# Suppose that Ashley likes 2 goods, good 1 (x_1) and good 2 (x_2). Her utility function is given...

## Question:

Suppose that Ashley likes 2 goods, good 1 ({eq}x_1 {/eq}) and good 2 ({eq}x_2 {/eq}). Her utility function is given by {eq}U(x_1, x_2) = x_1^2 + x_2 {/eq}. Ashley's income is denoted as {eq}M {/eq} and the cost of good 1 (per unit) is {eq}P_1 {/eq} and the cost of good 2 (per unit) is {eq}P_2 {/eq}.

1. Write down Ashley's utility maximization problem. Remember that she is constrained by her budget.

2. Given the above problem, assume that the budget constraint binds, i.e., set it as equal. Now solve for the optimal {eq}x_1 {/eq} and {eq}x_2 {/eq}. Label them as {eq}x_1^* {/eq} and {eq}x_2^* {/eq}.

3. From your answer in part 2, determine whether good 1 and good 2 are substitutes or complements. Show this mathematically and explain your answer.

4. What type of utility function has been used here? What is so special about this type of function?

## Utility Functions

In economics, the happiness or satisfaction a consumer gets from consuming a good or service is called utility, which can be modeled with utility functions. The goal of a consumer is to maximize their utility with respect to their budget constraints.

## Answer and Explanation: 1

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

View this answer1. Ashley's utility maximization problem with her budget constrain can be written down as:

