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8 CHAPTER 1 ANALYZING ECONOMIC PROBLEMS

LEARNING-BY-DOING EXERCISE 1.2
S
E D
Constrained Optimization: Consumer Choice
Suppose a consumer purchases only two (b) The constraint represents the amounts of food and
types of goods, food and clothing. The consumer has to clothing that she may choose while living within her in-
decide how many units of each good to purchase each come. If she buys F units of food at a price of P F per unit,
month. Let F be the number of units of food that she her total expenditure on food will be (P F )(F). If she buys
purchases each month, and C the number of units of cloth- C units of clothing at a price of P C per unit, her total ex-
ing. She wants to maximize her satisfaction with the two penditure on clothing will be (P C )(C). Therefore, her
goods. Suppose the consumer’s level of satisfaction when total expenditure will be (P F )(F) (P C )(C). Since her
she purchases F units of food and C units of clothing is total expenditure must not exceed her total income I, the
measured by the product FC, but she can purchase only constraint is (P F )(F) (P C )(C) I.
limited amounts of goods per month because she must live
within her budget. Goods cost money, and the consumer (c) The exogenous variables are the ones the consumer
takes as given when she makes her purchasing deci-
has a limited income. To keep the example simple, suppose
the consumer has a fixed monthly income I, and she must sions. Since her monthly income is fixed, I is exoge-
nous. The prices of food P F and clothing P C are also
not spend more than I during the month. Each unit of
food costs P F and each unit of clothing costs P C . exogenous, since she cannot control these prices. The
consumer’s only choices are the amounts of food and
Problem clothing to buy; hence, F and C are the endogenous
variables.
(a) What is the objective function for this problem?
(d) The statement of the constrained optimization prob-
(b) What is the constraint?
lem is
(c) Which variables (P F , F, P C , C, and I) are exogenous?
max FC
Which are endogenous? Explain.
(F,C)
(d) Write a statement of the constrained optimization subject to: (P F )(F) (P C )(C) I
problem.
The first line shows that the consumer wants to maximize
Solution
FC and that she can choose F and C. The second line de-
(a) The objective function is the relationship that the con- scribes the constraint: Total expenditure cannot exceed
sumer seeks to maximize. In this example she will choose total income.
the amount of food and clothing to maximize her satisfac-
tion, measured by FC. Thus, the objective function is FC. Similar Problems: 1.4, 1.16, 1.17

APPLICA TION 1.1
Generating Electricity: 8,760 Decisions • The company needs to generate enough power

per Year to ensure that its customers receive service dur-
ing each hour of the day.
Examples of constrained optimization are all around • To make good production decisions, the com-
us. Electric power companies typically own and oper- pany must forecast the demand for electricity.
ate plants that produce electricity. A company must The demand for electricity varies from one hour
decide how much electricity to produce at each plant to another during the day, as well as across seasons
to meet the needs of its customers. of the year. For example, in the summer the highest
The constrained optimization problem for a power demand may occur in the afternoon when cus-
company can be complex: tomers use air conditioners to cool offices and