Homework_4_solutions

Homework_4_solutions - Homework Assignment 4 1. Martin...

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Homework Assignment 4 1. Martin likes to consume wine(x) and beer (y). His preferences over the two commodities are represented by the following utility function; 1 3 4 4 ( , ) U x y x y = a) At home, Martin likes to distill his own wine and beer. Assume that Martin produces 4 bottles of wine and 2 kegs of beer. If Martin can buy and sell beer and wine at his local market for $10.00 and $5.00, respectively, then what is Martin's budget line? How much beer and wine will he consume? (Assume that he has no other income to allocate to his alcohol consumption). Include a well-labeled diagram in your answer. Tangency Condition: 3 3 4 4 1 1 4 4 1 4 3 4 1 5 3 10 W B B MRS W W B - - = = = . 2 3 B W = Feasibility Condition: 5 10 5 4 10 2 40 W B + = + = . 2 8 W B + = * * , . 2 3 W B = = b) If the price of a bottle of wine rises will Martin be better or worse off? In a diagram illustrate the income and substitution effects assuming that both goods are normal. Pw’ > Pw Substitution Effect Income Effect Total Effect Wine W W Ambiguous W B 4 8 (4, 2) (2, 3) W B 4 8 (2, 3) A B C
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Beer B B B c) If the price of a keg of beer rises will Martin be better or worse off? What is the income effect of a rise in beer prices? Illustrate an example of the effect of this price change on Martin's consumption of wine and beer. If the price of beer goes up, Martin will be worse off. He sells wine to buy more beer, so the relative price of beer rises means his real income decreases. The income effect of a rise in beer prices is negative. The new optimal bundle C could be located at the third quadrant of point B . The consumption of beer will certainly decrease because both the substitution and income effects are all negative. Pb’ > Pb Substitution Effect Income Effect Total Effect Wine W W Ambiguous Beer B B B 2. Bart purchases food (x) and lottery tickets (y). His preferences on units of food (x) and lottery tickets (y) can be represented by the utility function U(x,y) = min[2x,y]. a) If the price of a unit of food is P x , the price of a lottery ticket is P y and he has $I to spend then find his demand functions for food and lottery tickets. We want to solve the following optimization problem: [ ] I y P x P t s y x y x y x = + . . , 2 min max , Note that the tangency condition does not involve the MRS and price ratios because the preferences are given by a perfect complements function. The first order conditions are: “Tangency”: x y 2 = Feasibility: I y P x P y x = + W B 4 8 (2, 3) A B C
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Substitute the tangency condition into the feasibility condition: ( 29 y x y x y x y x P P I x I P P x I x P x P I y P x P 2 2 2 + = = + = + = + Now that we have x we can get y from: y x P P I y x y 2 2 2 + = = To summarize our results, the demand curves are: y x y x P P I y P P I x 2 2 2 + = + = b) Suppose that P x = $2, P y = $1 and I = $24. Using your answer to (a) find his demands for food and lottery tickets. Illustrate your answer in an indifference curve diagram on the
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Homework_4_solutions - Homework Assignment 4 1. Martin...

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