Consider the utility function u(x_1,x_2) = x_1 x_2. Suppose that the prices are given 1 for each...
Question:
Consider the utility function {eq}u(x_1,x_2) = x_1 x_2 {/eq}. Suppose that the prices are given 1 for each good and the income is 10.
(1) Find out the optimal choice(s). Figure out the utility level at this optimal choice.
(2) Suppose now that the price of good 2 increases from 1 to 2. Figure out the value of the bundle that you obtained in (1) under the new prices. Draw the budget set associated with this value and the new prices.
(3) Find out the optimal choice(s) given the budget set in (2).
(4) Find the optimal choice(s) under the new prices and the initial income of 10.
(5) What is the (Slutsky) substitution effect on good 2? What is the income effect on good 2?
(6) Consider the utility level you calculated in (1). What is the bundle that gives you the same utility while minimizing the expenditure under the new prices?
(7) What is the (Hicksian) substitution effect on good 2?
(8) Suppose that the price of good 2 is now given {eq}p_2 > 0 {/eq} while the price of good 1 is fixed at 1. Figure out the Marshallian demand curve of good 2.
(9) Continuing from (8), figure out the Hicksian demand curve of good 2 associated with utility level 25.
(10) Is the Hicksian demand curve from (9) always steeper than the Marshallian demand curve from (9) at {eq}x_2 = 5 {/eq} in this example?
Utility Maximization:
Utility is a fundamental concept in microeconomic theory. It measure the level of satisfaction of consumers while choosing and consuming a set of good, i.e. a product or a service. It measures preferences over a set of consumer choices. Utility function measures the satisfaction of a consumer as a function of consumption. It is widely used in microeconomics in order to analyse consumers' behaviours and consumers' preferences.
Answer and Explanation: 1
Become a Study.com member to unlock this answer! Create your account
View this answer(1) A consumer maximizes his utility given the constraint of his wealth, such that
{eq}\max {u(x_1, x_2) = x_1x_2} {/eq}
{eq}s.c. p_1x_1 + p_2x_2 =...
See full answer below.
Ask a question
Our experts can answer your tough homework and study questions.
Ask a question Ask a questionSearch Answers
Learn more about this topic:
Get access to this video and our entire Q&A library
Utility Theory: Definition, Examples & Economics
from
Chapter 1 / Lesson 27
40K
Learn about utility theory. Study utility in economics, examine utility economics examples, and discover how utility affects the decisions customers make.
Related to this Question
- Consider the utility function u(x_1,x_2) = x_1x_2. Suppose that the prices are given 1 for each good and the income is 10. Answer the questions.
- 1. Consider the utility function u x_1x_2 = x_1x_2. Suppose that the prices given are 1 for each good and that the income is 10.
- Consider the utility function u( x_1, x_2 ) = x_1 x_2 .Suppose that the prices are given 1 for each good and the income is 10.
- Consider the utility function u(x_1,x_2) = max(x_1,x_2). Suppose that the price of good 1 is fixed at 1. The price of good 2 is given p_2 greater than 0 and the income is m greater than 0. a) Draw the
- Consider the utility function u(x1, x2) = max{x1, x2}. Suppose that the price of good 1 is fixed at 1. The price of good 2 is given p2(>0) and the income is m(>0).
- Consider the utility function u(x1, x2) = max{x1, x2}. Suppose that the price of good 1 is fixed at 1. The price of good 2 is given p2 ( 0) and the income is m( 0). a) Find out the optimal choices at
- Consider a utility function u(x1; x2) = x1^{1/2}x2^{1/2} . Let the prices of good 1 and good 2 be p1 and p2, and of course consumer's income is m. Find the demand functions.
- Consider the utility function u(x_1, x_2) = max{x_1, x_2}. Suppose that the price of good 1 is fixed at 1. The price of good 2 is given p_2(gt 0) and the income is m(gt 0). Find out optimal choices a
- Suppose you have this utility function: u(x1, x2) = x^(1/2)x^(1/2). Prices for good 1 and good 2 are p1 and p2, and income is m. The demand functions are as follows: good 1: x1 = m/2p1 good 2: x2 = m/2p2 If the price per unit for the two goods is p1 = p2
- Consider the utility function u(x1,x2)=x1x2. Suppose that the price are given 1for each good and the income is 10. What is the Slutsky substitution effect on good 2? What is the income effect on goo
- Consider the following utility function: U = q_1^{0.2} q_2^{0.8} 1. Suppose Y (income) = $100, and prices are P_1= $20 and P_2= $ 10; calculate the utility maximizing quantities of q_1 and q_2 and the
- Consider the utility function U(x,y)=2x+3y. The marginal utility of good x is given by: 2/3 3/2 2 1/3 3
- Suppose your utility function is a function of goods x and y. Specifically, U(x,y) = min(3x,4y). The prices of good x and y are Px=2 and Py=3 respectively. The amount of money you have is 10 units. C
- Consider the following individual utility functions and answer the questions.
