# Suppose that a representative consumer has the following utility function: U(x, y) = x^{\alpha}...

## Question:

Suppose that a representative consumer has the following utility function: {eq}U(x, y) = x^{\alpha} y^{t - \alpha}. {/eq} The price of good x is p_x and the price of good y is {eq}p_r {/eq} The consumer's income is equal to I.

(a) Write down the constrained optimization problem, and form the Lagrangian function.

(b) Provide an economic interpretation to the first-order conditions. Solve for optimal values of x, y and the Lagrange multiplier.

(c) Is your solution a maximum or a minimum? Verify using the second-order condition.

(d) How would you interpret the Lagrange multiplier?

## LM

It is a tool to find out the maxima and minima. The condition includes that it should be subject to equality constraints. This concept was first brought and thus named after mathematician JL Lagrange.

## Answer and Explanation: 1

Become a Study.com member to unlock this answer! Create your account

View this answer

**Ans. (a)**

Maximize: {eq}U(x, y) = x^{\alpha} y^{t - \alpha} {/eq}

Subject to:

{eq}I = p_{x}x + p_{r}y {/eq}

Constrained Optimization Problem:

...

See full answer below.

#### Ask a question

Our experts can answer your tough homework and study questions.

Ask a question Ask a question#### Search Answers

#### Learn more about this topic:

from

Chapter 5 / Lesson 2One of the most important practical uses of higher mathematics is finding minima and maxima. This lesson will describe different ways to determine the maxima and minima of a function and give some real world examples.

