By: Brian Murphy
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Consumer has an income of $200 and wants
to buy two fixed goods: hats and guns.
Price of hats is $20 and price of shirts is $30.
Consumer wants to buy a certain number of
hats and shirts so that he spends all or nearly
all of his income while optimizing his
satisfaction.
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Key Variables:
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Income (I) = $200
Number of Hats Purchased (H)
Number of Shirts Purchased (S)
Price of Hat (PH) = $20
Price of Shirt (PS) = $30
Income Equation: HPH + SPS = 200
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In Microeconomics, consumer satisfaction is
mathematically represented by a utility
function.
Utility function is usually generated from
historical market trends.
Most common utility function is of the form
U=aXαYβ
For this problem, the consumer’s utility
function is U(H, S) = 2H1/2G1/2
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What we need to find optimum:
◦ Marginal Utility (MU) – the change in utility as a
result of a small change in quantity of one good
(calculated as the partial derivative of the utility
function with respect to the good).
◦ Marginal Rate of Substitution (MRS) – utility gain
from a small change in one good while the other
good is held fixed (Calculated as the ratio of the
two marginal utilities).
◦ Price ratio (PR) – ratio of the price of the two goods.
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Calculated Variables:
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MUH = H-1/2S1/2
MUS = H1/2S-1/2
MRSH,S = MUH/ MUS = S/H
PR = 20/30
Notes:
•Y-int = I/Py
•X-int = I/Px
•Slope = -Px/Py
S
Budget Line
20/3
Indifference Curve**
Optimal Bundle
0
10
H
**The Utility function projects outward in the third dimension in a bowl
shape. The indifference curve is simply a cross section of the utility
function.
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At Optimal Bundle:
◦ Nearly all the money is spent
◦ Slope of Indifference Curve = Slope of Budget Line
◦ Slope of Indifference Curve = -MUH/ MUS = -MRSH, S
Thus:
S/H = 20/30, H = 1.5S
Plug into Income Equation:
20(1.5s) + 30s = 200
S* = 3.33
H* = 1.5s* = 5