the tangency condition. A third method of looking for corner solutions is to draw the budget line and indi↵erence curve. See Figure 6 for this. Recall that as the indi...
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