(2) In (b)(2), several people said that M = U if P/R= 1 (should be M = PU= RU). For each of the following situations, decide whether Al has increasing, constant or diminishing marginal utility. This Problem set tests the knowledge that you accumulated in the lectures 5 to 8. 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. In microeconomics, the utility maximization problem is the problem consumers face: "how should I spend my money in order to maximize my utility? "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. 1. Yu$��wȀj !=$� $��f`bd�I00��� �� 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. Get help with your Utility maximization problem homework. 1.1 Commodity and Price "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. x + 4 y = 100 (a) Using the Lagrange multiplier method, find the quantities demanded of the two goods. 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 Let t represent the number of tetras and h represent the number of headstanders. a. Utility Maximization Problem. Maximization of a function with a constraint is common in economic situations. And the more classes he takes, the easier each one gets, making him enjoy each additional class more than the one before. Be very careful in writing the budget constraint as the consumer has many sources of income in this model. 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). the constraint optimization problem is max x 1;x 2 x 1 x 1 2 subject to p 1x 1 + p 2x 2 = I. d. Set this slope equal to the slope of the budget line and solve for the consumption in period 1 and 2. x + 4 y = 100 (a) Using the Lagrange multiplier method, find the quantities demanded of the two goods. 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 unconstrained, univariate optimization problem by eliminating the constraint. %PDF-1.5 %���� 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. Utility MaximizationConsumer BehaviorUtility MaximizationIndirect Utility FunctionThe Expenditure FunctionDualityComparative Statics His optimal consumption bundle is $(x_1, x_2) = (1,1)$. Utility maximization. Choose variables to represent the quantities involved. 285 0 obj <>stream Problem Set . h�bbd``b`:$��X[��C ��H�I�X�@�9 D�/A+�`] Problem 1. Then, we introduce the utility function without referring to preference.1 Finally, we state the consumer problem. A consumer has preferences over consump- tion bundles that are strongly monotone, strictly convex, and represented by the following (differentiable) utility function: It is focused on preferences, utility functions, and utility maximization. 1 COMMON ERRORS: (1) Some of you solved a utility maximization problem instead of the expenditure-minimization problem that is needed. Choose variables to represent the quantities involved. Output in each period Y 1 and Y 2 respectively, is given exogenously. Will she borrow or save in the first period. Then Lx 1 and qx 2. A representative consumer maximizes life-time utility U= u(C 1) + u(C 2) where C 1 and C 2 are consumption in the two periods and is a subjective 0 Consider the utility maximization problem max U (x, y) = √ x + y s.t. There two goods, X and Y , available in arbitrary non-negative quantities (so the consumption set is R2+). We consider three levels of generality in this treatment. (Or, after losing one unit of x The price of good xis pxand the price of good yis py.We denote income by M,as usual, with M>0.This endstream endobj startxref To solve this problem, you set up a linear programming problem, following these steps. 1. Fig. 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. To nd Pareto optimal allocation we need solve two maximization subproblems and then compare utility levels. Write an expression for the objective function using the variables. (52 points) In this exercise, we consider a standard maximization problem with an unusual utility function. Will Mainy be better or worse off? 3. Get help with your Utility maximization problem homework. unconstrained, univariate optimization problem by eliminating the constraint. ) is a global maximum strictly concave ⋅ unique global maximum Sufficient condition: ∗ is optimal if This is OK provided you then invert the indirect utility function to get the expenditure function, and some did not do this. 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. 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. 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 . Access the answers to hundreds of Utility maximization problem questions that are explained in a way that's easy for you to understand. e. = d, but the interest rate is 20%. 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 … h�bbd``b`f�@�i���s��9 ��bi��%�� Problem set 1 ECON 4330 Part 1 We are looking at an open economy that exists for two periods. 10.2.Utility maximization implies expenditure minimization. Problem Set 2 (Consumer Choice and Utility Maximization) 1. 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) 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# ��� The more economics classes Al takes, the more he enjoys the subject. Uncertainty Jonas Thern maximises expected utility: U(π 1, π 2,c 1,c 2) = π 1 c 1 + π 2 c 2 Problem Set . (52 points) In this exercise, we consider a standard maximization problem with an unusual utility function. 