{eq}M = P_{1}x_{1}+P_{2}x_{2} {/eq}

2. The optimal point...

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 3 / Lesson 2Learn 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 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
- Kathy's utility function is U = 4X + 3Y. Kathy has $180 to spend on goods X and Y. The price of good X is $6 and the price of good Y is $4. To maximize utility, Kathy should consume [{Blank}] units of good X and [{Blank}] units of good Y and will be recei
- You have an income of $150 to spend on two goods. Good 1 costs $10 per unit and good 2 costs $5 per unit. (1) Suppose now that a uniform sales tax rate is 0.5 (equivalently, 50% of the price) for bo
- You have an income of $150 to spend on two goods. Good 1 costs $10 per unit and good 2 costs $5 per unit. 1) If you want to buy 2 extra units of good 2, how many units of good 1 should be given up? 2)
- Consider the following information: Your income is equal to $12. You are to spend that income on Good X and Good Y. Good X costs $2 per unit and Good Y costs $3 per unit.
- Janet consumes two commodities x and y. Her utility function is U = min {x+2y, y+2x}. She chooses to buy 10 units of good x and 20 units of good y. The price of good x is $1. Janet's income is? a.50 b.30 c.40 d.20
- 1) The price of Good X is 1 for the first 50 units, and 0.50 thereafter. The price of Good Y is 1 for the first 50 units, and 0.50 thereafter. Income is 100. If the utility function is U = X^0.5Y^0.5,
- 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?
- Clara's utility function is U(x,y) = (x + 2)(y + 1). If her MRS = 0.5 and she's consuming 8 goods of Good X, how many units of good Y is she consuming? Show work.
- Suppose Country A can produce either 800 units of Good S or 600 units of Good T. Also, suppose there is an increasing opportunity cost to switching production from Good S to Good T. Use this informati
- Ester consumes goods X and Y, and her utility function is U(X,Y)=XY+Y. For this utility function: MU_x=Y MU_y=X+1 a. What is Ester's MRS_{xy}? b. Suppose her daily income is $20, the price of X is $4 per unit, and the price of Y is $1 per unit. What is h
- Consider a two person (A and B) exchange economy with two goods (1 and 2). There are ten units of good 1 and fifteen units of good 2 in the economy. Person A's utility function is U A(x1, x2) = min{2x1, x2} where xi is this person's consumption of good i
- Suppose we have a two good economy that produces beef and soy. The price of a unit of beef = $19 and the price of a unit of soy = $14. Suppose also that we produce 125 units of beef and 460 units of soy. A) What is the value of beef produced in our econom
- 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
- Consider a two person (A and B) exchange economy with two goods (1 and 2). There are ten units of good 1 and fifteen units of good 2 in the economy. Person A s utility function is UA(x1, x2) = min{2x1, x2} where xi is this person s consumption of good i &
- In a market economy, will units of a good be produced and purchased if consumers value them more than their cost of production? Explain. If the production cost per unit of a good exceeds the value der
- If a business produces and sells only one unit of a good, its profit would be the a. price received for the good b. price of the product minus the cost of the resources used to produce the product c.
- Assume a firm produces 500 units of a good by using two inputs, capital and labor, whose per unit prices are $10 and $4. Assume also that the marginal physical product of the last unit of capital is 30 and the marginal physical product of the last unit of
- Assume a firm is producing 1000 units of a good by using two inputs, capital and labor, whose per unit prices are $50 and $20. Assume also that the marginal physical product of the last unit of capita
- Suppose wages are 16, the rental rate of capital is 3, and the production function is F(L,K)=2L^1/4K^3/4 If the producer wants to make exactly 160 units of the good, which of the following is the tota
- Consumer X purchases a unit of a normal good for $300. The value-added of this good represents 90% of the perceived benefits. The supplier spends $50 on the production of a single unit. a. What is consumer's maximum willingness to pay for a unit? b. How
- Tom's income is $480 and he spends it on two goods, X and Y. His utility function is U = XY. Both X and Y sells for $8 per unit. a. Calculate Tom's utility-maximizing purchases of X and Y. b. How will his utility change if his income decreases by $2.00? c
- Eva's monthly income is 300 Euros. Her utility function is given by U(x,y) =x + 72y-3y^2, where x and y denote the quantity consumed of good x and good y, respectively. The market prices of both goods
- A consumer has budgeted a total of $225 to spend on two goods, X and Y. She likes to consume a unit of good X in combination with good Y. Any unit of X she cannot consume with Y is useless. Similarly, any unit of good Y she cannot consume with good X is u
- There are two goods, A and B. Suppose the production possibilities (PPF) of a country is given by Q_B = (2,500 - 0.25Q_A^2)^{1/2}. The slope of this PPF is -0.25Q_A/Q_B. 1. If the relative price of good A in terms of good B is 2/3, how much of good A will
- Bob can produce either 18 units of good X or 44 units of good Y in a day. If Bob wants to produce 4 units of good X then how many units of good Y can he produce? Assume Bob has constant opportunity
