Suppose that John has the utility function u(x, y) = 3x^{\frac{1}{2}} + y. Suppose that the...
Question:
Suppose that John has the utility function {eq}u(x, y) = 3x^{\frac{1}{2}} + y {/eq}. Suppose that the budget equation is given by {eq}P_xx + P_yy = M {/eq}. Suppose that the price of a unit of {eq}x {/eq} is 1 and the price of a unit of {eq}y {/eq} is 2 and John's income is 8. Find how many units of {eq}x {/eq} and {eq}y {/eq} John chooses to consume.
Budget Constraint:
In economics, the budget constraint determines the set of consumption goods that are feasible. Among the set of feasible consumption bundles, the one that yields the highest total utility is the optimal bundle.
Answer and Explanation: 1
Become a Study.com member to unlock this answer! Create your account
View this answerThe optimal consumption bundle is when the marginal rate of substitution is equal to the price ratio, i.e.,
- {eq}\dfrac{MU_x}{MU_y} =...
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 Maximization: Budget Constraints & Consumer Choice
from
Chapter 3 / Lesson 2
14K
Learn about utility maximization. Discover various types of utility, examine utility maximizing rules, and study examples of maximizing utilities in economics.
Related to this Question
- Suppose consumption is a linear function of disposable income: C(Y-T) = a+b(Y-T) where a is greater than 0 and 0 is less than b is less than I. Suppose also that investment is a linear function of the
- 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.
- Suppose that Patrick has a utility function of the form of U = Min(alpha x, beta y). Patrick has an income of $100 and currently faces prices of Px = $1.00 and Py = $2.00. Suppose that the price of X
- Assume that George has the following utility function: U = 4q1^(0.5)q2^(0.5) Prices are p1 and p2. Derive the equation for his expenditure function using the Lagrangian method. Derive the compensat
- 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
- 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. Answer the questions.
- 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
- Martha has the utility function u(x, y) = min{2x, y}. Write down her demand function for X as a function of variables m, pX, and pY , where m is income, pX is the price of x, and pY is the price of Y .
- a) What is John's long-run cost function? b) Suppose that John owns the regular shovel, and does not rent it anymore. What is the long-run cost function now?
- A professor has preferences over tea (x) and coffee (y), represented by the utility function U (x, y) = xy. The marginal utilities for this utility function are M Ux = y and M Uy = x. Her income is $48, and the price of y is given by $2 per unit. (a) Calc
- 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.
- Suppose that the linear equation for consumption in a hypothetical economy is: C = 40 + 0.75Y Also suppose that income (Y) is $400. Determine the following values: For parts a, b, d, and f, round your
- Suppose John's utility function is U = ln(2C) where C is the amount of consumption John has in any given period. John's income is $100,000 per year and there is a 3% chance that he will be involved in a catastrophic accident that will cost you $30,000 nex
- 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. Calculate the substitution effect and the income effect. Illustrate your answer with a graph.
- Advanced Analysis Suppose that the linear equation for consumption in a hypothetical economy is C = 40 + 0.75Y Also suppose that income (Y) is $400. Determine the following values. a. MPC b. MP
- Noah's utility function is: u(x, y)=x+2y. The price of y is py, the price of x is px, and Noah's income is m. Note that Noah's indifference curves are not strictly convex. We will use a different method to derive Noah's demand. Show that Noah always choos
- Suppose that the linear equation for consumption in a hypothetical economy is: C = 50 + 0.7 Y Also suppose that income (Y) is $500. Determine the following values: a. MPC. b. MPS. c. Consumption. d. APC. e. Saving. f. APS.
- 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
- Suppose that the linear equation for consumption in a hypothetical economy is C = 40 + 0.9Y Also suppose that income (Y) is $400. Determine the following values: For parts a, b, d, and f round your
- 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) = (Ax^p + By^p)^{frac{1}{p. Suppose p = 1. What is the MRS in this case? A. frac{A}{B} B. frac{Ax}{By} C. frac{Bx^2}{Ay^2} D. frac{x}{y}
- Harry's budget constraint is given by P_xX+P_yY = 60, and P_x = $5, P_y = $2. Suppose Harry's utility function is given by the equation U-XY, where U is the level of utility measured in utils and X an
- Suppose Consumer A has a utility function U (X, Y)= X + 3 Y. Consumer B's utility function is U (X, Y) = X*Y. Also there are a total of 18 units of good X and 9 total units of good Y. Using the Edgeworth Box, construct a contract curve.