- Suppose the utility function is U x 1 3 y 1 3 . Price of x is $2 and price of y is $5. Let the income be $50. Please derive the optimal quantity of x and y demanded.
- Consider a market with two goods, x and z that has the following utility function U = (x^.8)(z^.2) a. What is the marginal rate of substitution? b. As a function of the price of good x (px), the pr
- Suppose that income is m = 102, and prices are a = 2 and b = 5. Consider the following utility function: u(a,b) = (a + 2)(b +1) (i) Find the utility-maximizing quantities of a and b. (ii) What is t
- A. Assume that an individual with the utility function U(x, y) = ln(x) + y has an income of $100 and that price of good y is equal to $1. Derive the demand curve for good x as a function of P_x. B. Assume that an individual with the utility function U(x,
- Suppose there are two goods, X and Y. The utility function for these goods is given by U(X,Y) = 5X+2Y. Suppose I had $50 to spend on these two goods. Good X has a price of $5 per unit, while the price
- Consider a consumer who purchases only two good x and good y. The consumer's utility function is U(x,y) = 4xy. Suppose prices are Px = $2 and Py = $4. She has an income of $20. a) Solve the optimal b
- Consider a consumer who consumes two goods and has utility function u(x_1, x_2) = x_2 + \sqrt{x_1}. The price of good 2 is 1, the price of good 1 is p, and income is m. Show that a) both goods are n
- Assume the following utility function: U(x,y) = x^{0.5} y^{0.5}. Income equals $2 and the price of each good is initially $1. If the price of X was to increase to $9, and if we hold real income consta
- Suppose you have a utility function u(x1, x2) = x1^(1/2)x2^(1/2). The prices for good 1 and good 2 are p1 and p2, and the consumer's income is m. The prices per unit for the two goods are p1 = p2 = $2, and the consumer's income is m = $100. a. Calculate t
- Consider the utility function U(x,y) = 2x^1/2 + 4y^1/2: a) Find the demand functions for goods x and y as they depend on prices and income. b) Find compensated demand functions for goods x and y.
- Consider a person with the following utility function when consuming two goods, x and z. U = x 0.1 y 0.9 a) What is the marginal rate of substitution? b) As a function of the price of good x(px), the price of good z(pz), and the income level (Y) derive th
- Consider the following utility function: U(X, \: Y) = 5X + 2Y. Find the marginal utility of each good.
- Suppose that you know the utility function for an individual is given by the equation U = XY where U is the total amount of utility the individual gets when they consume good X and good Y. You are also told that this individual's income is $100 and that t
- Suppose that income is m =102, and prices are p_1 = 2 and p_2 = 5, and consider the following utility function: u(x_1 x_2) = (x_1 + 2)(x_2 + 1). i) Find the utility-maximizing quantities of x_1 and x_
- Consider a consumer who consumes two goods and has utility of function u(x_{1},x_{2})=x_{2}+ \sqrt{x_{1 Income is m, the price of good 2 is 1, and the price of good 1 changes from p to (1+t)p. Compu
- Suppose a consumer has a utility function U(X, Y) = min{X, 2Y}. Suppose the consumer has $945 to spend (M = 945) and the price of good Y is 1 (P_Y = 1). Fill in the table below.
- Suppose you consume two goods, whose prices are given by p_1 and p_2, and your income is m. Solve for your demand functions for the two goods if A. your utility function is given by U(x_1, x_2) = ax_1 + bx_2. B. your utility function is given by U(x_1, x_
- QUESTION 23 The utility function and the prices are the following: U = 11 x1 + 39 x2 P1=49, P2=10 and I =5,838 What is the optimal amount of x1?
- Consider a person with the following utility function when consuming two goods, x and z, U = x^{0.1} x^{0.9}. a) What is the marginal rate of substitution? b) As a function of the price of good r (
- Suppose a consumer's utility function is given by: U(x_1, x_2) = 1/3 ln (x_1) + 2/3 ln (x_2).The consumer's income is $100, the price of x_2 is p_2 and the price of good 1 is p_1 = 2p_2^2.
