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. 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. 1 Module 11: The Utility-Maximizing Model

2. 2 Objective 1 Use the utility-maximizing model to explain how consumers choose goods and services. Objective 1 Use the utility-maximizing model to explain how consumers choose goods and services. utility maximization.  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: preferences  knowledge of consumer’s preferences prices  the prices of the goods and services. budget  the consumer’s budget constraint equilibrium bundle  The consumer’s equilibrium bundle is 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. 3  Suppose a consumer has $I to spend on two goods, X and Y. Let P x = price of good X and P y = 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, Objective 1: Using the utility-maximizing model

4. 4  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. Cups of Soup Total Utility Number of Sandwiches Total Utility 1 40145 2 60275 3 723102 4 824120 5 885135 6 906145 Objective 1: Using the utility-maximizing model...an example

5. 5 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 Objective 1: Using the utility-maximizing model...

6. 6 1 40 40÷2=20145 45÷3=15 2 602020÷2=102753030÷3=10 3 721263102279 4 821054120186 5 88635135155 6 90216145103.33 Step 1: 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. (1) Cups of soup (2) Total Utility (3) Marginal Utility (4) Marginal Utility per dollar MU/Psoup (5) Number of Sandwiches (6) Total Utility (7) Marginal Utility (8) Marginal Utility per dollar MU/Psandwich Objective 1: Applying the utility-maximizing conditions Price of Soup = $2 per cupPrice of Sandwich = $3 per cup

7. 7 Step 2: Step 2: Identify the combinations of the goods that satisfy the marginal utility per dollar rule: Objective 1: Applying the utility-maximizing conditions (1) Cups of soup (4) Marginal Utility per dollar MU/Psoup (5) Number of Sandwiches (8) Marginal Utility per dollar MU/Psandwich 1 40÷2=20145÷3=15 2 20÷2=10230÷3=10 3 639 4 546 5 355 6163.33

8. 8  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 Objective 1: Applying the utility-maximizing conditions (1) Cups of soup (4) Marginal Utility per dollar MU/Psoup (5) Number of Sandwiches (8) Marginal Utility per dollar MU/Psandwich 1 40÷2=20145÷3=15 2 20÷2=10230÷3=10 3 639 4 546 5 355 6163.33

9. 9 Which of the three bundles is the optimal bundle? Step 3: 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) equilibrium bundle Kayla’s equilibrium bundle is 3 cups of soup and 4 sandwiches. Objective 1: Applying the utility-maximizing conditions

10. 10  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 prices is also called relative prices ratio of marginal utilities also called the marginal rate of substitution  In equilibrium, the consumer’s personal rate of exchange equals the rate of exchange required by the market. Objective 1:..more on the utility-maximizing model..

11. 11 Objective 2 Use the utility maximizing model to derive a demand curve 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. 12  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. Objective 2: ….deriving a demand curve.

13. 13 Step 1: 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 Price of Soup = $2 per cupPrice of Sandwich = $4 per cup (1) Cups of soup (2) Total Utility (3) Marginal Utility (4) Marginal Utility per dollar MU/Psoup (5) Number of Sandwiches (6) Total Utility (7) Marginal Utility (8) Marginal Utility per dollar MU/Psandwich 1 40 40÷2=20145 45÷4=10.8 2 602020÷2=102753030÷4=7.5 3 721263102276.75 4 821054120184.50 5 88635135153.75 690216145102.5 Objective 2: ….deriving a demand curve.

14. 14  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? (1) Cups of soup (4) Marginal Utility per dollar MU/Psoup (5) Number of Sandwiches (8) Marginal Utility per dollar MU/Psandwich 1 40÷2=20145÷4=10.8 2 20÷2=10230÷4=7.5 3 636.75 4 544.50 5 353.75 6162.5 Objective 2: ….deriving a demand curve

15. 15  Obviously, the item that gives her the highest marginal utility per dollar spent. What is the 2 nd item…and so forth until her budget is exhausted. Goods consumed Budget = $18 1 st cup of soup$18-$2 =$16 1 st sandwich$16-$4 =$12 2 nd cup of soup$12-$2 =$10 2 nd sandwich$10-$4=$6 3 rd sandwich$6-$4 =$2 3 rd cup of soup$2-$2 =$0 utility maximizing bundle  Her utility maximizing bundle is 3 cups of soup and 3 sandwiches. (1) Cups of soup (4) Marginal Utility per dollar MU/Psoup (5) Number of Sandwiches (8) Marginal Utility per dollar MU/Psandwich 1 40÷2 = 20145÷4 = 10.8 2 20÷2 = 10230÷4 = 7.5 3 636.75 4 544.50 5 353.75 6162.5 Objective 2: ….deriving a demand curve

16. 16  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. Objective 2: ….deriving a demand curve.

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

18. 18  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. Objective 2: ….how a consumer adjusts to a price change

19. 19  When the price of sandwiches rises to $4, the ratios of MU to price no longer hold with equality. We now have: 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

20. 20 equilibrium,  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. law of demand.  Note that when the price of sandwiches rises, quantity demanded falls – a result consistent with the law of demand. 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