- Harry's budget constraint is given by PX+PyY=60, and P=$5, Py=$2. Suppose Harry's utility function is given by the equation U XY, where U is the level of utility measured in utils and X and Y refer to
- Suppose your utility function over goods X and Y is given by U(X, Y) = X*Y. Sketch a graph of the indifference curve associated with 10 units of utility.
- 1. Suppose David spends his income (I) on two goods, x and y, whose market prices are px and py, respectively. His preferences are represented by the utility function u(x, y) = ln x + 2 ln y (MUx = 1
- Suppose that the linear equation for consumption in a hypothetical economy is C = 50 + 0.9 Y. Also suppose that income (Y) is $400. Required a. MPC b. MPS c. level of consumption d. APC e. APS.
- Suppose that Justin's preferences are represented by the utility function U(x,y) = (3/4)x^2y^2/3. The prices of x and y are px and py. Justin's income is equal to m.
- Suppose Vivian has a utility function of U = X^0.4*Y^0.6, where X and Y are two goods. The prices for X and Y are $4 and $6, respectively. She has $100 in her pocket. Explain why Vivian's utility function is a special c
- Suppose Jose's utility function for beer (x) and wings (y) is U=6x^{.4} y^{.6}. Jose has $50 to spend, and the price of beer is $4 and the price of wings is $.50. a. Show the LaGrangian equation for this problem. b. Show the first-order conditions. c.
- Suppose Vivian has a utility function U = X^(0.4)Y^(0.6), which X and Y are two goods. The prices for X and Y are $4 and $6, respectively. She has $100 in her pocket. a. Explain why Vivian's utility function is a special case of Cobb-Douglas function. b.
- If the ulitity function is: U=22x1 + 57x2 ; How many units of x2 is the consumer willing to pay to get 71 more unit of x1? (hint: remember how to obtain the opportunity cost of x1 using MRS. Enter thi
- 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
- Suppose Charles' utility over X and Y can be represented by the utility function U(x,y) = x^2+y^2. If the price of X is $25 and the price of Y is $20 and Charles has $200 to spend, what commodity bund
- Assume that the following log-linear estimated equation shows the relationship between the dependent variable Q_D (quantity demanded) and the independent variables P (price), M (disposable income) and A (advertisement expenditure) for Nike's tennis shoes:
- Consider the utility function, U=xy+x+y and suppose the consumer has budget constraint of pxx+pyy=m. a) form the lagrangian for this problem, derive the first order conditions for a utility maximum, a
- 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
- Diogo's utility function is U = 100X^0.9 Z^0.1 The price of X is px = $12, the price of Z is pz = $14, and his income is $1200. What is Diogo's optimal bundle? (Be sure to answer how many units for X_
- Suppose a demand equation was estimated using the Regression technique. The explanatory variables included in the equation were price of own good, price of substitute good, income of consumers and expected future price. What test will be used to test if e
- Suppose that the linear equation consumption in a hypothetical economy is: C = 50 + 0.75Y Also suppose that income (Y) is $500. Determine the following values: a. MPC = b. MPS =c. Consumption = d. AP
- Samira has a monthly income of $200 that she allocates among two goods: meat and potatoes. a. Suppose meat costs $4 per pound and potatoes $2 per pound. Write the equation for her budget line and draw it. b. Suppose also that her utility function is given
- Suppose Theon has a utility function of the form U=C11/3C22/3 and has an income of $100 today and $0 tomorrow. Find Theon's supply of savings function.
- A consumer's utility is defined by the function: u(x_1,x_2) = x_1^(1/3), x_2^(1/2). Assume prices of x_1 and x_2 are respectively defined by P_1 and P_2 and the consumer has W dollars in income. 1: F
- Nick has a utility function U(x, y) = x^2 y^2 and faces the budget constraint p_xx+p_yy=Y. a. Suppose that Y =120, py =1 and px =4. What are Nick's utility-maximizing amounts of good x and good y? What level of utility is Nick able to attain? b. If the
- 1. Consider the following economy: C = 3, I = 1.5 G = 2.65 T = 2 f = 0.5 d = 0.1 mpc = 0.8 A) Write the mathematical expression of the consumption function. B) Write the mathematical expression of the investment function. C) Find the IS curve and grap
- 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 a person's utility function takes the Cobb-Douglas form U(C,R) = C^0.4 R^0.6, where C is consumption, and R is recreation. Assume that the price of the consumption good is 1 and that the consu
- Consider the following cost function: c(y) = 4y^2 + 16. Write down the expression for the average cost function.
