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By: Brian Murphy
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.
Key Variables: ◦ 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
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
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.
Calculated Variables:◦ MUH = H-1/2S1/2
◦ MUS = H1/2S-1/2
◦ MRSH,S = MUH/ MUS = S/H◦ PR = 20/30
H
S
010
20/3
Budget Line
Optimal Bundle
Indifference Curve**
Notes:•Y-int = I/Py
•X-int = I/Px
•Slope = -Px/Py
**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.
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 = 1.5SPlug into Income Equation: 20(1.5s) + 30s = 200S* = 3.33H* = 1.5s* = 5