- Comparative advantage is defined as a situation in which one person can produce: a. more of all goods than another person. b. a good for a lower opportunity cost than another person. c. a good for a lower dollar cost than another person. d. more of a good
- In a two-commodity world, suppose that Winnie's preference can be characterized by the utility function U_W(X, Y) = lnX + lnY + 2018, and Tigger's utility is given by U_T(X, Y) = x^{1/3}Y^{1/3}. Good X costs $2 per unit, and Y is a composite good. Both Wi
- Suppose a firm charged $4 per unit when it produced 100 units of good X, and it charged $3 per unit when it produced 200 units. Furthermore, the firm earned the same profit per unit in both cases. How can this happen?
- I. Marginal benefit of a good is A. the cost of producing one more unit. B. dependent on a person's preferences for the good. C. the opportunity cost of producing one more unit. D. increasing as m
- Suppose a firm's production function is given by the equation Q = 12L^.5K^.5 . This firm operates in the short run where capital (K) is fixed at a quantity of 16. If the price per unit of the good is $1.9 and labor costs $10 per unit. Then the profit-maxi
- Cost functions, a part of the definition of profit, are useful to gauge the performance of the business. Suppose an economist estimated that the cost function of a single-product firm as: C(Q) = 40 +
- A company could produce 99 units of a good for $324 or produce 100 units of the same good for $330. What is the marginal cost of the 100th unit?
- In the country of Zana, a unit of labor can produce 4 units of good X or 2 units of good Y; in the country of Bren, a unit of labor can produce 2 units of good X or 1 unit of good Y. Which of the following is true? a. Bren has absolute advantage in prod
- A person's willingness to pay for a good is based on a. the availability of the good. b. the marginal benefit that an extra unit of the good would provide for that person. c. the marginal cost of producing an extra unit of the good. d. esoteric factors, t
- In the country of Zana, a unit of labor can produce 4 units of good X or 2 units of good Y; in the country of Bren, a unit of labor can produce 2 units of good X or 1 unit of good Y. Which of the following is true? a. Bren has an absolute advantage in pro
- Bottom of Form Cost functions, a part of the definition of profit, are useful to gauge the performance of the business. Suppose an economist estimated the cost function of a single-product firm as C(Q) = 40 + 4Q + 4Q^2 + 4Q^3. Define and compute the avera
- Bottom of Form Cost functions, a part of the definition of profit, are useful to gauge the performance of the business. Suppose an economist estimated the cost function of a single-product firm as C(Q) = 40 + 4Q + 4Q^2 + 4Q^3. Define and compute th
- Bottom of Form Cost functions, a part of the definition of profit, are useful to gauge the performance of the business. Suppose an economist estimated the cost function of a single-product firm as C(Q) = 40 + 4Q + 4Q^2 + 4Q^3. Define and compute
- Please show how to setup and solve. Assume a firm produces 500 units of a good by using two inputs, capital and labor, whose per unit prices are $10 and $4. Assume also that the marginal physical prod
- Bob can produce either 13 units of good X or 41 units of good Y in a day. If Bob wants to produce 5 units of good X then how many units of good Y can he produce? Assume Bob has constant opportunity c
- In the factor endowment model, suppose the price of the labor intensive good decrease relative to the capital intensive good. using unit isoquants and vector analysis, explain the effects on the wage
- Assume Holy consumes two goods (X and Y). Given his utility function U = min {2X, 3Y}, find Holy's optimum amount of X and Y as functions of the price of X (PX), price of Y (PY) and his Income (M)
- Firm A charged $4 per unit when it produced 100 units of good X, and it charged $3 per unit when it produced 200 units. Furthermore, the firm earned the same profit per unit in both cases. How can this happen?
- Firm A charged $4 per unit when it produced 100 units of good X, and it charged $3 per unit when it produced 200 units. The firm earned the same profit per unit in both cases. How can this happen?
- The budget constraint cuts the horizontal axis at 12 units of good X and it cuts the vertical axis at 20 units of good Y. If the price of good X is $20 and the price of good Y is $12, then what does income equal? a. $200. b. $240. c. $520. d. $280. e. Not
- Consider a profit-maximising, price-taking firm that only uses capital as input at a cost of v per unit. Its production function is given by f(k) = 11 - \frac{1}{1 + k}. What is its cost function (whe
- There are two goods, A and B. Suppose the production possibilities frontier (PPF) of a country is given by Q_B = (100 - 2Q_A^2)^{1/2}. The slope of this PPF is -2Q_A/Q_B. Finally, assume that the price of good A is 12 and the price of good B is 4. How muc
- Assume a firm can generate 5q1 units of profit for each unit of output q1. Assume that each unit of production produces 0.2 times q1 squared units of pollution. Suppose the firm is endowed with one em
- Suppose the total utility a person derives from two goods (x & y) can be described by the equation: U(x,y) = sqrt(xy) Units of y cost $10 each while the cost of each unit of x is dependent on the numb
