A consumer has preferences over consump- tion bundles that are strongly monotone, strictly convex, and represented by the following (differentiable) utility function: The first section consid-ers the problem in consumer theory of maximization of the utility function with a fixed amount of wealth to spend on the commodities. Will she borrow or save in the first period. For each of the following situations, decide whether Al has increasing, constant or diminishing marginal utility. endstream endobj 766 0 obj <>/Metadata 21 0 R/PageLayout/OneColumn/Pages 763 0 R/StructTreeRoot 78 0 R/Type/Catalog>> endobj 767 0 obj <>/ExtGState<>/Font<>/XObject<>>>/Rotate 0/StructParents 0/Type/Page>> endobj 768 0 obj <>stream It is the increase in the level of utility that would be achieved if income were to increase by one unit. Problem Set 2 (Consumer Choice and Utility Maximization) 1. General rules for problem sets: show your work, write down the steps that you use to get a solution (no credit for right solutions without explanation), write legibly. To solve this problem, you set up a linear programming problem, following these steps. 1.1 Commodity and Price 783 0 obj <>/Filter/FlateDecode/ID[<20ABC59884C0674C94CC958B65169113><47D85C1C77248E47A2863CF3B1107D9B>]/Index[765 61]/Info 764 0 R/Length 92/Prev 65750/Root 766 0 R/Size 826/Type/XRef/W[1 2 1]>>stream In particular, solve for C t+1 from the constraint: C t+1 = (1 + r t)(Y t C t) + Y t+1 Plug this back into the lifetime utility function, re-writing the maximization problem as just being over C t: max Ct U= u(C t) + u((1 + r t)(Y t C t) + Y t+1) Output in each period Y 1 and Y 2 respectively, is given exogenously. Example: Imagine that the utility function is U(x,y)=5xy2, p x=2 and py=8 and I=240. 1. Utility maximization. Currently, her marginal utility from one more flowbot would be 40 and her marginal utility from one more robotron would be 30.Which of the following statements … Get help with your Utility maximization problem homework. (52 points) In this exercise, we consider a standard maximization problem with an unusual utility function. His optimal consumption bundle is $(x_1, x_2) = (1,1)$. 0 Lecture 7: Utility Maximization Advanced Microeconomics I, ITAM, Fall 2020 Xinyang Wang 1 The Consumer Problem In this section, rst, we introduce the dual concepts of commodity and price. COMMON ERRORS: (1) Some of you solved a utility maximization problem instead of the expenditure-minimization problem that is needed. a) Solve the utility maximization problem for a representative consumer. The more economics classes Al takes, the more he enjoys the subject. It is focused on preferences, utility functions, and utility maximization. Will Mainy be better or worse off? Uncertainty Jonas Thern maximises expected utility: U(π 1, π 2,c 1,c 2) = π 1 c 1 + π 2 c 2 The utility function is u(x,y)= √ x+ √ y. the constraint optimization problem is max x 1;x 2 x 1 x 1 2 subject to p 1x 1 + p 2x 2 = I. Problem 1. Write an expression for the objective function using the variables. The price of good xis pxand the price of good yis py.We denote income by M,as usual, with M>0.This The utility function is u(x,y)= √ x+ √ y. Choose variables to represent the quantities involved. Utility Maximization . Let t represent the number of tetras and h represent the number of headstanders. This is OK provided you then invert the indirect utility function to get the expenditure function, and some did not do this. A Utility Maximization Example Charlie Gibbons University of California, Berkeley September 17, 2007 Since we couldn’t nish the utility maximization problem in section, here it is solved from the beginning. Utility Maximization . ... if freds marginal utility for pizza equals 10 and his marginal utility of salad equals 2, then a. he would give up 5 salads to get next pizza ... utility is the set of numerical values that In particular, solve for C t+1 from the constraint: C t+1 = (1 + r t)(Y t C t) + Y t+1 Plug this back into the lifetime utility function, re-writing the maximization problem as just being over C t: max Ct U= u(C t) + u((1 + r t)(Y t C t) + Y t+1) COMMON ERRORS: (1) Some of you solved a utility maximization problem instead of the expenditure-minimization problem that is needed. e. = d, but the interest rate is 20%. We consider three levels of generality in this treatment. Utility MaximizationConsumer BehaviorUtility MaximizationIndirect Utility FunctionThe Expenditure FunctionDualityComparative Statics Fig. h�bbd``b`f�@�i���s��9 ��bi��%�� For each of the following situations, decide whether Al has increasing, constant or diminishing marginal utility. (Or, after losing one unit of x Problem 1: Utility maximization. (52 points) In this exercise, we consider a standard maximization problem with an unusual utility function. %PDF-1.5 %���� This is OK provided you then invert the indirect utility function to get the expenditure function, and some did not do this. Answers to Problem Set 3 0. [14 points] b) Set up the firm’s profit maximization problem and find the FOCs. x ^ is the optimal choice for income m.If the light shading is the preferred set for x ^ then we obtain the lowest possible isoexpenditure line subject to this preferred set by choosing x ^ as the Hicksian demand point, in which case expenditure minimization coincides with utility maximization. %%EOF Write down the solution • copy directly from the solution to the firm’s problem 5. 