- Suppose that the utility function is u(x,y)=10x0.4y0.4. Further suppose that the consumer's budget constraint can be expressed as 30x+15y=2400. For this consumer, the optimal amount of good y to buy would be what?
- Suppose that the utility function is u(x, y) = Ax + By where A and B are constants. Also suppose that the marginal utility of x is 10 and the marginal utility of y is 20. Further suppose that the price of x is $20 and the price of y is $25 per unit. Suppo
- Consider a market with two goods, x and z that has the following utility function. U = x^.8z^.2 a. What is the marginal rate of substitution? b. As a function of the price of good x (px), the price
- Consider a consumer who consumes two goods and has utility function u(x1,x2)= x2 +\sqrt(x1). Income is m, the price of good 2 is 1, and the price of good 1 changes from p to (1+t)p Compute the compens
- Suppose that there are two goods, X and Y. The utility function is u=x^2y. the price of X is P, and the price of Y is $2. income is $90 Derive the demand for X as a function of P.
- Consider the general demand function: Qd= 680-9P+0.006M-4Pr, where P is the price of the good, M is income, and Pr is the price of a related good, R. a) Interpret the slope parameters of the demand function. b) Is the good inferior? why? c) Is the good
- Suppose that the price of good x is p_x and the price of good y is p_y. If Jim has I to spend on x and y and his utility function over two goods is given by, U(x,y)=radicalx+?y A.Find Jim?s demand fun
- 1. Suppose utility is given by U(x,m) = 100 \sqrt x + m. The price of good x is 5, the price of good m (money) is 1. The consumer has 1,000 dollars to spend. Write out the consumer's problem in full
- Consider the problem of a rational consumer with an experienced utility function given by 10 - x + m. Let q denote the market price of good x. What is the expression for the consumer surplus if the c
- Consider a general demand function for a good, X, that has been estimated as: Q(d) = 200 - 8*Px + 2*Py + 0.4*M, where Px is the price of the good being studied Py is the price of a related good, Y and M is per capita income. a) Are X and Y complement
- Suppose two goods have the following utility function: u(x1, x2)=x10.4, x20.6, what can be said about the income elasticity of demand for good x1? What about for good x2?
- Assume a consumer's utility function is U = (q_1)^0.5+ 2(q_2)^0.5 and her total income is $90. The price of both good 1 and good 2 is $1. (a) What is the bundle that maximizes this consumer's utility
- Assume you're given the utility function U(x,y) = 3x + 4y. Goods x and y are both priced at $2. Given an income of $12, what allocation of the two goods provides the highest utility?
- 1. Suppose you have the following hypothetical demand or sales function. Q X= 600- 6PX + 20I +0.4PY and PX = $80, (price of good X) PY =$1,300, (price of good Y) I = $30 (disposable per capita income)
- Suppose a consumer's utility function is given by U(X, Y) = X*Y. Also, the consumer has $360 to spend, and the price of X, P_x = 9, and the price of Y, P_y = 1. a) How much X and Y should the consume
- Consider a consumer with income of $220 and utility function U(x,y) =min{3x,5y). The prices of goods x and y are $5 and $10, respectively. Let x* and y* denote the utility-maximizing levels of consump
- Consider a consumer with utility function given by u(x_1, x_2) = x_1x_2 . (i) Find the demands for goods 1 and 2 when the consumer faces prices p_1 \enspace and \enspace p_2 , and income m . (i
- Suppose that Jack's utility function is U(x, y) = square root of xy. What would be Jack's optimal consumption of these two goods when his income is 72 dollars, and the prices of good x and y are 4 and 3 dollars, respectively?
- Question 12 The utility function and the prices are the following: U = 68 x1 + 37 x2 P1=29, P2=76 and I =1,790 What is the amount of maximized utility?Question 12 The utility function and the prices
- Suppose the consumer has a utility function of U(x_1 x_2) = x^alpha_1 x^beta_2, income y, and prices of the two goods are p_1, p_2 respectively. Let alpha + beta = 1. A) Sole the utility maximization
- Suppose we have the utility function U(x, y) = x^{0.1}y^{0.9}. A. What is the marginal utility of x? What is the marginal utility of y? What is the marginal rate of substitution? B. Suppose you have $200 to spend, the price of x s $1 and the price of y is
- Suppose that the marginal utility of Good X = 100, the price of X is $10 per unit, and the price of Y is $5. Assuming that the consumer is in equilibrium and is consuming both X and Y, what must the marginal utility of Y be?
