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