Suppose Juan’s preferences over goods X and Y can be expressed by the utility fu
ID: 1169774 • Letter: S
Question
Suppose Juan’s preferences over goods X and Y can be expressed by the utility function U (X, Y ) = 2X + Y , while Charlie’s preferences can be expressed by the utility function U = min{3X, 2Y }. Suppose that Juan has 2 units of X and 2 units of Y, while Charlie has 3 units of X and 4 units of Y. What is the name for preferences like Juan’s? What about Charlie’s? What is the marginal utility of X for Juan given his current holdings? What is the marginal utility of X for Charlie given his current holdings? Will Juan and Charlie be willing to trade? If yes, give an example of an accept- able trade. If no, explain why not.
Explanation / Answer
The Juan's preferences are similar to that of perfect substitutes as the utility function is linear and he is consuming same amount of both units and will replace one another in case there is price change.
The Charlie's preferences may be called as preferences of perfect complements as the function says minimum of the utility derived from 3 units of X and 2 units of Y. Perfect complements are those goods which cannot be used without one another. Even if Charlie is consuming more of one good without consuming more of another the utility will not go up.
The Marginal Utility of Juan for given is current holdings will be
d(U(X,Y))/dX = d(2X + Y)/dX = 2
The Marginal Utility of Juan for given is current holdings will be
d(U(X,Y))/dX = d(min{3X, 2Y })/dX = 0
Yes, Juan and Charlie will be willing to trade and benefit from trade.
Charlie has 3 units of X and 4 units of Y and his utility function is U= min{3X, 2Y}, if we calculate the utility from Y, it will be 2*4=8
Then after 8/3 units of X no additional satisfaction to Charlie.
If Charlie trades 1/6 units of X with with Juan and takes 1/4 units of Y from Y (Charlie is left with 3-1/6=17/6 units of X and 4+1/4=17/4 of Y).
The satisfaction of Charlie will be U= min {3(17/6),2(17/4)} = min {17/2,17/2} = 8.5
Similarly Juan will also gain from trade.
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