Question 1

Georgina consumes only grapefruits and pineapples. Her utility function is U(x, y) = x²y⁸, where x is the number of grapefruits consumed and y is the number of pineapples consumed. Georgina's income is $105, and the prices of grapefruits and pineapples are $1 and $3, respectively. How many grapefruits will she consume?

Accepted Answer

To maximize utility subject to the budget constraint, we need to use the Lagrange method. Our objective function is U(x, y) = x²y⁸, and our constraint is P1x + P2y = M. Taking the partial derivative of the objective function with respect to x and y gives: dU/dx = U1P1^2x^U1-1(yP2)^U2 / (P1P2)^U1 + U1P1^2x^U1(yP2)^U2-1 / (P1P2)^U2 = λP1 dU/dy = U1P1(xP2)^U2 / (P1P2)^U1 + U1(xP2)^U2-1(yP2)^U2 / (P1P2)^U2 = λP2 Solving for x and y gives: x = 21 y = 14 Therefore, Georgina will consume 21 grapefruits.

Question 2

Darlene's utility function is U(x, y, z) = x³y⁷z. If her income doubles and prices remain unchanged, her demand for good y will more than double.

Accepted Answer

Darlene's utility function suggests that her preferences are Cobb-Douglas, where the demand for each good is proportional to her income. Therefore, if her income doubles, her demand for good y will also double, not more than double.

Question 3

Natalie consumes only apples and tomatoes. Her utility function is U(x, y) = x²y⁸, where x is the number of apples consumed and y is the number of tomatoes consumed. Natalie's income is $320, and the prices of apples and tomatoes are $4 and $3, respectively. How many apples will she consume?

Accepted Answer

To maximize Natalie's utility subject to her budget constraint, we need to set up the following optimization problem: Maximize U(x, y) = x²y⁸ Subject to the budget constraint: 4x + 3y = 320 First, let's simplify the utility function by taking the natural logarithm of both sides: ln(U) = ln[x²y⁸] ln(U) = ln(x)² + ln(y)⁸ Next, we can use the method of Lagrange multipliers to maximize ln(U) subject the the budget constraint: L(x, y, λ) = ln(x)² + ln(y)⁸ + λ(320 - 4x - 3y) Taking the partial derivatives with respect to x, y, and λ, and setting them equal to zero, we get: ² / x - 4λ = 0 ⁸ / y - 3λ = 0 320 - 4x - 3y = 0 Solving the first two equations for x and y in terms of λ, and setting them equal to each other, we get: ln(x)² / (⁸y) = 4 / 3 ln(x) / ln(y) = 4 / 3 Solving for y in terms of x, we get: y = (4/3)^(3/4) * x^(1/4) Substituting this expression for y into the budget constraint and solving for x, we get: 4x + 3[(4/3)^(3/4) * x^(1/4)] = 320 x = 16 Therefore, Natalie will consume 16 apples.

Question 4

For m > p₂, the demand functions for goods 1 and 2 are given by the equations, - 1 and , where m is income and p₁ and p₂ are prices. Let the horizontal axis represent the quantity of good 1. Let p₁ = 1 and p₂ = 2. Then for m > 2, the income offer curve is

Accepted Answer

The answer of For m > p₂, the demand functions...

Question 5

Quasilinear preferences are homothetic when the optimal amount of one of the goods is not affordable.

Accepted Answer

Quasilinear preferences are not necessarily homothetic because homothetic preferences imply that the shape of indifference curves is consistent across all levels of income, whereas quasilinear preferences allow for the optimal amount of one good to be constant regardless of income levels, not affecting the affordability of goods.

Question 6

Mickey is considering buying a tape recorder. His utility function is U(x, y, z) = x + f(y)z^.5, where x is the amount of money he spends on other goods, y is the number of tape recorders he buys, and z is the number of tapes he buys. Let f(y) = 0 if y < 1 and f(y) = 8 if y is greater than or equal to 1. The price of tape recorders is $20, the price of tapes is $1, and Mickey can easily afford to buy a tape recorder and several tapes. Will he buy a tape recorder?

Accepted Answer

The answer of Mickey is considering buying a tape recorder....

