A household has the budget constraint y = px1 + x2 where the price of good is normalized to one....
Question:
A household has the budget constraint {eq}y = px_1 + x_2{/eq} where the price of good is normalized to one. The utility function is given by:
{eq}U(x_1, x_2) - Min \left [ x_1, 4x_2 \right ]{/eq}
Solve the demand function {eq}x_1 {/eq} and {eq}x_2 {/eq} and utility {eq}U(x_1, x_2) {/eq} as functions of y and p.
Utility Function
A utility function comprises a set of goods that define the utility level of the consumer. Any rational consumer seeks to maximise his utility under a given budget constraint. A point of tangency between the utility function and the budget constraint gives the optimum bundle to be consumed.
Answer and Explanation: 1
Become a Study.com member to unlock this answer! Create your account
View this answerThe given utility function can be also represented as:
{eq}{x_1} = 4{x_2} {/eq}
Because the given function represents the perfect complimentary...
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
- A consumer's utility function is U = ln(xy^4). A) Find the values of 'x' and 'y' which maximise utility subject to eh budgetary constraint 10x + 5y = 150. Use the method of substitution to solve this
- 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
- Maximize the utility of two good x and y given by the utility function U(x, y) = xy subject to the budget constraint where the price of x is $3, the price of y is $6 and the individual has an income b
- The utility function and the prices are the following: u= 10 x1 + 49 x2 p1= 37 p2=4 and I =2141 What is the optimal amount of x1?
- Suppose a consumer maximizes the utility function U=ln (xy) subject to the budget constraint 4x +2y =12. Find the consumption bundle (x*,y*) that maximizes utility for the consumer.
- 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?
- Given a consumer has a utility function of: U(X,Y) = 5XY^2 And the following budget constraint: 2X + 8Y = 240 A) Find the best feasible bundle. B) Suppose your income doubles. What is your new best fe
- The utility function of a consumer who consumes two products (X1, X2) is: u(x1,x2) = x1x2 + 2x1 The consumer's budget constraint is 4x1 + 2x2 = 60 You are required to: a) Find the expression for marginal utility from consuming each of the two products
- U(x,y)=2x+y Assuming you are rational and maximize utility: What is the demand for y if the budget constraint is given by x+y = 10 What is the demand for y if the budget constraint is given by 2x+y=
- a) A consumer has utility given by U(x,y) = square root of xy subject to the budget constraint 100 = 4x + y. Find the bundle that maximizes utility. b) Suppose the original utility is now written as
- Max's utility function is U=12XY, where the MUX=12Y and MUY=12X. The prices of good X and good Y are $24 and $30, respectively. Suppose that Max's indifference curve is tangent to his budget constraint, where he is consuming 40 units of good X. How many u
- 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 utility function for an individual is given by u(x, y) = x + y, where x and y are two goods. The budget constraint of an individual is 2x + 5y = 10, where I = 10 is the income of an individual, P_x = 2 - the price of good x, and P_y = 5 - the
- 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?
- 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,
- If there is no budget constraint, is utility maximization achieved when marginal utility is zero?
- 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.
- Rafael has the utility function U = x - y with the marginal utilities MUx = y and MUy = x. The price of x is $2, the price of y is unknown and equal to the variable py, and his income is $40. When Rafael maximizes utility subject to his budget constraint,
- Let U=Y�(X+12) and the budget constraint be 10=X+Y. Find the optimal consumption point using calculus and illustrate your solution on a graph, indicating the coordinates of the relevant points. [Hint:
- 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 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
- 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
- Max has a utility function u(x, y) = 16x - 2x^2 + 4y. The prices of x and y are p_x and p_y, respectively. Max has an income m. How much of each good will he demand? Find the algebraic expression for the marginal rate of substitution between goods x and y
- Suppose that an individual maximizes U(x,y) = x^{\alpha}x^{\beta}(\alpha,\beta greater than 0) subject to the budget constraint y = p_1x_1+p_2x_2 . Solve for x_1 and x_2 (the optimal values of x_1 a
- 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
- The utility function and the prices are the following: U=49 x1 + 84 x2 p1=78 p2 =10 and I = 3320 What is the optimal amount of x2?
- The utility function and the prices are the following: u = 10 x1 - 49 x2 p1=37 p2=4 I =2141 What is the optimal amount of x2 ?
- Suppose an individual consumers two goods, with utility function U (x1, x2) = x1 + 6{\sqrt{x1x2 +9x2 Formulate the consumers utility maximization problem when she faces a budget line p1x1 + p2x2
- Ann's utility function is U(x, y) = x + y, with associated marginal utility functions MUx = 1 and MUy = 1. Ann has income I = 4. a) Determine all optimal baskets given that she faces prices Px = 1 an
- Find the expression for the following: The income that is needed to achieve a utility level U0 > 0, as a function of prices and U0 (NOTE: I(Px, Py, U0) is called the expenditure function).
