# Microeconomic Theory

## Quiz 3 :Utility Maximization and Choice

Each day Paul, who is in third grade, eats lunch at school. He likes only Twinkies (t) and soda (s), and these provide him a utility of a. If Twinkies cost $0.10 each and soda costs$0.25 per cup, how should Paul spend the $1 his mother gives him to maximize his utility? b. If the school tries to discourage Twinkie consumption by increasing the price to$0.40, by how much will Paul's mother have to increase his lunch allowance to provide him with the same level of utility he received in part (a)?
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a) P's utility function is
His budget constraint is:
Setting the Lagrange
Yields the first order conditions
...... (1)
...... (2)
Dividing equation (1) by equation (2), we get
Thus,
...... (3)
Substitution of this value of s into the budget constraint
We get,
Solving for t yields
Substituting the above value of t in equation (3),
As shown above, P should buy 5 twinkies and 2 cups of soda.
b) P's utility when the price of twinkies was $0.10 each is found by substituting the optimal values t = 5 and s = 2 into the utility function. Now, we will maximize P's utility When the price of twinkies is$0.40
The new budget constraint is
Where $x is the increased money his mother gives him so that his utility remains at the initial level of . Setting the Lagrange Yields the first order condition ...... (4) ...... (5) Taking the ratio of the first two terms shows that Thus, ...... (6) Substitution of this value of s into the budget constraint gives: Solving for t yields Substituting the above value of t in equation (6) we get The utility level when t is 1.25 x and s is 2 x Equating this utility level with his initial utility level of We get, As his initial allowance was$1, his mother will have to increase his allowance by $1. a. A young connoisseur has$600 to spend to build a small wine cellar. She enjoys two vintages in particular: a 2001 French Bordeaux ( w F ) at $40 per bottle and a less expensive 2005 California varietal wine ( w C ) priced at$8. If her utility is then how much of each wine should she purchase? b. When she arrived at the wine store, our young oenologist discovered that the price of the French Bordeaux had fallen to $20 a bottle because of a decrease in the value of the euro. If the price of the California wine remains stable at$8 per bottle, how much of each wine should our friend purchase to maximize utility under these altered conditions? c. Explain why this wine fancier is better off in part (b) than in part (a). How would you put a monetary value on this utility increase?
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a.
Utility function is a mathematical function which gives the level of utility achieved by an individual at different amounts of commodities.
Individual's utility is represented by
The Budget constraint is given by
Setting the Lagrange
The first order conditions for the Lagrange equation are given below:
…… (1)
…… (2)
Dividing equation (1) by equation (2), the value of
can be calculated as follows:
Thus,
Putting this value into budget constraint, the value of
can be calculated as follows:
Substituting the above value of
in equation:
To maximize her utility, she should purchase
bottles of French Bordeaux and
bottles of California wine.
b.
The utility function of the individual isgiven below:
When the of price
falls to $20, the new budget constraint is given below: The Lagrange equation is given below: The first order condition for the Lagrange equations are given below: …… (3) …… (4) Dividing equation (3) by equation (4), the value of can be calculated as follows: Thus, Putting this value into budget constraint, the value of can be calculated as follows: Substituting the above value of in equation the value of can be calculated as shown below: To maximize her utility, she should purchase bottles of French Bordeaux and bottles of California wine. c. As price of French Bordeaux decreases from$40 to $20 and individual can buy more of French Bordeaux she is better off in second situation.She is consuming same amount of California wine and more French Bordeaux wine. The monetary value that can be put on this utility increase is the additional money that the individual would have had to spend to purchase the additional 10 bottles of French Bordeaux at the initial price of$40.
The monetary value of utility increase is calculated below:
Therefore, the monetary value of increase in utility is
.

a. On a given evening, J. P. enjoys the consumption of cigars (c) and brandy (b) according to the function How many cigars and glasses of brandy does he consume during an evening? (Cost is no object to J. P.) b. Lately, however, J. P. has been advised by his doctors that he should limit the sum of glasses of brandy and cigars consumed to 5. How many glasses of brandy and cigars will he consume under these circumstances?
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a) The utility function of the individual is
As cost is not object, there does not exist a budget constraint.
First order conditions of the utility function is
...... (1)
...... (2)
From equation (1) we get
c = 10
From equation (2) we get,
b = 3
The individual should consume 3 units of b and 10 units of c to maximize utility.
b) The utility function of the individual is given by
As instructed by the doctor, his constraint is given by
Thus, b = 5 - c
Substituting this in the utility function, we get
The first order condition is
Therefore,
Putting this back into equation
The individual should consume 1 unit of b and 4 units of c to maximize his utility.

