Case Study - Consumer Theory
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Section 1 - Preferences and Utility: Make a table containing utility values for each house (bundle) that are consistent with these preferences.
Preferences and utility:
Combining the given conditions, we get C < A < D ~ B < F ~ E
Table:
|
S. NO.
|
House
|
Utility
|
|
1
|
E,F
|
F(30,6)=f(20,7)
|
|
2
|
B,D
|
F(25,3) = f(13,6)
|
|
3
|
A
|
F(10,5)
|
|
4
|
C
|
F(20,2)
|
Utility function = f(x,y)
Section 2 - Indifference Curves
a) Create a graph with Size on the x-axis and Amenities on the y-axis. Plot a point on the graph for each of the houses in the table above. Draw 4 indifference curves that are consistent with the preferences (and utilities) identified in Section 1.
b) In general, do we expect the marginal utility of size to be higher for house D or for house E? Explain.
Answer - Marginal utility of size is greater for D since the slope is higher at D as compared to E.
c) Is the marginal rate of substitution higher for house B or for house D? Is the marginal rate of substitution higher for house E or for house F? Explain.
Answer - Rate of substitution is greater for D since the slope is higher at D as compared to B.
Rate of substitution is greater for F since the slope is higher at F as compared to E.
Slope = Marginal Rate of substitution
Section 3 - Consumer Maximization
Budget: 25000 x + 10000 y = B
Equations:
25000 y + 10000 x = 200,000 => Points (20,0) and (0,8)
25000 y + 10000 x = 400,000 => Points (40,0) and (0,16)
25000 y + 10000 x = 600,000 => Points (60,0) and (0,24)
a) Create a graph with Size on the x-axis and Amenities on the y-axis. Amenities cost $25,000 and Size costs $10,000. Draw the budget curve for a consumer with a budget of $200,000. Also draw the budget curves for a consumer with a budget of $400,000 and a consumer with a budget of $600,000. When drawing the curves be sure to indicate a couple points on each curve.
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b) Now take the graph with your indifference curves from section 2 and add in a couple budget curves. You can put the budget curves in any of several places, but they should meet a couple constraints. The budget curves should be parallel because they are for constant prices. Each budget curve should be tangent to a single indifference curve. The resulting optimal combination of features (amenities and size) should be marked. This point need not correspond to one of the 6 houses in the original table.
Section 4 - Indifference Curves and Demand
S= size, A = Amenities
Utility function = U(S,A) = S½ A½
a) Take the derivative with respect to Size (S) to find the general form of the consumer's marginal utility of Size (S).
Answer - General form of consumer's marginal utility of size, dU/dS = A½/ 2S½.
b) Take the derivative with respect to Amenities (A) to find the general form of the consumer's marginal utility of Amenities (A).
Answer - General form of consumer's marginal utility of amenities, dU/dA = S½/2A½.
c) Use the marginal utilities to find the marginal rate of substitution (MRS).
Answer - MRS , Marginal Rate of substitution = Marginal utility of size/ Marginal utility of amenities
= dU/dS / dU/dA = A/S
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d) Suppose the consumer has an income (M) of 10. Assume the price of Amenities (pA) is 2.5 and the price of Size (pS) is 1. Write the consumer's budget equation by substituting the prices and income into the general budget equation: M = pSS + pAA.
Answer - Consumer's Budget Equation: M = pSS + pAA
10 = 1 x S + 2.5 x A
S + 2.5 A = 10
e) What amounts of size and amenities will the consumer choose?
S + 2.5 A = 10
MRS = Price Ratio
A/S = 2.5 / 1 = 2.5
A= 2.5 S
S + 2.5 x 2.5 S = 10
S = 10/7.25 = 1.38
A = 2.5 S = 2.5 x 1.38 = 3.45
Hence Size = 1.38 and Amenities = 3.45 must be chosen.
f) Finally, find the aggregate demand curve for the two consumers for prices of 1, 2, 3, and 4.
Aggregate Demand Curve:
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