- Consider the following cost function: c(y) = 4y^2 + 16. Write down the expression for the average variable cost.
- Wolf's utility function is U = aq_1^{0.5} + q_2. For given prices and income, show how whether he has an interior or corner solution depends on a. M
- Suppose that U(x, y) = 2 min (5x, 3y). Further suppose that Px = $10 per unit and Py = $15 per unit and income is I = $420. For this consumer, the optimal basket contains (blank) units of y.
- 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 function U(x, y) = x + ln y. This is a function that is used relatively frequently in economic modeling as it has some useful properties. a. Find the MRS of the function. Now, interpret
- Wolf's utility function is U=aq_{1}^{0.5}+q_{2}. For given prices and income, show how whether he has an interior or corner solution depends on a.
- Consider the utility function \upsilon (x_1, x_2) = x_1 + x_2. (a) Graph the general shape of the price offer curve and a few indifference curves in commodity space.
- Suppose you have a budget of $1,300 to spend of beer and pizza this year. Your utility function is U = 200B + 150P - 0.5B^2 - 0.25P^2. The price of a beer is $5 and the price of a pizza is $10. a. Using calculus, set up a constrained optimization problem
- Suppose that the consumer's preferences are described by the utility function U(x; y) = 5x + y, where y denotes the quantity of good Y and x denotes the quantity of good X that the consumer consumes. Suppose also that the consumer's income is 600 euros, t
- Graphing the BC. Suppose that an individual has a weekly income of $180 to spend on goods X and Y, which have prices of $2 and $6, respectively. Graph the budget constraint. Suppose that the price of
- Suppose that John's MPC is constant at 3/4. If his breakeven point occurs at $7,000, how much will John have to borrow when his income is $3,000?
- Suppose the consumption function is C=127+0.79YD. If disposable income is $400, consumption is what?
- Assume that a consumer has the utility function U(x,y)=(3x+1)y where x and y represent the quantity of two goods X and Y.(not sure if this part is relevant) Now assume that the consumer has a $31 to s
- Suppose that a consumer's preferences can be represented by the utility function u(x1, x2) = ln(x1) +ln(x2). Suppose that the consumer's income, m, suddenly increases. Use calculus to show what will
- 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
- 3. Suppose we are studying the market for burgers. Bob is the sole provider of burgers in the market; he faces the following cost function: C(Q) = Q^2 + 9, and the following3. Suppose we are studying
- Suppose you are given the following consumption and income data: |Consumption |100 |190| 280| 370 |460 |550 |Income| 0 |100 |200 |300| 400 |500 Obtain an equation for the consumption function. Use your function to predict the value of consumption wh
- Yvonne consumes only X and Y, which have prices PX = 3 and PY =1, and her income is 12. a. Solve for Y on her budget line (as a function of X). b. Differentiate Y with respect to X to find the slope of her budget line. c. If Yvonne s utility is repre
- Suppose that X falls in price to $1.75 and good Y still costs S1. Suppose the individual still has $22 to spend solely on X and Y a) Sketch carefully the new budget line. Where does it lie compared t
- Suppose that the utility function is given as u(x) = 2ln(x_1) + 3ln(x_2). (Note: The preferences represented by this utility function is strictly convex and monotonic.) How much of commodity 1 (x_1) and commodity 2 (x_2) will the consumer purchase as a fu
- Consider the estimate demand equation of: Qx =1000-3.3Px - 0.2Pz +0.001Y (3.5) (2.1) (0.5) t values in parenthesis, where Pz is the price of another good Z, and Y is income. Is good Z a substitute
- Suppose that a = \frac{1}{2}. The cost function should now be linear in the output. Explain why it is not surprising that this is the case if a = frac(1)(2).