- Suppose a consumer's utility function is U(X, Y) = X + 2Y. The consumer has $8 to spend (M = 8). The price of good Y is P_Y = $2. What are the respective demands of good X when P_X = 1/4, 1/2, 2, and 4 dollars?
- Suppose that Ashley likes 2 goods, good 1 (x_1) and good 2 (x_2). Her utility function is given by U(x_1, x_2) = x_1^2 + x_2. Ashley's income is denoted as M and the cost of good 1 (per unit) is P_1 and the cost of good 2 (per unit) is P_2. 1. Write down
- Consider the following utility function: U(X, \: Y) = 10X^{0.5} + 5Y. Find the marginal utility of each good.
- Consider a consumer with utility function given by u(x_1, x_2) = x_1x_2. A) Find the demands for goods 1 and 2 when the consumer faces prices p_1 and p_2, and income m. B) Are goods 1 and 2 normal goo
- A consumer has a utility function, given by u(x1,x2)=rad x1 x2. Suppose the price of good 1 falls, from $5 to $2, while the price of good 2, and the consumer's income remain constant, at $10 and $100,
- Consider the following utility function (referred as a quasi-linear utility function as it is linear in the second element): u(x,y)=ln(x)+y. with prices and income given by: p_x=1,p_y epsilon R_+ and
- For the below question, consider a consumer that consumes two goods, x and z with the following utility function. U =( x^.25)(z^.75) 1. What is the marginal rate of subsitution? a. -3x/z b.-x/3z c. -3
- Answer the question on the basis of the following marginal utility data for products X and Y. Assume that the prices of X and Y are $4 and $2 respectively and that the consumers' income is $18. What quantities of X and Y should be purchased to maximize u
- Suppose that a consumer's preferences can be represented by the utility function u(x1, x2) = min {2x1 ,x2}. Suppose that the originally the price of good one is $2, the price of good two is $2 and the
- Consider the following utility function (referred as a quasi-linear utility function as it is linear in the second element): u(x, y) = \ln(x) + y, with prices and income given by: px = 1, p_y \in R_
- Suppose that you have the following utility function: U(x, y) = 6 x^0.5 + y.The price of x is px and the price of y is 1. Your income is Y = $36. a. Find the uncompensated demand for good x. That is, find the amount of x which maximizes her utility subje
- Suppose that John has the utility function u(x, y) = 3x^{\frac{1}{2 + y. Suppose that the budget equation is given by P_xx + P_yy = M. Suppose that the price of a unit of x is 1 and the price of a unit of y is 2 and John's income is 8. Find how many uni
- Consider the following utility function: U(X, \: Y) = 5X + 2Y. Determine whether the marginal utility decreases as consumption of each good increases.
- Consider the following utility function: U(X, \: Y) = X^{0.33}Y^{0.67}. Determine whether the marginal utility decreases as consumption of each good increases.
- Suppose a consumer's utility function is U(X, Y) = X + 2Y. The consumer has $8 to spend (M = $8). The price of good Y is PY = $2. Fill in the table below which gives price/quantity combinations on the Demand Curve for Good X:
- Suppose a consumer has a utility function for two goods, X and Y, given by U(X, Y) = 2X + 3Y. The consumer has $30 to spend and the prices of good X and good Y are P_X = $2 and P_Y = $5, respectively. Draw the consumer's budget constraint and indicate the
- Suppose the weighted marginal utility for two goods, X and Y at a position of consumer equilibrium is 70. if the price of good X is R 10 and the relevant marginal utility for Y is 140, what is the price of good Y and the relevant marginal utility for good
- Suppose that the marginal utility of good A is 4 times the marginal utility of good B, but the price of good A is only 2 times the price of good B. Is this point consumer equilibrium? If not, what will occur?
- Suppose a worker has the utility function U(Consumption, Leisure). Assume that leisure is an inferior good for a worker and answer the questions.
- Suppose a consumer's utility function is given by U(X,Y) = X*Y. Also the consumer has $288 to spend, and the price of X, Px=16, and the price of Y, Py=1. How much X and Y should the consumer purchase
- Consider the following demand equation: Q = 206 - 15 p - 10 p_0 + 0.18Y Here, p is the price of the good, p_0 is the price of a related good, and Y is income. If we assume the price of the related good is held fixed at $10 and that income is held fixed at
- Suppose that a representative consumer has the following utility function: U(x, y) = x^{\alpha} y^{t - \alpha}. The price of good x is p_x and the price of good y is p_r The consumer's income is equal
- Suppose that the utility function is given U(x)=x_1^2 x_2^3. How much of commodity 1 (x_1) and commodity 2 (x2_) will the consumer purchase as a function of p_1, p_2, and m?