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) Problem 1. Write an expression for the objective function using the variables. Utility Units 0 1 2 3 4 5 6 7 Total Utility 0 20 35 45 50 50 45 35 (2) In (b)(2), several people said that M = U if P/R= 1 (should be M = PU= RU). This is OK provided you then invert the indirect utility function to get the expenditure function, and some did not do this. Utility Maximization . To solve this problem, you set up a linear programming problem, following these steps. The robust utility maximization problem for this set Q was studied by Baudoin [2002], who coined the terminology weak information.The interpretation behind the set Q is that an investor has full knowledge about the pricing measure P * but is uncertain about the true distribution P of market prices and only knows that a certain functional Y of the stock price has distribution v Define Q 0 by Solution. [14 points] b) Set up the firm’s profit maximization problem and find the FOCs. Answers to Problem Set 3 0. In fact, in this sort of problem, λ has the interpretation of being the marginal utility of income. an interior solution to a consumer's utility maximization problem implies. 825 0 obj <>stream 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.. The price of good xis pxand the price of good yis py.We denote income by M,as usual, with M>0.This Ҧ$��@�I@Bj*Ȕl��X������ d100ҙ���� � #^X 1. The utility function is u(x,y)= √ x+ √ y. The utility function is u(x,y)= √ x+ √ y. 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 For 0 x 1 20, the problem is max x 1;x 2 logx 1 + logx 2; s.th. (b) Suppose income increases from 100 to 101. utility the consumer can achieve when facing a given set of prices with income I? Problem Set . This means that the demands for goods 1 and 2 are x1 = 0 and x2 = Y. L = labor q = consumption. Let t represent the number of tetras and h represent the number of headstanders. 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? the constraint optimization problem is max x 1;x 2 x 1 x 1 2 subject to p 1x 1 + p 2x 2 = I. a. 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. COMMON ERRORS: (1) Some of you solved a utility maximization problem instead of the expenditure-minimization problem that is needed. 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 The Engel curve for good 2 is the graph of Y = x2, which is the 45-degree line. In microeconomics, the utility maximization problem is the problem consumers face: "how should I spend my money in order to maximize my utility? h�b```f``����� ��π �@V�8ǃ��F�� 5��iA �Lb�唜|�����J��3Y*�i`���V���1j.+Cf �fb`� �Y;�4'C+H #� 5`� Erin has $30 to spend on robotrons and flowbots, which each cost $2. 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 . Jack has a utility function for two perfectly divisible goods, x and y. Jack’s utility function is u(x;y) = (x+y)2. 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? 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. %%EOF Utility Maximization . 3.2 Utility-maximizing worker Convert to a problem with positive variables. 241 0 obj <> endobj %PDF-1.5 %���� Write down the solution • copy directly from the solution to the firm’s problem 5. 765 0 obj <> endobj Problem 1: Utility maximization. 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. Utility Maximization . 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. utility maximization problem. Example: Imagine that the utility function is U(x,y)=5xy2, p x=2 and py=8 and I=240. The more economics classes Al takes, the more he enjoys the subject. Use the table below to answer questions 1-2. Show that the solution is equivalent to another problem • the dual problem 3. 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. It is the increase in the level of utility that would be achieved if income were to increase by one unit. (b) Suppose income increases from 100 to 101. Problem 1. endstream endobj 242 0 obj <. For Q 5 : Utility maximization problem (with free disposal) of the consumer is : (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. Utility maximization. And the more classes he takes, the easier each one gets, making him enjoy each additional class more than the one before. ... 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 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 MaximizationConsumer BehaviorUtility MaximizationIndirect Utility FunctionThe Expenditure FunctionDualityComparative Statics Consider the utility maximization problem max U (x, y) = √ x + y s.t. 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. a) Solve the utility maximization problem for a representative consumer. Set out the basic consumer optimisation problem • the primal problem 2. Notice that production set is linear over some range and then starts to exhibit increasing returns to scale. Preview this quiz on Quizizz. Set up the Lagrangian 2. Fig. 0 For each of the following situations, decide whether Al has increasing, constant or diminishing marginal utility. 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. 10.2.Utility maximization implies expenditure minimization. Show that this problem is identical to that of the firm 4. %%EOF (c) Given Y, utility is maximized at (x1;x2) = (0;Y). endstream endobj startxref