- A consumers utility is defined by the function, U(x1,x2) = x1^1/2 x2^1/2 Assume the prices of x1 and x2 are respectively defined by $5 and $10, and the consumer has 20 dollars in income. a. Find the c
- Suppose that Sally s preferences over baskets containing milk (good x), and coffee (goody ), are described by the utility function U(x, y ) = xy + 2x. Sally s corresponding marginal utilities are, MUx = y + 2 and MUy = x. Use Px to represent the price of
- Marcus's utility function is U(X, Y) = (X + 2) (Y + 1). a. Write an equation for the indifference curve that goes through the point (2, 8). b. Graph the indifference curve for U=36 c. Let PX=PY=1 a
- Suppose an individual has a preference represented by the utility function U(X_{1},X_{2} = X_{1} + lnX_{2}. The individual consumes two goods,X_{1} and X_{2} and faces the prices P_{1} and P_{2}. Con
- Suppose a consumer maximizes the utility function U= \ln(xy) subject to 4x + 2y = 12. Find the consumption bundle (x*,y*) that maximizes utility for the consumer. (Hint: use the Lagrangian method, co
- Suppose that Colin's preferences for goods X and Y can be represented by the utility function U(x,y) = min[x,y], where x (respectively, y) is the amount of X (respectively, Y) that he consumes. min[x,
- Suppose that individual A's utility function, which only depends on wealth, is defined as U(W)= W^(1/2), where W stands for wealth. a. What is his level of utility when W= 1000;
- How do I solve this "Tom spends all his $100 weekly income on two goods, X and Y. His utility function is given by U(X,Y) XY. If PX = 4 and PY = 10, how much of each good should he buy?"
- 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
- Consider the utility function U(x, y) = xy. a. Is the assumption that "more is better" satisfied for both goods? b. What is the MRSx, y for this utility function? c. Is the MRSx, y diminishing, con
- Assume that the level of autonomous consumption in Mudville is $400. If the marginal propensity to consume is 0.9, what is the consumption function? The saving function? Give an equation for each and show each graphically. At what level of income is savin
- Consider a Hotelling model with consumers distributed uniformly along the line [0, 1]. There are two firms, each with cost function C(q_i) = 2q_i. Let x epsilon [0, 1] be a location of consumer and z
- Assume that Sally has the following utility function: U = B2 + 4Z2. Sally's income is $200, PB equals 1 and PZ equals 4. Determine the quantity demanded of Z. Show all of your work. (Hint: draw a coup
- Suppose the following equation. Y= C + I + G + Ex - Im yd = y + tr - ta C = 1000 + 0.65YD I = 725 G = 1000 EX = 1000 IM = 400 + 0.25Y TR = 1000 TA = 1500 a) What is the value of equilibrium income? b) What is the value of investment spending mul
- Compute the expenditure function from the Cobb-Douglas utility function and quasi linear utility function with steps.
- 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 a class of utility functions which are "additively separable," i.e., a. Find the first and second order conditions for utility maximization for these utility functions. Show that diminishing
- Suppose a person has a utility function U=YX. What is the minimum level of income required to get a utility of 20? Your answer should be a function of, among other things, Px and Py.
- 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.
- The utility function and the prices are the following: U = 14 x_{1} + 21x_{2} P1=47, P2=9 and I =2,545 What is the optimal amount of x1?
- Suppose that the utility function of an individual can be described as U(X, Y) = X + Y. For this utility function the MRS A) is always XY. B) is always X - Y. C) is always X/Y. D) is always X + Y. E) is always constant.
- An individual utility function is given by U(x,y) = x \ cdot y. Let U = 200, p_x = $20 and p_y = $40. Consider this individual dual constrained expenditure-minimization problem using the Lagrangian te
- Suppose your utility function is U=ln(4I) where I is the amount of income you make in a year. Suppose that you typically make $30000 per year, but there is a 5 percent chance that, in the next year,
- An individual has the utility function a) Find his consumption on x_1 and x_2 when his budget constraint is 3x_1+4x_2=100. b) Year later his budget constraint is 3x_1+2x_2=100, find his consumption
- Suppose a consumer's utility function is given by U = (q_1,\; q_2,\; ...,\; q_n). Derive Engel's Aggregation Condition.
- Consider an economy that is described by the following equations: C = 140 + 0.80(Y - T) - 200r Consumption Function T = 400 + 0.1Y Tax Function I = 1000 - 700r Investment Function L = 0.5Y - 1000i
- Consider the following utility function: U(X, \: Y) = 10X^{0.5} + 5Y. Find the marginal utility of each good.
- Consider the consumption-savings problem in a two-period model without government. Suppose the lifetime utility function is given by U(c,c')= \ln (c) + b \In (c'), where 0 is less than b is less than
- How would I derive a linear regression equation 'under the logarithm' with respect to output per worker in an augmented standard Solow model?
- Suppose that Anne is facing two options a and b. a gives Anne $4 on Tuesday, while b gives Anne $9 on Wednesday. Anne's utility function over money is given by u(x) = \sqrt x. Suppose that Anne is a
Explore our homework questions and answers library
Browse
by subject