Microeconomic Theory
12th Edition
ISBN: 9781337517942
Author: NICHOLSON
Publisher: Cengage
expand_more
expand_more
format_list_bulleted
Concept explainers
Question
error_outline
This textbook solution is under construction.
Students have asked these similar questions
Khan lives in a world with two consumption goods x and y. Her utility function is U (x, y) = √x² + y². a. If px = $3,py = $4, and her income, I, is equal to $50, what will be the quantities of x and y that Maya should buy to maximize her utility?
Make sure that you write out the Lagrangian and the first-order conditions. (Hint: It may be easier to maximize U² than U).
Have you found a true maximum? Explain your answer.
Rafe is optimally choosing to consume 6 apples and 3 bananas. The prices of apples
and bananas are p. = 7 and pb - 7. Which of the following utility functions over
quantities of apples (a) and bananas (b) could represent Rafe's preferences?
=
u(a, b)-a4/5 61/5
Ou(a, b) = a¹/5 64/5
Ou(a, b)-a2/3 b1/3
Ou(a, b)-a¹/3 2/3
Two friends, Minrui and Jing, share a flat and both consume internet (i) and all other goods (g). The
utility function for Minrui is U_m=i^(0.3)g^(0.7) whereas the utility function for Jing is U_m=i^(0.1)g^(0.9).
Considering that the income of both individuals are the same $500 and that the prices are p_i = $80
and p_g = $5, what would be the optimal allocation of public good (i) and private good (g)? Interpret
your answer.
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, economics and related others by exploring similar questions and additional content below.Similar questions
- Your preferences are represented by the utility function U = (x9.5 + x2:5)², where the price of good 1 is 4 PHP and the price of good 2 is 3 PHP. Your income is given by 50 PHP. • Set up the Lagrangean function for this utility-maximization problem with constraint. Compute for the utility-maximizing quantities of x₁ and 22. • What is the maximum level of utility you can attain given your utility function and budget con- straint?arrow_forward1.Suppose that Chris's utility function is given by UC=QC1/2 RC1/2 , where QC and RC are his consumption of Q and R, respectively. Dana's utility function is given by UD=QD1/3 RD2/3, where QD and RD are her consumption of Q and R, respectively. Write an equation for the marginal rate of substitution (MRS) between Q and R for each of the two agents. 2.Suppose that the price of good R is pR=1 and the price of good Q is pQ=2. How much is Chris's and Dana's initial income, given his endowments and given these prices? 3. At these prices, how many units of Q would Chris and Dana want to consume? 4. At these prices, how many units of R would Chris and Dana want to consume?arrow_forwardIf a and ß are just parameters and if the direct utility function is u (x1, X2) = a · In(x1) + ß · In(x2) %3D then what is the marginal utility of x1 and what is the marginal utility of x2 ? Demonstrate whether these marginal utilities are increasing, diminishing, or constant. Assume that the following indifference curve belongs to the direct utility function specified above. u - 1.565 G (5, 4) 4 Н (10, 0.25) 0.25 5 10 1 What is the value for M RS12 evaluated at bundle G(5,4) if a = 0.8 and B = 0.2 ? What is the value for M RS12 evaluated at bundle H(10, 0.25) ? Discuss what these calculated values for M RS12 imply regarding whether preferences are convex or not (regardless of what the figure depicts above). Show your work!arrow_forward
- A consumer chooses non-negative consumption levels of two goods x1 and x2 and has a utility function given as: U = = 0xfx% where 0 < a <1 and 0 is a constant. Assume that the prices of goods x1 and x2 are respectively Piand p2 Set up the expenditure minimization problem of the consumer and derive the Compensated (Hicksian) demand function for goods 1 and 2 and investigate whether the two goods are complements or substitutes. i.arrow_forwardSuppose Jimi has reference dependent preferences over guitars and money as in Tversky and Kahneman (1991). His utility functions are given below. Gains Gains 400 -2 -2 2 Guitars Losses Losses |-600 -2 What is the least amount of money Jimi is willing to accept to sell one of his guitars? (just enter a dollar amount, i.e., "1000", not "$1000"arrow_forwardConsider an economy with one consumption good, 100 identical consumers and 100identical firms. Each consumer is endowed with one unit of time and one unit ofcapital. The agent can spend time either working or enjoying leisure. A represen-tative consumerís utility function is u (x; l) = ln x + 2 ln l, where x is consumptionof goods and l is leisure. Each firm hires consumers to work and rents capital fromconsumers to produce goods: y = f (K; L) = K^1/2L^1/2, where L is the amount oflabor hired and K is the amount of capital rented. Both consumers and firms takegoods price p, wage rate w and rental rate r as given. Normalize p = 1. 1. Set up a representative consumerís utility-maximization problem.Derive the Marshallian demands for consumption and leisure as functions ofwage w and rental rate r 2. Set up a representative firm's profit-maximization problem. Writedown the first-order conditions regarding the choices of K and L 3. Set up all the market-clearing conditions. 4. Use the…arrow_forward
