Objectives: Use the utility-maximizing model to explain how consumers choose goods and services.

1. Objectives: Use the utility-maximizing model to explain how consumers choose goods and services. Use the concept of utility to explain how the law of demand results from consumers adjusting their consumption choices to changes in prices. Module 11: The Utility-Maximizing Model

2. Objective 1 Use the utility-maximizing model to explain how consumers choose goods and services. • Economists assume that consumers try to allocate their limited incomes to maximize their satisfaction, a goal referred to as utility maximization. • The utility-maximizing model is used to determine the optimal amounts of goods and services a consumer will purchase given: • knowledge of consumer’s preferences • the prices of the goods and services. • the consumer’s budget constraint • The consumer’s equilibrium bundleis a combination of goods and services consumed which gives the consumer the maximum total utility, subject to a budget constraint or an income constraint.

3. Objective 1: Using the utility-maximizing model • Suppose a consumer has $I to spend on two goods, X and Y. Let Px = price of good X and Py = price of good Y. • How will the consumer allocate her $I towards these two goods so that she gets the most satisfaction? • The equilibrium bundle satisfies two conditions: Condition 1: Income should be allocated so that the last dollar spent on each good yields the same amount of marginal utility. In terms of an equation, where MU = marginal utility, and P = price,

4. Objective 1: Using the utility-maximizing model ...an example • The table below shows Kayla's utility from soup and sandwiches. • The price of a cup of soup is $2 and the price of a sandwich is $3. • Kayla has $18 to spend on these two goods.

5. Objective 1: Using the utility-maximizing model ... Condition 2: The consumer must spend the total income allocated to the consumption of goods and services. In terms of an equation: Where = Price of good X ×Quantity of good X = expenditure on good X, and = Price of good Y × Quantity of good Y = expenditure on good Y 5

6. Objective 1: Applying the utility-maximizing conditions Step 1:Calculate the marginal utility per dollar spent on each good using the formula: marginal utility per dollar = marginal utility ÷ price of the good *If you are given total utility figures, you will have to calculate the marginal utility before using the equation above. For example, see columns 3 and 7. 6

7. Objective 1: Applying the utility-maximizing conditions Step 2:Identify the combinations of the goods that satisfy the marginal utility per dollar rule:

8. Objective 1: Applying the utility-maximizing conditions • The marginal utility per dollar rule holds for these three combinations: • 2 cups of soup and 2 sandwiches • 3 cups of soup and 4 sandwiches • 4 cups of soup and 5 sandwiches 8

9. Objective 1: Applying the utility-maximizing conditions Which of the three bundles is the optimal bundle? Step 3: Identify the optimal bundle by applying the condition 2: that Kayla’s expenditure on the two goods must exhaust her budget of $18. 2 cups of soup and 2 sandwiches will cost her $10 ($2×2 cups of soup + $3×2 sandwiches) 3 cups of soup and 4 sandwiches will cost her $18 ($2×3 cups of soup + $3×4 sandwiches) 4 cups of soup and 5 sandwiches will cost her $23 ($2×4 cups of soup + $3×5 sandwiches) Kayla’s equilibrium bundle is 3 cups of soup and 4 sandwiches. 9

10. Objective 1: ..more on the utility-maximizing model.. • The utility maximizing model applies a key economic principle: optimal decisions are made at the margin. • Examine the marginal utility per dollar rule again: Rearrange to get ratio of marginal utilities also called the marginal rate of substitution ratio of prices is also called relative prices • In equilibrium, the consumer’s personal rate of exchange equals the rate of exchange required by the market.

11. Objective 2 Use the utility maximizing model to derive a demand curve • To derive Kayla’s demand for sandwiches curve, we must change the price of sandwiches and observe what happens to her quantity demanded of sandwiches, holding all else constant. • We already have one price-quantity combination: At a price of $3, Kayla’s optimal quantity was 4 sandwiches.

12. Objective 2: ….deriving a demand curve. • To construct a demand curve we need at least one other price-quantity combination. • Suppose the price of sandwiches rises to $4.00. How would Kayla’s quantity demanded of sandwiches change? • Since the price of one good has changed we have to recalculate the marginal utility per dollar for that good.

13. Objective 2: ….deriving a demand curve. Step 1:Calculate the marginal utility per dollar spent on each good using the formula: marginal utility per dollar = marginal utility ÷ price of the good 13

14. Objective 2: ….deriving a demand curve • How to determine the optimal combination of soup and sandwiches in the case where the rule of equal marginal utility per dollar does not hold? • Apply the principle of marginal analysis. Ask the question: what is the first item Kayla should buy? 14

15. Objective 2: ….deriving a demand curve • Obviously, the item that gives her the highest marginal utility per dollar spent. What is the 2nd item…and so forth until her budget is exhausted. • Her utility maximizing bundle is 3 cups of soup and 3 sandwiches. 15

16. Objective 2: ….deriving a demand curve. • Now that we have two price-quantity combination points on Kayla’s demand for sandwiches curve we can trace her demand curve. • The resulting demand curve is downward-sloping. It obeys the law of demand.

17. Some key points: • The utility-maximizing choices lead to a demand curve. • Each price-quantity combination on a demand curve is a utility-maximizing quantity, given the price. • If people seek to maximize utility, then the law of demand follows. • When price changes, there a substitution effect and an income effect on the quantity of sandwiches demanded.

18. Objective 2: ….how a consumer adjusts to a price change • Initially, when the price of soup = $2 and the price of sandwich = $3, Kayla’s equilibrium bundle was 3 cups of soup and 4 sandwiches. • The ratio of marginal utility to price was the same for soup and for sandwiches. • When a consumer is in equilibrium, she is maximizing utility.

19. Objective 2: ….how a consumer adjusts to a price change • When the price of sandwiches rises to $4, the ratios of MU to price no longer hold with equality. We now have: A dollar spent on soup gives Kayla more utility than a dollar spent on sandwiches 19

20. Objective 2: ….how a consumer adjusts to a price change A dollar spent on soup gives Kayla more utility than a dollar spent on sandwiches • To restore equilibrium, Kayla buys more soup and fewer sandwiches, subject to her budget constraint. • In my example, given her budget, Kayla buys fewer sandwiches but is not able to increase her soup consumption. • Note that when the price of sandwiches rises, quantity demanded falls – a result consistent with the law of demand. 20