1.
Microeconomics HW1 Solution
a. Marvin loves pepperoni pizza and hates anchovy pizza.
b. Mavis hates anchovy pizza and is completely indifferent about pepperoni pizza.
2. Suppose that he has bundle C right now and prefers A to B, B to C, and C to A. If you offer him a trade
that leaves him at B instead of C, he will accept the deal. If you now offer him a trade that leaves him at A
instead of B, he will accept that. But he will prefer to be back where he originally was to where he is. So
you could offer to give him back his original bundle, minus a reward to you for your efforts, and he would
accept the deal.
3. a. U(x, y) = x + 2y
b. U(x, y) = min{x,2y}.
c. U(x, y) = max{x,2y}
4.
a. U(x, y) = min{2x + y,2y + x}
b. U(x, y) = max{2x + y,2y + x}
c. U(x, y) = x + min{x, y}
5. Max has the utility function U(x, y) = x(y + 1). The price of x is $2 and the price of y is $1.
Income is $10. How much x does Max demand? How much y? If his income doubles and prices
stay unchanged, will Max’s demand for both goods double?
To set his MRS equal to the price ratio, Max sets
. His budget constraint is 2x + y = 10.
Solve these two equations to find that
and
.
If his income doubles and prices stay unchanged, his demand for both goods does not double. A quick
way to see this is to note that if quantities of both goods doubled, the MRS would not stay the same and
hence would not equal the price ratio, which has stayed constant.
6. Max has a utility function U(x, y) = 2xy + 1. The prices of x and y are both $1 and Max has an
income of $20.
a. How much of each good will he demand?
b. A tax is placed on x so that x now costs Max $2 while his income and the price of y stay the
same. How much of good x does he now demand?
c. Would Max be as well off as he was before the tax if when the tax was imposed, his income
rose by an amount equal to $1 times the answer to part (b)?
a.
2y/2x=1---(1)
x+y=20---(2) x=10, y=10
b. y/x=2/1---(1)
2x+y=20---(2) - x=5
c. y/x=2/1 ----(1)
2x+y=20+5*1=25 y=25/3, x=25/6.
No. U(x, y)<2*10*10+1
7. Casper consumes cocoa and cheese. Cocoa is sold in an unusual way. There is only one
supplier, and the more cocoa you buy from him, the higher the price you have to pay per unit. In
fact y units of cocoa will cost Casper y2 dollars. Cheese is sold in the usual way at a price of 2
dollars per unit. Casper’s income is 20 dollars and his utility function is U(x, y) = x + 2y, where
x is his consumption of cheese and y is his consumption of cocoa.
a. Sketch Casper’s budget set and shade it in.
b Sketch some of his indifference curves and label the point that he chooses.
c. Calculate the amount of cheese and the amount of cocoa that Casper demands at these prices
and this income.
(This problem is different from those in the text and is designed to see whether you can use the tools
presented there in a creative way.)
2
a. Budget: 2x+y =20, including points: (18,1) (16,2) (11,3)(4,4)
|slope|=2/2y
b. The solution is a point of tangency.
c. Casper demands 2 units of cocoa and 16 units of cheese.
U(x, y) = x + 2y
|slope| of indifference curve=|MRS|=1/2
|slope| of budget curve=2/2y
1/2=2/2y ----(1)
2
2x+y =20----(2)
From (1) & (2), x=8, y==2