- Suppose that production for good X is characterized by the following production function, Q=4(K)^{0.5} (L)^{0.5}, where K is the fixed input set at K=9 in the short run.If the per-unit rental rate of
- John consumes two goods, X and Y, given the following utility function U = X^3Y^2. The price of X is P_X = $10 and the price of Y is P_Y = $20. A. If John's income (M) is $1,500, what is his maximizing utility bundle? B. Say that P_X falls by $5. What is
- A firm's fixed costs are $600 regardless of the output; Variable costs are $5 per unit of output. TC = FC + VC. The selling price of the good is $10 per unit. A) State the FC function, VC function,
- Suppose that a firm has the following production function: Q(K, L) = 2LK^{1/2} a. If the price of labour is $2/unit and the price of capital is $4/unit, what is the optimal ratio of capital to labou
- Real income in terms of a good is defined as: a) dollar income. b) income divided by the price of a good. c) income divided by the quantity consumed of a good. d) the income of a producer of that good. e) the price of one good divided by the price of anot
- Katie's utility function is U(x_{1}, x_{2}) = 2(lnx_{1})+ x_{2}. Given her current income and the current relative prices, she consumes 10 units of x_{1} and 15 units of x_{2}. If her income doubles,
- The slope of the PPF can also be expressed as: a) The ratio of abundance of labor to capital, b) The ratio of the marginal products of labor to the marginal product of capital, c) Consumer utility, d) The opportunity cost of the good measured on the ve
- George has $49 which he decides to spend on x and y. Commodity x costs $5 per unit and commodity y costs $11 per unit. He has the utility function U(x, y) = 3x^2 + 6y^2 and he can purchase fractional
- Suppose the production function is given by Q=3K+4L. What is the average product of capital when 10 units of capital and 10 units of labor are employed? Answer choices are a. 3 b. 4 c. 7 d. 45
- Cost functions, a part of the definition of profit, are useful to gauge the performance of the business. Suppose an economist estimated that the cost function of a single-product firm as: C(Q) = 40 + 4Q + 4Q^2 + 4Q^3. Based on this information, determine
- Consider a commodity money model economy but with the following features: there are 100 identical people in every generation. Each individual is endowed with 10 units of the consumption good when youn
- A company could produce 99 units of a good for $316 or produce 100 units of the same good for $320. The marginal cost of the 100th unit A. is $320. B. is $3.20. C. is $4.00 D. The cost cannot be calculated with this information.
- What is the value of economic rent given in the diagram below? Show all your work. Also, notate (in the appropriate segment) whether the income is from transfer earnings or economic rent.
- Sarah's utility function for housing (H) and other consumable goods (C) is U = H.4 C.6 a. Show mathematically that her marginal utility from each good is diminishing. b. If her monthly income is $30
- An economy's production possibilities boundary is given by the mathematical expression 45 = A + 2B, where A is the quantity of good A and B is the quantity of good B. In what way does the combination of 30 units of good A and 7 units of good B represent
- Adam has $38 and he decides to spend on X1 and X2. Commodity one costs $5 per unit and commodity two costs $11 per unit. His utility function is U(X1,X2)= min(3X1,X2) . a. Write down the budget line.
- Suppose that those in charge of this economy want to maximize the value of the output produced. Each unit of X sells for $6 and each unit of Y sells for $1, so that the value of the output can be represented by the equation V = 6X + Y. The point on the PP
- Individuals A and B can both produce good X. We say that A has a comparative advantage in the production of good X if: a. A has a higher opportunity cost of producing X than B. b. A can produce less units of X in a given time period than B. c. A can produ
- Suppose the total cost function for production in the short run is: TC(q) = 100 + 25q^2 Also, the production function is: q = 10L If a producer wants to make 100 units of the good, then she would ne
- If the marginal product of a worker is 10 units and each unit of the good is sold for $5, the value of the marginal product of the worker is [{Blank}]. a. $50 b. $2 c. $10 d. $5
- Sarah's utility function for housing (H) and other consumable goods (C) is U = H^4 C^6 a. Show mathematically that her marginal utility from each good is diminishing. b. If her monthly income is $3000
- A perfectly competitive firm produces three types of goods, A, B, and D. The weekly cost of producing a units of good A, b units of good B and d units of good D is: The firm sells its output at $102 p
- 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^
- Sarah's utility function for housing (H) and other consumable goods (C) is a. Show mathematically that her marginal utility from each good is diminishing. b. If her monthly income is $3000, housing co
- 1. It would be good practice to be able to compute revenue and profit as well. Such as: If the firm sells 5 units for $12 each what is the economic profit based on the above costs of production. a.
- Consider a firm who uses 'L' units of labor and 'K' units of capital to produce a good. Let its production function be denoted as: F(K, L) = (alpha)ln (K) + (beta)ln(L), where alpha is greater than 0