285 0 obj <>stream The Engel curve for good 2 is the graph of Y = x2, which is the 45-degree line. a. %%EOF Problem Set 2: Solutions ECON 301: Intermediate Microeconomics Prof. Marek Weretka Problem 1 (Marginal Rate of Substitution) ... utility level), a consumer is willing to give up 9=10 of x 2 for one additional unit of x 1. 1. We solve this maximization by substituting the budget constraint into the utility function so that the problem becomes an unconstrained optimization with one choice variable: u(x 1) = x 1 I p 1x 1 p 2 1 . (c) Given Y, utility is maximized at (x1;x2) = (0;Y). In microeconomics, the utility maximization problem is the problem consumers face: "how should I spend my money in order to maximize my utility? And the more classes he takes, the easier each one gets, making him enjoy each additional class more than the one before. Utility Maximization . The problem of finding consumer equilibrium, that is, the combination of goods and services that will maximize an individual’s total utility, comes down to comparing the trade-offs between one affordable combination (shown by a point on the budget line in Figure 1, below) with all the other affordable combinations.. Solution. Jack has a utility function for two perfectly divisible goods, x and y. Jack’s utility function is u(x;y) = (x+y)2. h�bbd``b`:$��X[��C ��H�I�X�@�9 D�/A+�`] Problem 1. the constraint optimization problem is max x 1;x 2 x 1 x 1 2 subject to p 1x 1 + p 2x 2 = I. 1 Show that the solution is equivalent to another problem • the dual problem 3. 260 0 obj <>/Filter/FlateDecode/ID[<629F7BB8BCA47347A66496A906E6B75E><1855F8D0D7557F4EA3666F839EFFDE29>]/Index[241 45]/Info 240 0 R/Length 94/Prev 34718/Root 242 0 R/Size 286/Type/XRef/W[1 2 1]>>stream Problem 1. Consider the utility maximization problem max U (x, y) = √ x + y s.t. 10.2.Utility maximization implies expenditure minimization. Set out the basic consumer optimisation problem • the primal problem 2. Write an expression for the objective function using the variables. an interior solution to a consumer's utility maximization problem implies. utility maximization problem. Example of duality for the consumer choice problem Example 4: Utility Maximization Consider a consumer with the utility function U = xy, who faces a budget constraint of B = P xx+P yy, where B, P x and P y are the budget and prices, which are given. Utility MaximizationConsumer BehaviorUtility MaximizationIndirect Utility FunctionThe Expenditure FunctionDualityComparative Statics Utility Units 0 1 2 3 4 5 6 7 Total Utility 0 20 35 45 50 50 45 35 Notice that production set is linear over some range and then starts to exhibit increasing returns to scale. Problem set 1 ECON 4330 Part 1 We are looking at an open economy that exists for two periods. Here is the constraint set of the consumer, along with a few indifference curves: Observe that the constraint set is convex and the consumer does not spend all his income in optimum. Problem Set . Get help with your Utility maximization problem homework. To nd Pareto optimal allocation we need solve two maximization subproblems and then compare utility levels. Utility maximisation must be seen as an optimisation problem regarding the utility function and the budget constraint.These two sides of the problem, define Marshallian demand curves.. An individual is therefore faced with the following problem: faced with a set of choices, or baskets of goods, and a fixed budget, how to choose the basket which maximises their utility? 0 x ^ is the optimal choice for income m.If the light shading is the preferred set for x ^ then we obtain the lowest possible isoexpenditure line subject to this preferred set by choosing x ^ as the Hicksian demand point, in which case expenditure minimization coincides with utility maximization. L = labor q = consumption. A Utility Maximization Example Charlie Gibbons University of California, Berkeley September 17, 2007 Since we couldn’t nish the utility maximization problem in section, here it is solved from the beginning. unconstrained, univariate optimization problem by eliminating the constraint. 3.2 Utility-maximizing worker Convert to a problem with positive variables. There two goods, X and Y , available in arbitrary non-negative quantities (so the consumption set is R2+). Derive Jack’s demand function for the two goods as a function of px (the price of good x), py (the price of good y), and I, (Jack’s total income to be allocated to the 2 goods). (a) By solving the following utility maximization problem, max x 1 2 1 x 1 2 2 s:t: p1x1 +p2x2 = Y we have x1 = Y=2p1 and x2 = Y=2p2. unconstrained, univariate optimization problem by eliminating the constraint. 