a. Mr. Odde Ball enjoys commodities x and y according to the utility function Maximize Mr. Ball's utility if , and he has $50 to spend. Hint: It may be easier here to maximize U 2 rather than U. Why will this not alter your results? b. Graph Mr. Ball's indifference curve and its point of tangency with his budget constraint. What does the graph say about Mr. Ball's behavior? Have you found a true maximum? Essay Answer: Mr. A derives utility from martinis (m) in proportion to the number he drinks: Mr. A is particular about his martinis, however: He only enjoys them made in the exact proportion of two parts gin (g) to one part vermouth (v). Hence we can rewrite Mr. A's utility function as a. Graph Mr. A's indifference curve in terms of g and v for various levels of utility. Show that, regardless of the prices of the two ingredients, Mr. A will never alter the way he mixes martinis. b. Calculate the demand functions for g and v. c. Using the results from part (b), what is Mr. A's indirect utility function? d. Calculate Mr. A's expenditure function; for each level of utility, show spending as a function of p g and p v. Hint: Because this problem involves a fixed-proportions utility function, you cannot solve for utility-maximizing decisions by using calculus. Essay Answer: Suppose that a fast-food junkie derives utility from three goods-soft drinks (x), hamburgers (y), and ice cream sundaes (z)- according to the Cobb-Douglas utility function Suppose also that the prices for these goods are given by and that this consumer's income is given by I = 8. a. Show that, for z = 0, maximization of utility results in the same optimal choices as in Example 4.1. Show also that any choice that results in z 0 (even for a fractional z) reduces utility from this optimum. b. How do you explain the fact that z ¼ 0 is optimal here? c. How high would this individual's income have to be for any z to be purchased? Essay Answer: The lump sum principle illustrated in Figure 4.5 applies to transfer policy and taxation. This problem examines this application of the principle. a. Use a graph similar to Figure 4.5 to show that an income grant to a person provides more utility than does a subsidy on good x that costs the same amount to the government. b. Use the Cobb-Douglas expenditure function presented in Equation 4.52 to calculate the extra purchasing power needed to increase this person's utility from U = 2 to U = 3. c. Use Equation 4.52 again to estimate the degree to which good x must be subsidized to increase this person's utility from U = 2 to U = 3. How much would this subsidy cost the government? How would this cost compare with the cost calculated in part (b)? d. Problem 4.10 asks you to compute an expenditure function for a more general Cobb-Douglas utility function than the one used in Example 4.4. Use that expenditure function to re-solve parts (b) and (c) here for the case ? = 0.3, a figure close to the fraction of income that low-income people spend on food. e. How would your calculations in this problem have changed if we had used the expenditure function for the fixed proportions case (Equation 4.54) instead? Equation 4.52 Essay Answer: Two of the simplest utility functions are: 1. Fixed proportions: 2. Perfect substitutes a. For each of these utility functions, compute the following: • Demand functions for x and y • Indirect utility function • Expenditure function b. Discuss the particular forms of these functions you calculated-why do they take the specific forms they do? Essay Answer: Suppose that we have a utility function involving two goods that is linear of the form U(x, y) = ax + by. Calculate the expenditure function for this utility function. Hint: The expenditure function will have kinks at various price ratios. Essay Answer: Cobb-Douglas utility In Example 4.1 we looked at the Cobb-Douglas utility function This problem illustrates a few more attributes of that function. a. Calculate the indirect utility function for this Cobb-Douglas case. b. Calculate the expenditure function for this case. c. Show explicitly how the compensation required to offset the effect of an increase in the price of x is related to the size of the exponent ? Essay Answer: CES utility The CES utility function we have used in this chapter is given by a. Show that the first-order conditions for a constrained utility maximum with this function require individuals to choose goods in the proportion b. Show that the result in part (a) implies that individuals will allocate their funds equally between x and y for the Cobb- Douglas case as we have shown before in several problems. c. How does the ratio p x x/p y y depend on the value of ?? Explain your results intuitively. (For further details on this function, see Extension E4.3.) d. Derive the indirect utility and expenditure functions for this case and check your results by describing the homogeneity properties of the functions you calculated. Essay Answer: Stone-Geary utility Suppose individuals require a certain level of food (x) to remain alive. Let this amount be given by x 0. Once x 0 is purchased, individuals obtain utility from food and other goods (y) of the form where ? + ? = 1. a. Show that if on good y. Interpret this result. b. How do the ratios p x x/I and p y y/I change as income increases in this problem? (See also Extension E4.2 for more on this utility function.) Essay Answer: CES indirect utility and expenditure functions In this problem, we will use a more standard form of the CES utility function to derive indirect utility and expenditure functions. Suppose utility is given by [in this function the elasticity of substitution a. Show that the indirect utility function for the utility function just given is where . b. Show that the function derived in part (a) is homogeneous of degree zero in prices and income. c. Show that this function is strictly increasing in income. d. Show that this function is strictly decreasing in any price. e. Show that the expenditure function for this case of CES utility is given by f. Show that the function derived in part (e) is homogeneous of degree one in the goods' prices. g. Show that this expenditure function is increasing in each of the prices. h. Show that the function is concave in each price. Essay Answer: Altruism Michele, who has a relatively high income I, has altruistic feelings toward Sofia, who lives in such poverty that she essentiallyhas no income. Suppose Michele's preferences are represented by the utility function where c 1 and c 2 are Michele and Sofia's consumption levels, appearing as goods in a standard Cobb-Douglas utility function. Assume that Michele can spend her income either on her own or Sofia's consumption (through charitable donations) and that$1 buys a unit of consumption for either (thus, the ''prices'' of consumption are p 1 = p 2 = 1). a. Argue that the exponent a can be taken as a measure of the degree of Michele's altruism by providing an interpretation of extremes values a = 0 and a = 1. What value would make her a perfect altruist (regarding others the same as oneself )? b. Solve for Michele's optimal choices and demonstrate how they change with a. c. Solve for Michele's optimal choices under an income tax at rate t. How do her choices change if there is a charitable deduction (so income spent on charitable deductions is not taxed)? Does the charitable deduction have a bigger incentive effect on more or less altruistic people? d. Return to the case without taxes for simplicity. Now suppose that Michele's altruism is represented by the utility function which is similar to the representation of altruism in Extension E3.4 to the previous chapter. According to this specification, Michele cares directly about Sofia's utility level and only indirectly about Sofia's consumption level. 1. Solve for Michele's optimal choices if Sofia's utility function is symmetric to Michele's: Compare your answer with part (b). Is Michele more or less charitable under the new specification? Explain. 2. Repeat the previous analysis assuming Sofia's utility function is U 2 (c 2 ) = c 2.
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