Consumer Choice With Certainty Part III: Utility Functions Agenda: I. Review of the key consumption relationship II. Utility functions – play dough economics III. Our FIRST numerical example – good luck! IV. The key relationship 2 Ways!

Putting it Together: The Optimal Bundle! Why not here? Tangent: where a line touches a curve at one point. The rate of change of the curve = the rate of change of the line Why not here? This is the KEY slide for Objective #2: First the intuition: consume at the highest utility you can! People prefer more to less. Then the math: Tangency occurs where the rate of change of the indifference curve = the rate of change of the budget line Marginal rate of substitution Slope of the budget line Check your understanding: Why is it Px/Py and not Py/Px?

Utility Functions Functional forms Utility is ordinal, not cardinal! A function is a mapping from X to one Y Functional forms Key point p. 88 in Frank Utility is ordinal not cardinal. P&R have some good practice problems that as you to graph indifference curves given a utility function. A lot depends on specific functional forms!! Note that convex preferences lead to concave utility functions Utility is ordinal, not cardinal! We can draw indifference curves from a utility function BUT we can NOT draw a utility function from indifference curves!

Indifference Curves & Utility Functions Budget constraint Utility curves There are really THREE things going on here – two goods and utility – three dimensional (N-dimensional with N goods!) Feasible budget is a plane, binding budget constraint is the top line of the plane. Be careful – ratio changes based on what measure you use for MRS. Opportunity cost notion – marginal utility of X is the change (what you’re willing to give up) in Y. Different notation Same meaning!

A little math…a lot of different interpretations! The ratio of marginal utilities equals the price ratio, or Marginal benefit = Marginal cost The marginal utility per dollar for each good must be equal Utility per dollar Opportunity cost The marginal rate of substitution equals the slope of the budget line The opportunity cost per dollar for each good must be equal

Putting it Together Again: Optimal Bundle Example You like coffee (Y-axis) and bagels (X-axis) with the marginal utility of coffee = 1/2 bagels and the marginal utility of bagels = 2 coffee. The price of coffee is $1 and the price of a bagel is $2 (with cream cheese). You have $6 to spend on breakfast and you can buy fractions. What is your optimal bundle? How are you feeling? KEY: Using marginal utilities allows us to more generally substitute into the budget constraint.

$1C + $2B = 6 REMEMBER the Process of Problem Solving! Step #1: Where do you need to go? What does the answer look like? A QUANTITY of Coffee AND a QUANTITY of Bagels (B,C) Step #2: What do you know? The budget & key relationship! $1C + $2B = 6 Step #3: Use what you know to get where you need to go! This is just math: two equations & two unknowns!

1. Cross multiply & simplify to express C in terms of B. 2. Substitute into the Budget Constraint! 3. Substitute the answer for B into the Budget Constraint. 2 suggestions: Use $ for prices so you keep your units clear! Check your answer by substituting your answer into the budget constraint!

Hint: The MRS must stay constant at 2 no matter what the prices are! What if you were ONLY willing to have coffee and bagels for breakfast at a ratio of one (1) bagel to two (2) cups of coffee? In other words, coffee and bagels are perfect complements to you! What is your optimal breakfast? Hint: The MRS must stay constant at 2 no matter what the prices are! What do these indifference curves look like? MRS is the rate at which we substitute Y (coffee) for X (bagels) Perfect complements and substitutes will be VERY important when we do Slutsky! Your answer MUST have a ratio of 2 coffee per bagel!

The Key Relationship THE INTUITION We consume to the point where the marginal cost (price) = the marginal benefit Next up… Slutsky!