Question 7

Clarissa's utility function is U(r, z) = z + 160r - r², where r is the number of rose plants she has in her garden and z is the number of zinnias. She has 250 square feet to allocate to roses and zinnias. Roses each take up 4 square feet and zinnias each take up 1 square foot. She gets the plants for free from a generous friend. If she acquires another 100 square feet of land for her garden and her utility function remains unchanged, she will plant

Accepted Answer

To maximize her utility function, we need to find out the marginal utility of roses and zinnias. Taking the partial derivative of the utility function with respect to r and z, we get: ∂U/∂r = 160 - ² ∂U/∂z = 1 Since she has 250 square feet to allocate, we know that 4r + z = 250. Rearranging this equation, we get: z = 250 - 4r Substituting this into the utility function, we get: U(r) = 250 - 3r + 160r - r² U(r) = 250 - r(3 - ²) + 160r To find the optimal number of roses, we need to set the derivative of the utility function with respect to r equal to zero and solve for r: ∂U/∂r = 160 - (3 - ²) = 0 ² = 157 Substituting this value back into the equation for z, we get: z = 250 - 4r = 250 - 4(157/3) ≈ 47 Therefore, she should plant max(1, min(99, 47 + 100)) = 99 more zinnias and some more roses (depending on how much space is left after planting the additional zinnias).

Question 8

Where x is the quantity of good X demanded, the inverse demand function for X

Accepted Answer

The inverse demand function expresses price as a function of quantity demanded, not as a function of income or other variables.

Question 9

Daisy received a tape recorder as a birthday gift and is not able to return it. Her utility function is U(x, y, z) = x + z f(y), where z is the number of tapes she buys, y is the number of tape recorders she has, and x is the amount of money she has left to spend. f(y) = 0 if y < 1 and f(y) = 24 if y is 1 or greater. The price of tapes is $4 and she can easily afford to buy dozens of tapes. How many tapes will she buy?

Accepted Answer

Since Daisy cannot return the tape recorder, she must maximize her utility function given by U(x, y, z) = x + z + f(y). 
Since f(y) does not change with the purchase of tapes, we only need to focus on maximizing x + z. 
Let the price of the tape recorder be denoted as P. Then Daisy's initial budget is P, and she can spend 4 dollars on each tape she buys. 
With each tape she purchases, her budget is reduced by 4 dollars. Thus, in order to maximize x + z, she should buy as many tapes as possible until her budget is exhausted. 
If she spends all her money on tapes, she can buy (P-4)/4 tapes, since she starts with one tape recorder. 
Substituting P = 24 (from f(y)) gives (24-4)/4 = 5 tapes. 
Therefore, Daisy will purchase 5 tapes, leaving her with 4 dollars to spend on other items, giving a utility score of 28. 
For option A)where she purchases 9 tapes, she will have a utility score of 37, which is greater than 28. 
Therefore, the best choice is to buy 9 tapes.

Question 10

Mike consumes two commodities, x and y, and his utility function is min{x + 2y, y + 2x}. He chooses to buy 8 units of good x and 16 units of good y. The price of good y is $.50. What is his income?

Accepted Answer

The answer of Mike consumes two commodities, x and y,...

Question 11

Daisy received a tape recorder as a birthday gift and is not able to return it. Her utility function is U(x, y, z) = x + z f(y), where z is the number of tapes she buys, y is the number of tape recorders she has, and x is the amount of money she has left to spend. f(y) = 0 if y < 1 and f(y) = 8 if y is 1 or greater. The price of tapes is $1 and she can easily afford to buy dozens of tapes. How many tapes will she buy?

Accepted Answer

Since Daisy can afford to buy dozens of tapes, we know that the marginal utility of tapes is positive. To maximize her utility, she should keep buying tapes until the marginal utility of tapes equals the price of tapes. The marginal utility of tapes is the partial derivative of her utility function with respect to z: MUz = ∂U/∂z = 11ea6c55_6e0e_6b37_9875_89a57022877f_TB6162_11 8. Since the price of tapes is $1, she should keep buying tapes as long as the marginal utility of tapes is at least $1. In this case, her marginal utility of tapes is $8, so she will continue buying tapes until MUz = $1. Solving for z, we get: 11ea6c55_6e0e_6b37_9875_89a57022877f_TB6162_11 8 = $1 z = 1/8 She can only buy a whole number of tapes, so she will buy 16 tapes (since she cannot buy 17 tapes with the money she has left).

Question 12

Clarissa's utility function is U(r, z) = z + 120r - r², where r is the number of rose plants she has in her garden and z is the number of zinnias. She has 250 square feet to allocate to roses and zinnias. Roses each take up 4 square feet and zinnias each take up 1 square foot. She gets the plants for free from a generous friend. If she acquires another 100 square feet of land for her garden and her utility function remains unchanged, she will plant

Accepted Answer

The answer of Clarissa's utility function is U(r, z) =...

Question 13

If there are two goods and if income doubles and the price of good 1 doubles while the price of good 2 stays constant, a consumer's demand for good

Accepted Answer

The answer of If there are two goods and if...

Question 14

Madonna buys only two goods. Her utility function is Cobb-Douglas. Her demand functions have which of the following properties?

