Temple University

Department of Economics

Intermediate Microeconomic Theory

Consumer Choice

Name Key

1. Gore Monde is a food lover who lives on bread and wine. Let his marginal utility of bread, MU_{b}, be given algebraically by the expression MU_{b} = 40 - 5b. Similarly, let his marginal utility of wine be MU_{w} = 30 - w. The current prices of bread and wine are Pb = 5 and Pw = 1, respectively. Gore's income is 40.

a. What is the greatest quantity of bread that Gore can consume? b = 40/5 = 8

b. What is the greatest quantity of wine that Gore can consume? w= 40/1 = 40

c. What is the slope of Gore's budget constraint when you put wine on the horizontal axis and bread on the vertical axis? slope = -8/40 = -1/5

d. Given the prices and Gore's income, how much bread and wine should he consume? Bread = , Wine =

From the hint we know MRS_{wb} = MU_{w}/MU_{b} = P_{w}/P_{b} . Or, (30-w)/(40-5b) = 1/5. This can be rearranged as w = 22 + b. Whatever w,b choice makes the hint true must be a choice that also satisfies Gore's budget constraint. Therefore 40 = 5b+(22+b). Solving for b we get b = 3 and w = 25.

e. Suppose that Gore's income falls to 10. How much bread and wine will he consume now? Bread = , Wine = When we use the same reasoning we find 10 = 6b+22, or b = -2. This is not possible, so it must be the case that Gore consumes no bread at all, and therefore spends all of his money on wine; w = 10.

2. Al Fabette consumes x's and y's. With y on the vertical and x on the horizontal, his marginal rate of substitution in consumption is given by MRS = y/x. At a shop on Sesame Street he can buy his x's and y's at the prices P_{x} = 5 and P_{y} = 1.

a. Fill in the boxes below to create Al's income expansion path.

From the hint we know that MRS_{xy} = MU_{x}/MU_{y} = P_{x}/P_{y } . Making the appropriate substitutions we can write y/x = 5/1. This must be true regardless of income, therefore y = 5 x + 0 is the income consumption path.

intercept

y =

x

b. Suppose that the price of a unit of x falls to P_{x}=4. Then Al's income expansion path will become

flatter steeper no change

3. Put y on the vertical and x on the horizontal. Let Pepe Smith have the marginal rate of substitution MRS = y. His income is 120 and the price of y is P_{y} = 1.

a. In the box below write an expression for Pepe's price expansion path:

Again, the hint is MRS_{xy} = MU_{x}/MU_{y} = P_{x}/P_{y}. This time, upon substitution y = P_{x}/1; this must always be true for every tangency between budget constraint and indifference curve. Whatever choice of x and y makes this true must also be an x,y pair that lies on Pepe's budget constraint. His budget is 120 = P_{x}X + P_{y}Y = P_{x}X + Y. Now substitute in for P_{x} to get 120 = YX + Y. The x,y pairs that lie on this line are common to both the MRS_{xy} = P_{x}/P_{y} rule and the budget constraint. Rearranging gives us the price expansion path y = 120/(x+1)

b. In the box below write an expression for Pepe's demand curve associated with the price expansion path you found in part a of the question. Write your expression in the box below:

The demand curve shows Pepe's reservation price as a function of quantity. Since we know that y = Px, it must be the case that P_{x} = 120/(x+1) is Pepe's demand curve.

Hints: MRS_{xy} = MU_{x}/MU_{y} = P_{x}/P_{y}