- Consider the utility function u(x, y) = 2lnx + lny. Initially, the prices are p_x = $2/unit and p_y = $1/unit. The consumer has an income of $18. Illustrate answers with graphs. A. Derive the consumer's optimal consumption bundle. B. Now, suppose the pric
- Question: Janet's utility function is u(x1,x2)=min(4x1,x2) if the price of good 1 is $2, the price of good 2 is $3 and Janet's income is $28, what is the optimal consumption bundle for Janet?
- a.Tom spends all his $100 weekly income on two goods, X and Y. His utility function is given by U(X, Y) = X^2Y^2. If PX= 4 and PY=10, how much of each good should he buy? b.Same as part a, except no
- Suppose that marginal utility of Good X = 100, the price of X is $10 per unit, and the price of Y is $5 per unit. Assuming that the consumer's in equilibrium and is consuming both X and Y, what must the marginal utility of Y be?
- Suppose that marginal utility of Good X = 100, the price of X is $10 per unit, and the price of Y is $5 per unit. Assuming that the consumer is in equilibrium and is consuming both X and Y, what must the marginal utility of Y be?
- Phil's utility function is U = XY , where MU_ X = Y and MU _Y = X . In Lancaster, Pennsylvania, Phil's income would equal $500, and the prices of good X and good Y are $5 and $5, respectively. In Washington, D.C., Phil's income would equal $800, and the p
- A household has the budget constraint y = px1 + x2 where the price of good is normalized to one. The utility function is given by: U(x1, x2) - Min (x11, 4x2) Solve the demand function x1 and x2 and utility U(x1, x2) as functions of y and p.
- Suppose that a person's preferences can be described by the utility function U = ? L n q 2 where 0 is less than ? is less than 1 . They have income Y and face prices of p1 and p2. a).What is the
- Consider the utility function u(x, y) = 2 ln x + ln y. Initially, the prices are px = $2/unit and py = $1/unit. The consumer has an income of $18. Suppose the price of good x increases to px = $3/unit. What is the new optimal consumption bundle? Illustr
- Show that the utility function is homothetic if and only if all demand functions are multiplicatively separable in price and income and of the form x(p,y) = \phi(y)x(p,1).
- Set up the utility maximization problem for each consumer. Then solve for the Marshallian demand functions of good x and good y for each consumer as a function of prices. Consider a two consumer endowment economy. Consumer 1 and consumer 2 come into the e
- Suppose a consumer's utility function is given by U(X,Y) = X^ 2 \times Y. The Price of Y is P Y = 3, and the consumer has M = $18 to spend. Draw the Price Consumption Curve for the following values of P_x: P_x=1, P_x =2, P_x=3.
- Consider the following three utility functions: U(X,Y) = 2X + 2XY + 2Y U(X,Y) = X^2 + 3Y U(X,Y) = 0.5ln(X) + 0.5ln(Y) a. Is the assumption that more is better satisfied for both goods? Justify. (6 points) b. Does the marginal utility of good X diminish, r
- Suppose you have a total income of I to spend on two goods x_1 and x_2, with unit prices p_1 and p_2 respectively. It can be represented by the utility function u(x_1, x_2) = x_1x_2 + x_2. a) What is
- Suppose that you know the utility function for an individual is given by the equation U= XY where U is the total amount of utility the in and good Y. You are also told that this individual's income is
- Consider the utility function u(x, y) = 2 ln x + ln y. Initially, the prices are px = $2/unit and py = $1/unit. The consumer has an income of $18. Derive the consumer s optimal consumption bundle. Illustrate your answer with a graph.
- Suppose a consumer's utility function is given by U(X, Y) = X*Y. Also, the consumer has $180 to spend, and the price of X, P_X = 4.50, and the price of Y, P_Y = 2. a) How much X and Y should the consumer purchase in order to maximize her utility? b) How
- Consider a situation where a consumer demands two goods, x and z with the utility function: U=x^{0.3}z^{0.7} (a) Derive the marginal rate of substitution (b) Derive the demand functions for x and z a
- Suppose the marginal cost to produce a good is $10. There is only one person who is willing to purchase the good, and she is willing to pay $90 each for two units, and $30 for the third unit. Accordin
- Consider an individual making choices over two goods x and y with prices p x and p y with income I and the utility function u x y 2 x y Find the compensated Hicksian demand for x
Explore our homework questions and answers library
Browse
by subject