kinked line with the kink at P = 50,000

1. Sally consumes two goods, X and Y. Her utility function is given by the expression U =  . The current market price for X is $10, while the market price for Y is $5.00. Sally's current income is $500.

1. 
   Sketch a set of two indifference curves for Sally in her consumption of X and Y (Use Y = 2 and Y = 3) (4)
   

To draw indifference curves, pick 2 levels of utility and find the values of x and y that hold the total utility constant:

Let U = 60

for Y = 2

60 =

60 =

x = 5 y = 2, x = 5

y = 3

60 =

x = 2.2 y = 3, x = 2.2

Let U = 72

for y = 2

72 =

72 =

x = 6 y = 2, x = 6

y = 3

72 =

x = 2.67 y=3, x=2.67

2. 
   Write the expression for Sally's budget constraint. Graph the budget constraint and determine its slope. (3)
   

I = P_xx + P_yy

500 = 10x + 5y

Slope =

3. 
   Determine the X,Y combination which maximizes Sally's utility, given her budget constraint. Show her optimum point on a graph. (Partial quantities are possible.) (Hint: Use MU) (8)
   

To maximize utility, Sally must find the point where

MRS is equal to .

recall:

MRS =

set MRS =

Y = 4X

To determine exact quantities, substitute Y = 4X into

I = P_XX + P_YY

500 = 10X + 5Y

500 = 10X + 5(4X)

500 = 30X

X = 16.67

Y = 4(16.67)

Y = 66.67

4. 
   Calculate the impact on Sally's optimum market basket of an increase in the price of X to 15. (5)
   

MRS remains

Equating MRS to

Substitute Y = 6X into the equation

500 = 15X + 5Y

500 = 15X +5(6X)

500 = 45X

X = 11.11

Y = 6(11.11)

Y = 66.67

2. Suppose that a consumer's increase in nominal income from the base year exceeds the inflation level given by a Laspeyres cost of living index for their level of purchases . Show that this information implies that the consumer is strictly better-off as compared to the base year. (HINT: Use a revealed preference argument and the assumption that total expenditure is equal to total income!) (7)

The information given above implies that This expression says that the commodity bundle purchased in the base period is affordable in the new period. However, the consumer selected a more expensive bundle. Thus, the new commodity bundle is revealed preferred to the base year commodity bundle.

3. The following data pertain to products A and B, both of which are purchased by Madame X. Initially, the prices of the products and quantities consumed are:

Madame X has $100 to spend per time period. After a reduction in price of B, the prices and quantities consumed are:

Assume that Madame X maximizes utility under both price conditions above. Also, note that if after the price reduction enough income were taken away from Madame X to put her back on the original indifference curve, she would consume this combination of A and B:

Determine the change in consumption rate of good B due to (1) the substitution effect and (2) the income effect. (3 + 3)

The total effect of the price change is the difference in the quantities before and after the price change, or 15 - 7 = 8. This change of 8 includes the income and substitution effects. The reduction in consumption that resulted from the reduction in income to put Madame X back on the original indifference curve represents the income effect. This difference is

15 - 9 = 6. The difference between 15 - 7 = 8 and

15 - 9 = 6 is the substitution effect, i.e. 8 - 6 = 2.