1. By: Brian Murphy

2.  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.

3.  Key Variables: ◦ Income (I) = $200 ◦ Number of Hats Purchased (H) ◦ Number of Shirts Purchased (S) ◦ Price of Hat (P H ) = $20 ◦ Price of Shirt (P S ) = $30 ◦ Income Equation: HP H + SP S = 200

4.  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=aX α Y β  For this problem, the consumer’s utility function is U(H, S) = 2H 1/2 G 1/2

5.  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.

6.  Calculated Variables: ◦ MU H = H -1/2 S 1/2 ◦ MU S = H 1/2 S -1/2 ◦ MRS H,S = MU H / MU S = S/H ◦ PR = 20/30

7. H S 0 10 20/3 Budget Line Optimal Bundle Indifference Curve** Notes: Y-int = I/P y X-int = I/P x Slope = -P x /P y **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.

8.  At Optimal Bundle: ◦ Nearly all the money is spent ◦ Slope of Indifference Curve = Slope of Budget Line ◦ Slope of Indifference Curve = -MU H / MU S = -MRS H, 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