- If a consumer's budget constraint has a slope that is less than -1: a) the consumer gets more utility from good X than from good Y. b) the price of good X is less than the price of good Y. c) the consumer gets less utility from good X than from good Y. d)
- Let U=Y(X+4) and the budget constraint be 10=X+Y (i.e., prices of X and Y are both Find the optimal consumption point using calculus and illustrate your solution on a graph, indicating the coordinat
- A consumer has a utility function U(X, Y) = X^{1/4}Y^{3/4}. The consumer has $24 to spend and the prices of the goods are P_X = $2 and P_Y = $3. Note that MU_X = (1/4)X^{-3/4}Y^{3/4} and MU_Y = (3/4)X^{1/4}Y^{-1/4}. Draw the consumer's budget constraint a
- 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
- 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
- Maximum consumer utility is found where: a. MU = TU b. the budget constraint line is the steepest c. the budget constraint line is tangent to the indifference curve d. the budget constraint line has the slope - price ratio
- A consumer wishes to maximize utility; U=(X1^2)+(X2^2) subject to a budget constraint: l= p1x1+p2x2. a. Show graphically the bundle of x1 and x2 that will maximize utility. b.Solve mathematically for the bundle of goods that will maximize utility. c. S
- 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?
- The marginal utility associated with the additional consumption of X is given by the partial derivatives of the utility function with respect to good X. How is this related to optimization of utility in business economics?
- 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,
- The utility function and the prices are the following: U = 100 x_1 + 18 x_2 P_1=7, P_2=99 and I =2,011 What is the optimal amount of x_2?
- Suppose a consumer's utility function is given by U(X, Y) = XY. Also, the consumer has $720 to spend. The price of X is PX = $9, and the price of Y is PY = $9. a. How much X and Y should the consumer purchase in order to maximize her utility? b. How much
- 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
- 1. When a consumer has maximized their utility: a. What is the relationship between the budget constraint and their utility maximizing indifference curve? b. What is the relationship between the margi
- Tiffany's utility function is given by the formula U(x1, x2) = min{20x1, 10x2}. If x1 = x2, which commodity will have a greater marginal utility? A. Both x1 and x2 have the same marginal utility B. It
- 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
- Suppose utility is given by U(x,y) = 2X + Y (a) Draw an indifference curve for this utility function. (b) Suppose Px = 3 and Py = 2 and I = 12. Draw the budget constraint and determine where this per
- What is the relation between the aggregated demand function and the marginal utility function?
- The utility function and the prices are the following: U = min\left \{ 84 x_1 , 65 x_2 \right \} P_1=18, P_2=6 and I =1,605 What is the optimal amount of x_2?
- Consider a consumer who has the utility function u(x1, x2) = a ln(x) + B ln(y) where x1 and x2 and the quantities consumed of the two goods and a, B > 0. Suppose the consumer is constrained to spend l
- The price of good X is $1.50 and that of good Y is $1. If a particular consumer's marginal utility for Y is 30, and he is currently maximizing his total utility, then what must be his marginal utility of X?
- The indirect utility function is given by: U = 2In[2 / (3P1)] + ln[Y / (3P2)] Proove that the indirect utility function is homogeneous of degree zero in prices (P) and income (Y). Also, the Hicksian demand function for good 1 is: H (good 1) = (u + 2lnP1 +
- A firm has the total cost function C(q) = F + 25q + 5q^2. Its marginal cost function is MC(q) = 25 + 10q. The inverse demand for the firm's market is P(q) = A - 0.5q and F = 200. A) What are the long-run average and avoidable cost functions? B) What is th
- 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
- If a consumer's income is $300, the price of good X is $3, and the price of good Y is $6, then what is the algebraic expression for the consumer's budget constraint?
- Ginger's utility function is U(x, y) = x^2y, with associated marginal utility functions MU_x = 2xy and MU_y = x^2. She has income I = $240 and faces prices P_x = $8 and P_y = $2. a) Determine Ginger's optimal basket given these prices and her income. b) I
- I am given utility function. U=10XY^2. a) general form of budget constraint b) marginal utilities given the total utility curve. that is, derive MUx and MUy. c) find MRS.
- 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.
- Derive the demand curve good X for the utility function U=X^{2}Y^{3}. Show your work.
- If the cost function for John's Shoe Repair is C(q) =100+10q-q^2+(1/3)q^3. A. What is the firm's marginal cost function? B. What is its profit-maximizing condition if the market price is p? C. What is its supply curve?
- Consider the indirect utility function: v(p1; p2; m) = m /(p1 + p2) a. Derive the Marshallian demand functions. b. What is the expenditure function? c. What is the direct utility function?