- Utility is a balance between ________ and ________ factors. A. economic; personal B. transitive; intransitive C. capital; labor D. good; bad E. production; consumption
- Suppose that Nadine has a production function 3x1 + x2. If the factor prices are 3 for factor 1 and 3 for factor 2, how much will it cost her to produce 20 units of output?
- When the costs of a good are paid for by nonconsumers, _______ of the good is produced. A) an inefficiently small amount B) an efficient amount C) an inefficiently large amount D) none
- The max units of good A are 180. The max units of good B are 60. In general terms, what happens to the opportunity cost of good A as the output of good A increases? In general terms, what happens to t
- 4) A firm produces Good X, using inputs 'L' and 'K.' The production function is x = K^0.5 L^0.5. Assume that the price of L is $15 and the price of K is $20. Find the firm's average total cost of prod
- Which of the following is most appropriately measured along one axis of the production possibilities frontier diagram: a) Society's welfare and satisfaction, b) The price of a produced good, c) The quantity of a good consumed, d) The quantity of a good pr
- Suppose a firm produces an output measured in units Q. The cost of producing Q units is given by the cost function C(Q) = aQ^2 + bQ + c, where you can assume a 0,b 0,c 0. In Economics we also think about cost per unit (average cost) given by: AC(Q) = C(
- Suppose it has been determined that the marginal revenue associated with the production of x units of a particular commodity is R'(x) = 240 - 4x dollars per unit. What is the revenue function R(x)? You may assume R(0) = 0. What price will be paid for each
- Suppose the production function for good q is given by q=3k+2l where k and l are capital and labor inputs. a. What is the return to scale for this function? b. What is the RTS of this function?
- 1) A result of a positive externality in the production of a good is that: a) The price system will over-allocate resources to the production of that good or service. b) The price system will under-
- Suppose that the price of labor is $40 per unit and the price of capital is $120 per unit. In the short run, when L = 20, the marginal cost (computed to the nearest penny) is: A) $0.10 B) $1.00 C) $8.00 D) $20.00 E) $40.00 F) $60.00 G) $80.00 H) $120.00 D
- Consider an economy with the following Cobb-Douglas production function: Y = F(K, L) = K^1/3 L^2/3. A. Derive the equation describing labor demand in this economy as a function of the real wage and the capital stock. B. The economy has 27,000 units of cap
- 1. Suppose that a firm's production function is q = 10L^{1/2}K^{1/2}. The cost of a unit of labor is $20 and the cost of a unit of capital is $80.a. If the firm wishes to produce 100 units of output,
- Suppose a firm with a production function given by Q=4K^0.25L^0.75 produces 100 units of output. The firm pays a wage of $30 per units and pays a rental rate of capital $10 per unit. The minimum cost
- Assume a firm has a Cobb-Douglas production function Y = L^{0.5} K^{0.5} . Assume (w) wage = $1, (r) rental = $2 and price of output (p) = $5 and firm has linear cost function. What is the marginal
- John consumes two goods, X and Y, given the following utility function: U = X^3Y^2 The price of X (Px) = $10 , and the price of Y (Py) = $20. a) If John's income (M) is $1,500, then his maximizing utility bundle is: X = Y = b) Say the Px falls by $5. The
- Suppose we are given a profit function Q = 12L0.5K0.5. The price of labor (L) is $6 per unit and the price of capital (K) is $6 per unit. The firm is interested in the optimal mix of inputs to minimize the cost of producing any level of output Q. In the o
- Suppose we are given a profit function Q = 12L^.5K^.5. The price of labor is $6 per unit and the price of capital (K) is $7 per unit. The firm is interested in the optimal mix of inputs to minimize the cost of producing any level of output Q. In the optim
- A person has an absolute advantage in producing a good when he/she: a) can produce the good at a lower opportunity cost than anyone else. b) has a comparative advantage in producing that good. c) can produce more of that good than anyone else, using the s
- Suppose that production for good X is characterized by the following production function, Q = K0.5L0.5, where K is the fixed input in the short run. If the per-unit rental rate of capital, r, is $15 a
- Suppose the marginal product of labor is currently equal to its average product. If you were one of 10 new workers the firm was about to hire, would you prefer to be paid the value of your average pro
- Opportunity cost is defined as: a. the value of the least desired alternative sacrificed to obtain another good or service, or to undertake another activity. b. the monetary value of obtaining a good
- In terms of the factor proportions model, Nation 1 can produce good X with 2 units of labor and 3 units of capital while Nation 2 produces good Y with 3 units of labor and 4 units of capital. It can be concluded: A. That good Y requires more capital and i
- Suppose that an economy produces 300 units of output, employing the 50 units of input, and the price of the input is $9 per unit. The level of productivity and the per-unit cost of production are, respectively: a. 1.50 and $6.00. b. 6 and $1.50. c. 5 and