- An agent has income m that can be spent on frequent flier miles f or on other goods – a “composite good” g. Their respective prices are: pf = 10 per mile and pg = 1 per unit. The flier miles have a stepwise price schedule. After the first 25 miles the price is reduced by 20% and after 50 miles the price is further reduced by another 50%.1. Put g on the vertical axis and f on the horizontal axis. Assume m = 200, and draw the budget constraint with all the intercepts and appropriate slopes. 2. On a separate graph, repeat part (1) for m = 300. 3. On a separate graph, repeat part (1) for m = 600.arrow_forward2. Jie has the utility function U, (a, b) = 2a + b and Daokui has the utility function U,(a, b) = v2ab, where a is the number of apples consumed and b is the number of bananas consumed. There are 10 units of apple and 10 units of banana in total. We draw an Edgeworth box with apple on the horizontal axis and banana on the vertical 4 axis and we measure Jie's consumptions from the lower left corner of the box. We assume that both apple and banana are infinitely divisible. Please answer the following questions. (a) Write down all pareto optimal allocations (you do not need to explain why). (b) If social welfare function is W min{U,(a, b), U(a, b)}, derive the social optimal allocation. Is this allocation pareto optimal? Suppose the endowment of Jie is 7 units of apple and 0 units of banana, and the endowment of Daokui is 3 units of apple and 10 units of banana. Please answer the following questions. (c) Assume that the price of apple is p, and the uniform price of banana is 1. Derive the…arrow_forward1. For each of the following scaling functions for a von Neumann-Morgenstern utility func- tion, determine the Marginal Rate of Substitution between X1 and X2 and the equation for an indifference curve through the consumption bundle (100,100) (solve for X2 on the left hand side of the equation). State 1 is the bad outcome that occurs with probability 0.2 and State 2 the good outcome that occurs with probability 0.8. Graph these indif- ference curves and comment on what you see. Is a consumer with these preferences risk averse, risk neutral, or risk loving? (a) V(X) = InX (b) V(X)= VX (c) V(X)= X (d) V(X)= X²arrow_forward
- 2. Consider a consumer whose preference relation over the consumption set X = R can be repre- sented by the utility function: u (11, 12, 13) = A (x1 – #1)° (x2 – #2)® (r3 – 3)" where A, a, B and y are all strictly positive. (a) Why can you assume A = 1 and a + 3 + y = 1 without loss of generality? Do so for the %3D rest of the question. (b) Solve the UMP and derive the consumer's Walrasian demand. (c) Find the indirect utility function, the minimum expenditure function, and the Hicksian demand function.arrow_forward2. Consider an individual who not only consumes goods x and y, but also needs a mini- mum amount of x = xo to survive. Her (Stone-Geary) utility function is given by U(x, y) = (x — xo)¹/²y¹/² Let prices be pa, Py and income is I > Pro. (a) Show that this individual maximizes utility by spending (I-Pro) + Pro on good rand(I - Pro) on good y. Provide an economic interpretation for this result. [Hint: It might help to define z = x - xo and I = I - Pro. Then find the optimal z (and therefore x = z + xo) and y]arrow_forwardA consumer can consume two goods, A and B, and has the utility function U-15A1/281/2, The consumer's budget is $900, the price of good A is $15 per unit and the price of good B is $45 per unit. (Assume A is the horizontal axis good and B is the vertical axis good. Both goods are infinitely divisible, but round numerical answers to 2 decimal places as necessary) What is the consumer's marginal rate of substitution? What is the consumer's marginal rate of transformation? What is the formula for the consumer's budget constraint? What is the consumer's utility-maximizing bundle given the utility function and budget constraint? A B Harrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