- A Consumer has preferences represented by the utility function: U = xy; the Prices are: Px = 1 and Py = 2. I. Expenditure minimization problem: determine the optimal consumption vector and the minimu
- 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
- Given the following utility function: U = (2x_(1)x_(2) + 3x_(1)^(2))^(3)x_(2) a) Find the marginal utility for x_(1) and x_(2). Do not simplify the function. Work with it in the form given. b) Show m
- Suppose that the price of X is twice the price of Y. Mr. Cleary is a utility maximizer who allocates his budget between the two goods. a. What must be true about the equilibrium relationship between the marginal utility levels of the last unit consumed o
- 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
- Given that the utility function for an individual is: and Income 72, the price of good one 4, and the price of good two 4, and the new price of good one = 77, What is the change in consumer surplus fo
- For the following utility function, * Find the marginal utility of each good. * Determine whether the marginal utility decreases as consumption of each good increases (i.e., does the utility function
- 1. A consumer's spending is restricted because of? A) marginal utility. B) total utility. C) a budget constraint. D) utility maximization. 2. Utility is maximized in the consumption of two goods by equating the? A) marginal utility of one good to the pric
- Price of good X is $1.50 and the price of good Y is $3. You have $45 to spend and your preferences over X and Y are defined as: U(x, y) = x2/3y1/3 a. Calculate the marginal utility of X (remember, this is the change in utility resulting from a slight incr
- 67. A firm has a demand function P = 200 - Q and cost function: AC=MC=20 and a potential entrant has a cost function: AC=MC=30. D. Determine the optimal price, quantity and economic profit for the fi
- 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
- Let income be I = $100, P_x = $1, P_y = $1 and utility U(x, y) = xy \enspace with \enspace MU_x = y \enspace and \enspace MU_y = x 1.Write the budget constraint and find the optimal consumption
- Suppose that the inverse demand function is given by p = 120 - 2q and the cost function is given by TC = q2 + 10. a. What is the optimum amount of output and the market price in a monopolistic market? (Hint: a monopolist can influence the price) b. What i
- 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.
- Regarding consumer choice and utility maximization, what is meant by the application of the "Equimarginal Rule" and its relevance to the marginal principle ? In your analysis, discuss the concepts of utility (total & marginal), budget constraints and indi
- The consumer's budget constraint is $6 = 0.50G + P, where G is packs of gum and P is bags of pretzels. The marginal utility of pretzels is MUP = G 0.5, and the marginal utility of gum is MUG = 0.5G-0.5P. The consumer's utility function is U = G0.5P. the u
- 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.
- Given a firm has a demand function of: P = 45 - 4Q and average cost function of: AC = 1.5Q + 25 + 26/Q a. Find the level of output (Q) which: i. Maximizes total revenue (TR) ii. Makes marginal cost
- Suppose that a competitive firm has a total cost function of C(q) = 410 + 15q + 2(q^2) and a marginal cost function of MC(q) = 15 - 4q. If the market price is P = $131 per unit, find: a) The level
- Suppose there is a fully discriminating monopoly whose cost function is C=1+3X+0.1X^2, and this monopoly is facing a demand function of X =100 - 0.5P. a. What would be the maximal and minimal prices c
- Simple monopoly problem: A monopolist faces the demand function Q^D=1,000-0.5P and the cost function TC=200,000+2Q+0.1Q^2. Find the pro
- 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?
- Assume that the operation has the cost function C(Q) = 200 + 3Q + 8Q^2+ 4Q^3 and the marginal cost function MC(Q) = 3 + 16Q + 12Q^2. Determine the following:
- 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
- a. Use the method of Lagrange to find the conditional factor demands for cost minimization. b. Find the firm's cost function. c. Is this a short-run function or a long-run function? Explain. d. Write the equations for the firm's average cost function and
- The utility function and the prices are the following: U = 62 x1 + 85 x2 P1=68, P2=13 and I =2,655 What is the optimal amount of x2?
- What is the consumer equilibrium condition as an equation in terms of marginal utility and price?
- One goal of consumers is to maximize utility. We analyze utility maximization graphically using indifference curves and budget constraints. Explain and show graphically whether you know the prices o
- The utility function and the prices are the following: U=min (105 x_1, 74x_2) p1 =20 p2 = 44 I = 1803 What is the optimal amount of x2?
- 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
- The intercept of the demand function is 878,000 and the intercept for the marginal revenue is 100,000. Vertical. a) Determine the price and quantity that will maximize revenue. b) What is the price
- Given the production function f (xl, x2) = min{x1, x2), calculate the profit-maximizing demand and supply functions, and the profit function. What restriction must it satisfy?
- 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
- Suppose consumer preferences are given by the utility function U(X, Y) = XY^2. The marginal utility functions are MU_X = Y^2 and MU_Y = 2XY. Suppose the consumer has $300 to spend (M = 300). A. Derive an expression for the demand for good X B. Derive an
- Consider a linear demand function for oranges given as: Qd = 431.6 - 80.7P. At what price will the total expenditure on oranges be largest? (Hint: total expenditure is maximized where price elasticit
- Consider a monopolist with a cost function pf c(Q)=4+2Q. Suppose the market demand function is QD=24-2P. a) Under uniform pricing (basic monopoly problem), find the monopolist's optimal price and quan
- Q = KL TC = wL + rK K is fixed, K = 100 L = Q/100 - short run demand function SRTC = Q/100 + 100 What is the marginal cost equation?
- Consider the cost function (C) of a firm that under the perfectly competitive market as C=q^{3}-14q^{2}+69q+128. (Hint: The last term in the cost function "128" is the fixed cost.) The marginal cost f
- 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.
Explore our homework questions and answers library
Browse
by subject