#### Related to this Question

- 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
- 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
- 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
- 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.
- 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
- 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 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
- A consumer has the utility function U(x,y) = x^{\alpha} y^{\beta}, where x is greater or equal than 0 and y is greater or equal than 0 represent her consumption of goods X and Y, and \alpha is greater
- A consumer has the utility function U(x,y) = x^alpha y^beta, where x is greater than or equal to 0 and y is greater than or equal to 0 represent her consumption of goods X and Y, and alpha 0 and bet
- The consumer is choosing between two goods: good 1 and good 2. Good 1 is priced at $2 and good 2 is priced at $3. The consumer has a total of $600 to spend on both the goods. 1. Let the utility function of the consumer be U (X1, X2) = X1(1/2)X2(1/2). a)
- 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
- The consumer is choosing between two goods- good1 and good2. Good 1 is priced at $2 and good 2 is priced at $3. The consumer has a total of $600 to spend on both the goods. Let the utility function of the consumer be U (X1, X2) = Min (X1,X2) a) Calculat
- 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 that a consumer is spending all of their income on goods A and B. The price per unit of good A is 25 Euros and the price per unit of good B is 10 Euros. Assume the consumer's income is 2000 Eu
- A consumer's utility function is U(x; y) = (x + 2)(y + 1). The prices are p_x = $8; p_y = $2. The consumer consumes both goods. If his consumptions of good x is 14 units, what is his consumption of good y? Why?
- The consumer is choosing between two goods- good1 and good2. Good 1 is priced at $2 and good 2 is priced at $3. The consumer has a total of $600 to spend on both the goods. Let the utility function of the consumer be U (X1, X2) = 2X1+3X2. a) Calculate t
- The consumer is choosing between two goods- good1 and good2. Good 1 is priced at $2 and good 2 is priced at $3. The consumer has a total of $600 to spend on both the goods. Let the utility function of the consumer be U (X1, X2) = 2X1 + X22. a) Calculate
- The consumer is choosing between two goods- good1 and good2. Good 1 is priced at $2 and good 2 is priced at $3. The consumer has a total of $600 to spend on both the goods. Let the utility function of the consumer be U (X1, X2) = X1*X2 a) Calculate the
- The consumer is choosing between two goods- good1 and good2. Good 1 is priced at $2 and good 2 is priced at $3. The consumer has a total of $600 to spend on both the goods. Let the utility function of the consumer be U (X1, X2) = X1+X2. a) Calculate the
- 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 goods 1 and 2 are perfect complements. All consumers have utility function U(q1,q2) = min{q1,q2}. The sum of consumers' income is 100. Supply of the two goods is given by: S1(p1) = 2p1 and S2(
- 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
- Suppose a consumer has a utility function given by U = 4X +12Y. The price of X is $2 and the price of Y is $1. The consumer has $24 (his income) to spend on the two goods. Plot good X on the X-axis and good Y on the Y-axis.
- 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 there are two goods (X and Y). The price of X is $2 per unit, and the price of Y is $1 per unit. There are two consumers ( A and B). The utility functions for the consumers are: for consumer A: U (X,Y)= X^.5Y^.5 and for consumer B: U(X,Y)=X^.
- Suppose that a representative consumer has the following utility function for leisure l and consumption C. [{MathJax fullWidth='false' U(L, C) = C^{3/4}l^{1/4} }] The real wage rate, w, is competit
- Assume that a person's utility over two goods is given by: U(x,y) = x^(alpha)y^(1 - alpha). The price of Good X is p_x and the price of Good Y is p_y. The total income of the individual is given by M.
- Suppose consumers A and B have the following demand functions for x. x^A(I, p_x, p_y) = \frac{1 + p_y}{p_x}, x^B(I, p_x, p_y) = \frac{p_y}{1 + p_x} For each consumer, is x a normal good or an inferior good? Explain why.
- 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 consumer is choosing between two goods- good1 and good2. Good 1 is priced at $2 and good 2 is priced at $3. The consumer has a total of $600 to spend on both the goods. Let the utility function of the consumer be U (X1, X2) = X1(1/4)X2(3/4). a) Calc
- Consider the following utility function: U = 100X^{0.50} Y^{0.10 } A consumer faces prices of P = $1 and P =$5. Assuming that graphically good X is on the horizontal axis and good Y is on the vertical axis, suppose the consumer chooses to consume 13 unit
- 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 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
- A consumer treats goods x and y as perfect complements always combining one unit of good x with two units of good y. Suppose prices are p_x=$6 and p_y=$2. The consumer's income is i=$60. Find this con
- 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 a consumer with utility function defined over consumption of two goods, X and Y, U(cX, cY) = s ln cX + (1 - s) ln cY where cX is the consumption of good X; cY is the consumption of good Y; s is a number between 0 and 1. The consumer has income I
- A consumer has the utility function U(x1, x2) = x1x2 and an income of 24 pounds. Initially, the price of good 1 was 1 pound and price of good 2 was 2 pounds. Then the price of good 2 rose to 3 pounds while price of good 1 stayed the same. Find the consume
- 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?
- In a hypothetical economy, consumers consume two goods - Good 1 and Good 2. Assume that the utility function of a consumer is given by U = Q1Q2 (Qi referring to the amount of consumption of the good), Income (Y) of the consumer=100; prices of the goods P1
- Suppose that a consumer is currently spending all of her income on 10 units of good A and 5 units of good B. The price of good A is $4 per unit, the price of good B is $10 per unit, the marginal utility of the last unit of good A consumed is 20, and the m
- 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
- 1. A consumer has the utility function U(x,y) = x^4y, where x greater or equal 0 and y greater or equal 0 represent her consumption of goods X an Y, and alpha >0 is an exogenous parameter. (a) Calcul
- 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.
- Assume that a consumer can buy only two goods X and Y, and has an income of $120. The price of X is $10, and the price of Y is $20. If the consumer spends all of her money on X and Y, which of the fol
- If a consumer buys a good, the expected: A. total utility derived from the consumption of the good is less than its price B. marginal utility of per dollar spent on the good equals its price C. mar
- 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
- 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, 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.
- Assume that there are two goods (X and Y). The price of X is $2 per unit, and the price of Y is $1 per unit. There are two consumers ( A and B). The utility functions for the consumers are: for consumer A: U (X,Y)= X^.5Y^.5 and for consumer B: U(X,Y)=X^.8
- Consider a consumer that consumes two goods, x and z, with the following utility function. U = x^{0.125}z^{0.875} Suppose initial values for income and the prices of goods x and z are y = 100, P_x =5, and P_z = 15, respectively, then the price of good x f
- 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 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
