Consumer Choice Under Certainty Part II Agenda: Review of the key consumption relationship Our FIRST numerical example! Utility functions – play dough economics So where DO demand curves come from? Demand vs. quantity demanded Whew…That’s it!

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?

Putting it Together: The Optimal Bundle Numerical Example You love coffee and bagels for breakfast. Currently you’re willing to give up two cups of coffee for one bagel (with cream cheese). The price of coffee is $1 and the price of a bagel is $2. You have $6 to spend on breakfast and you can buy fractions. 1. What is the marginal rate of substitution? A: 2 coffee/1 bagel 2. What is the budget constraint? A: 1C + 2B = 6 Test yourself: draw the budget line – what are the X and Y intercepts and what do they mean? NOTE: Take the time here to draw the indifference curves for complements and substitutes! 3. What your optimal breakfast?

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? 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! NOTE: This is an easy case because the goods are perfect complements. More general using Marginal Utilities that change with quantity. 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!

TEST YOURSELF! What if you were ONLY willing to have coffee and bagels for breakfast at a ration of one (1) bagel to two (2) cups of coffee? In other words, coffee and bagels are perfect complements to you! Hint: The MRS must stay constant at 2. What do these indifference curves look like? Answer: 1.5 Bagels and 3 cups of coffee NOTE: optimal bundle MUST have a ratio of 2:1!

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

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 revisited You like coffee and bagels 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. You have $6 to spend on breakfast and you can buy fractions. What is your optimal bundle? What is the key relationship that defines the optimal bundle? KEY: Using marginal utilities allows us to more generally substitute into the budget constraint. How can you check that this is correct?

Pay attention to what is on the X and Y axes! So Where DO Demand Curves Come From? composite good Indifference Curves Budget Constraints Price-consumption curve quantity Price What is the price of shelter at this budget constraint? KEY: Pay attention to what is on the X and Y axes! What’s changing in the top graph? – only the price of X Hammer skill of looking at what is on the X and Y axis of a graph. This shows the CHANGE IN QUANTITY DEMANDED for a given change in PRICE

What is Elasticity? Point elasticity Arc elasticity quantity Price Point elasticity Change Mid-point Arc elasticity Key things to remember about elasticity: Elasticity is NOT CONSTANT along the demand curve. If quantity does not change, demand is perfectly ELASTIC. If price does not change, demand is perfectly INELASTIC. Percent change in quantity for a percent change in price Arc is used when there is no known functional form for demand – can’t use calculus to get instantaneous change at a point. Emphasize that the only thing that causes a shift in the QUANTITY DEMANDED is PRICE. Everything else is a shift in DEMAND – the whole line. TEST YOURSELF: 1)Compute the arc elasticities between each of the points in the graph above. 2) Draw demand curves for points 2 & 3.

What would cause a shift in Demand? A change in income Put it together – MRS = slope of the budget line A change in preferences (shape of the indifference curves) A change in population (aggregate demand) Check your understanding: Write some examples in your notes of shifts in demand. Try to draw the graphs too!

Yikes! That was a lot of stuff! The key relationship: The intuition: We consume to the point where the marginal cost (price) = the marginal benefit PRACTICE!!