Accepted Answer

Since Madonna's utility function is Cobb-Douglas, her demand functions will have the property of being homogenous of degree zero, meaning that changing her income will not change the ratios of the quantities of the two goods that she consumes. Therefore, her demand for one of the goods does not depend on income, but her demand for each of the goods does depend on the prices of both goods. This is captured in choice D. The other choices are incorrect as they do not fully capture the properties of a Cobb-Douglas utility function.

Question 15

Regardless of his income and regardless of prices, Smedley always spends 25% of his income on housing, 10% on clothing, 30% on food, 15% on transportation, and 20% on recreation. This behavior is consistent with which of the following?

Accepted Answer

The answer of Regardless of his income and regardless of...

Question 16

Walt consumes strawberries and cream but only in the fixed ratio of three boxes of strawberries to two cartons of cream. At any other ratio, the excess goods are totally useless to him. The cost of a box of strawberries is $10 and the cost of a carton of cream is $10. Walt's income is $200.

Accepted Answer

To optimize his utility, Walt must consume the goods in the fixed ratio of 3 boxes of strawberries to 2 cartons of cream. Let X be the number of boxes of strawberries and Y be the number of cartons of cream that Walt buys. Then the utility function is:
U(X, Y) = min(X/3, Y/2)
Since the price of both strawberries and cream is $10, Walt can spend at most $200 on them. Therefore, the budget constraint is:
10X + 10Y = 200
Solving for X and Y, we get:
X = 20 - Y
Substituting this into the utility function, we get:
U(Y) = min((20-Y)/3, Y/2)
To maximize his utility, Walt must choose Y such that the marginal utility (MU) of cream is equal to the marginal disutility (MDU) of strawberries. In other words, MU(Y) = dU(Y)/dY = -1/6 = MDU(X) = dU(X)/dX = -1/3.
Differentiating the utility function with respect to Y, we get:
dU(Y)/dY = -1/6 - 1/3 = -1/2
Setting this equal to zero, we get:
-1/2 = 0, which is not possible.
Therefore, the utility function U(Y) is maximized at the boundary point Y = 10. This means Walt must buy 10 cartons of cream and 12 boxes of strawberries (since X = 20 - Y). This is the best choice of goods for Walt to maximize his utility given his income and his fixed ratio of consumption.

Question 17

Fred has a Cobb-Douglas utility function with exponents that sum to 1. Sally consumes the same two goods, but the two goods are perfect substitutes for her. Despite these differences, Fred and Sally have the same price offer curves.

Accepted Answer

Fred's Cobb-Douglas utility function implies that his preferences exhibit diminishing marginal rates of substitution between the two goods, leading to a convex indifference curve. Sally's perfect substitutes utility function implies a linear indifference curve, resulting in different price offer curves for the two individuals.

Question 18

Harry has $10 to spend on cans of Coke and Pepsi, which he regards as perfect substitutes, one for one. Pepsi costs $.50 a can and Coke costs $.60 a can. Harry has 20 coupons, each of which can be used to buy 1 can of Coke for $.40. Which of the following bundles will Harry buy?

Accepted Answer

The answer of Harry has $10 to spend on cans...

Question 19

Darlene's utility function is U(x, y, z) = x⁴y⁸z. If her income doubles and prices remain unchanged, her demand for good y will more than double.

Accepted Answer

Darlene's utility function suggests a Cobb-Douglas preference, where the demand for each good is proportional to income. Therefore, if her income doubles, her demand for good y will also double, not more than double.

Question 20

Seppo consumes brandy and saunas. Neither is an inferior good. Seppo has a total of $30 a day and 6 hours a day to spend on brandy and saunas. Each brandy costs $2 and takes half an hour to consume. Each sauna costs $1 and takes 1 hour to consume. (It is, unfortunately, impossible to consume a brandy in the sauna.) Seppo suddenly inherits a lot of money and now has $50 a day to spend on brandy and saunas. Since Seppo is a rational consumer, he will

Accepted Answer

The answer of Seppo consumes brandy and saunas. Neither is...

Quiz 5: Demand

Darlene's utility function is U(x, y, z) = x3y7z. If her income doubles and prices remain unchanged, her demand for good y will more than double.

Natalie consumes only apples and tomatoes. Her utility function is U(x, y) = x2y8, where x is the number of apples consumed and y is the number of tomatoes consumed. Natalie's income is $320, and the prices of apples and tomatoes are $4 and $3, respectively. How many apples will she consume?

For m > p2, the demand functions for goods 1 and 2 are given by the equations, - 1 and , where m is income and p1 and p2 are prices. Let the horizontal axis represent the quantity of good 1. Let p1 = 1 and p2 = 2. Then for m > 2, the income offer curve is