- Suppose a consumer has references re resented by the utility function U(X, Y) = MIN[X, 3Y]. Suppose P_x = 2 and P_y = 4. Draw the Income Consumption Curve for this consumer for income values M = 100, M = 200, and M = 300. Graph should accurately draw the
- 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
- A utility function is U(x,y) = min(x,y^2). If the price of x is $25, the price of y is $10, and consumer chooses 5 units of y. How much is the consumer's income?
- A consumer is making purchases of products Alpha and Beta such that the marginal utility of product Alpha is 30 and the marginal utility of product Beta is 40. The price of product Alpha is $5, and the price of product Beta is $10. The utility-maximizing
- A consumer has the following utility function: U(x, y) =x(y+1), where x and y are quantities of two consumption goods whose prices are P_x and P_y, respectively. The consumer also has a budget of B. Therefore, the consumer's Lagrangian is x(y+1)+Lambda(B
- 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
- 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 a consumer who allocates her income m to the consumption of goods 1 and 2. Denote by pi the price of good i = 1, 2. The consumer's preferences are such that there exists a bundle x = (x1, x2)
- 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.
- Assume a consumer's utility function for goods x_1 and x_2 is of the following form: U(x_1,x_2) = x_1^2/3 x_2^1/3 Next, assume p_1 = 1, p_2 = 1, and Y = 120. a. Find the expressions for the consumer's
- Consider a market with two goods, x and z, and two consumers, A and B. The utility functions for consumers A and B are as follows \bar{U}_A = x_A^{0.75} z_A^{0.25} \bar{U}_B = x_B^{0.25} z_B^{0.75} and the initial endowments for each consumer are e_A
- 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 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
- 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.
- 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
- Two consumer Exchange economy: Consumer 1 utility u1(x,y)= x^2/3 y^1/3 with endowment e1=(18,7). Consumer 2 utility u2(x,y)= xy with e2=(15,16) a.Compute each consumer's demand ?function for good x a
- Consumer surplus is: A. The ratio of the price of a good to the proportion of income spent on the good B. The difference between the most a consumer would be willing to pay for a quantity of a good and what a consumer actually has to pay C. The sum of qua
- 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
- A consumer has utility u.(x, y) = .6 In x + .4 In y where prices are p_x = 2 and p_y = 2. (a) If the consumer has income of $100 per period. how much of each good should be consumed to maximize utili
- 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
- Consumers A and B consume good 1 and good 2 in an exchange economy. A's utility function is U^A = min(q_1^A, 2q_2^A) while B's utility function is U^B = q_1^B(q_2^B)^2. Their endowments are (e_1^A ,
- A consumer consumes good x and good y. She initially has an income of I = $1,000 and faces prices of px = $10 and py = $20. Then, the price of good x doubles. In response to the rising prices, the government gives the consumer a lump-sum payment of $300.
- Assume that a consumer derives more utility by spending an additional dollar on Good A rather than on Good B. We can assume that: A. the price of Good A is less than the price of Good B. B. the marginal utility per dollar spent on Good A is equal to the m
- Suppose for a given consumer X is an inferior good and Y is a normal good. a. Focusing on the income effect, show that an increase in the price of X will cause the consumer to "buy more" X (that is, s
- Suppose a consumer is willing to pay $20 for one good, $10 for a second good, and $5 for a third good, and the market price is $4. What is the consumer surplus? a. $16 b. $6 c. $1 d. $23
- The consumer's utility function for goods X and Y is U = 3X + 15Y. Good X is placed on the x-axis and good Y is placed on the y-axis. Which of the following statements is true? I. The marginal utility of good Y is 15. II. The MRSXY = 5. III. The consumer
- 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
- 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 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
- There is a 2 good, 2 consumer, no production economy. Consumer 1 has utility function U1(x1, y1) = x^.25y^.75 and currently has the bundle (30, 35). Consumer 2 has utility function U2(x2, y2) = x^.6y^
- 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
- If a consumer has a utility function of u(x_1,x_2)= x_1x_2^4 where MU_1 = x_2^4 and MU_2 = 4x_1x_2^3 , what fraction of her income will she spend on good 2? Assume that the person faces a downwar
- Suppose there are two goods, labeled x, y. A consumer has a budget of w = 220. Suppose the following two consumption bundles: (x = 4, y = 10), (x = 6, y = 4) all cost the consumer 220. (a) What's the
- Suppose there are only two goods, A and B, and that consumer income is constant. If the price of good A falls and the consumption of good B rises, we can conclude that: A) A is a normal good. B) B is a normal good. C) A is an inferior good. D) B is an
- Which of the following is NOT true about consumer utility maximization? a) a consumer maximizes utility by exhausting all available income b) consumer utility shows how much more of each good a consum
- Consider a consumer whose preferences are given by U(x_1, x_2) = log(x_1) + x_2 with an income, m, and prices for the two goods, p_2 = 1 and p_1 = 1. A. Neither of the two goods is a Giffen good. B. At the utility-maximizing choice, MRS necessarily equals
- A consumer with an income of $240 is spending it all on 12 units of good X and 18 units of good Y. The price of X is $5 and the price of Y is $10. The marginal utility of the last X is 20 and the marginal utility of the last Y is $30. What should the cons
- Suppose a consumer's preferences can be represented by the utility function: U(X,Y)=Y + 3X. a. Draw an indifference curve. b. Under what conditions would the consumer consume good Y? Good X?
- Consider three consumers indexed by i ∈ {1, 2, 3) with the following demand functions for a public good G. p1 = 20 - (1/10) G, p2 = 20 - (1/10) G, p3 = 30 - (2/10) G Where pi is the price consumer i is willing to pay for a quantity of G. If marginal
- 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)
- 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
- Suppose there are two consumers, A and B. The utility functions of each consumer are given by: UA(X,Y) = X*Y UB(X,Y) = 2X + Y The initial endowments are: A: X = 4; Y = 2 B: X = 4; Y = 6 a) Usin
- Suppose there are two consumers, A and B. The utility functions of each consumer are given by: U_A(X,Y) = X + Y U_B(X,Y) = Min(X,Y) The initial endowments are: A: X = 4; Y = 4 B: X = 2; Y = 2 I
- Suppose that a consumer consumes only two goods, good x and good y. Assume that (q = 20, l = 50) and (q = 23, l = 60) are two points on the consumer's Engle curve for good x. Assume that good y is a normal good. Which of the following statements is true?
- Suppose a consumer's preference are represented by the utility function \\ U(X,Y) = X^3Y^2 \\ Therefore, \\ MU_x = 3X^2Y^2\\ MU_y = 2X^3Y \\ Also, suppose the consumers has $200 to spend (M = $200) , P_y = 1, and that they spend all of their money on goo
- Suppose a consumer has preferences represented by the utility function U(X,Y) = MIN[2X,Y]. Suppose P_X = 1 and P_Y = 2. Sketch the graph of the consumer's Engel Curve for good X.
- A consumer's preferences can be represented by the utility function U(X,Y) = Min (2X,Y). The consumer has $300 to spend and the price of Good X is PX = $2 and the price of Good Y is PY = $5. If the consumer maximizes their utility subject to their budget