241 0 obj <> endobj Choose variables to represent the quantities involved. 1. endstream endobj startxref The set of available bundles for the consumer is given by: B p;m = fx 2X : px mg Then, the utility maximization problem is expressed as, max x u(x) subject to px m and x 2X. Problem Set . Utility maximization. Fig. Yu$��wȀj !=$� $��f`bd�I00��� �� The more economics classes Al takes, the more he enjoys the subject. And the more classes he takes, the easier each one gets, making him enjoy each additional class more than the one before. utility the consumer can achieve when facing a given set of prices with income I? Show that this problem is identical to that of the firm 4. Utility Maximization Problem. x + 4 y = 100 (a) Using the Lagrange multiplier method, find the quantities demanded of the two goods. x + 4 y = 100 (a) Using the Lagrange multiplier method, find the quantities demanded of the two goods. Be very careful in writing the budget constraint as the consumer has many sources of income in this model. In microeconomics, the utility maximization problem is the problem consumers face: "how should I spend my money in order to maximize my utility? This Problem set tests the knowledge that you accumulated in the lectures 5 to 8. endstream endobj 242 0 obj <. (b) Suppose income increases from 100 to 101. Let t represent the number of tetras and h represent the number of headstanders. For 0 x 1 20, the problem is max x 1;x 2 logx 1 + logx 2; s.th. We solve this maximization by substituting the budget constraint into the utility function so that the problem becomes an unconstrained optimization with one choice variable: u(x 1) = x 1 I p 1x 1 p 2 1 . 825 0 obj <>stream For Q 5 : Utility maximization problem (with free disposal) of the consumer is : A consumer has utility function over two goods, apples (A) and bananas (B) given by U(A, B) = 3A +5B (a) What is the marginal utility of apples? (2) In (b)(2), several people said that M = U if P/R= 1 (should be M = PU= RU). Maximization of a function with a constraint is common in economic situations. endstream endobj startxref Problem Set . This means that the demands for goods 1 and 2 are x1 = 0 and x2 = Y. "It is a type of optimal decision problem.It consists of choosing how much of each available good or service to consume, taking into account a constraint on total spending as well as the prices of the goods. Ҧ$��@�I@Bj*Ȕl��X������ d100ҙ���� � #^X First, in order to solve the problem, we need more information about the MRS. As it turns out, every utility function has its own MRS, which can easily be found using calculus. Consider the utility maximization problem max U (x, y) = √ x + y s.t. Use the table below to answer questions 1-2. Then, we introduce the utility function without referring to preference.1 Finally, we state the consumer problem. h�b```f``������Y��π �@V�8��n00900HhpM��L�h�@��20���X,R��˩����ը�oO,�R�D�ƀ�2R�d��O@,�c`8���TB�4k�"q�{�4# ��� ) is a global maximum strictly concave ⋅ unique global maximum Sufficient condition: ∗ is optimal if Erin has $30 to spend on robotrons and flowbots, which each cost $2. Set up the Lagrangian 2. 3. (2) In (b)(2), several people said that M = U if P/R= 1 (should be M = PU= RU). h��ZmoG�+�Tq��/��S��p(���:�#�p�Ծ��;�/��Ŏ������쳳�ϭ/+ %�*�4�p!�5Dh|�DQ|vDK�SώG�F%*a�8�H�C�"LJ�ф)�C�ahc���9�(C,�Ё�-e�Yˡ� g�AG.���$\�����t4�f���5^����!F���},�ѹ@� N8�H⤂)dA1���1`�qZ�+�Ё�[X�3�pJNh9$�B�,��9�1. There are three equivalent ways to formulate the consumer’s utility maximization problem.2 (i) In class, you have seen that the problem can be stated as max.x1;x2/2R2 C.x 1C2/x 2 subject to p 1x 1Cp 2x 2 I: (ii) Note that .x 1;x 2/must be an element of R2 C (b) Suppose income increases from 100 to 101. Then Lx 1 and qx 2. %PDF-1.5 %���� a. Preview this quiz on Quizizz. The price of good xis pxand the price of good yis py.We denote income by M,as usual, with M>0.This 10.2.Utility maximization implies expenditure minimization. To solve this problem, you set up a linear programming problem, following these steps. h�b```f``����� ��π �@V�8ǃ��F�� 5��iA �Lb�唜|�����J��3Y*�i`���V���1j.+Cf �fb`� �Y;�4'C+H #� 5`� The set of available bundles for the consumer is given by: B p;m = fx 2X : px mg Then, the utility maximization problem is expressed as, max x u(x) subject to px m and x 2X. 765 0 obj <> endobj Access the answers to hundreds of Utility maximization problem questions that are explained in a way that's easy for you to understand. In fact, in this sort of problem, λ has the interpretation of being the marginal utility of income. d. Set this slope equal to the slope of the budget line and solve for the consumption in period 1 and 2. "It is a type of optimal decision problem.It consists of choosing how much of each available good or service to consume, taking into account a constraint on total spending as well as the prices of the goods. 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