Utility Maximization Problem. 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. 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? For each of the following situations, decide whether Al has increasing, constant or diminishing marginal utility. Problem 1. Consider the utility maximization problem max U (x, y) = √ x + y s.t. e. = d, but the interest rate is 20%. 765 0 obj <> endobj 1 utility maximization problem. 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) ... 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 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 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.. To nd Pareto optimal allocation we need solve two maximization subproblems and then compare utility levels. 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). Ҧ\$��@�I@Bj*Ȕl��X������ d100ҙ���� � #^X And the more classes he takes, the easier each one gets, making him enjoy each additional class more than the one before. Jack has a utility function for two perfectly divisible goods, x and y. Jack’s utility function is u(x;y) = (x+y)2. 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. Erin has \$30 to spend on robotrons and flowbots, which each cost \$2. Example: Imagine that the utility function is U(x,y)=5xy2, p x=2 and py=8 and I=240. To solve this problem, you set up a linear programming problem, following these steps. The utility function is u(x,y)= √ x+ √ y. Utility MaximizationConsumer BehaviorUtility MaximizationIndirect Utility FunctionThe Expenditure FunctionDualityComparative Statics Problem Set . Then, we introduce the utility function without referring to preference.1 Finally, we state the consumer problem. Solution. [14 points] b) Set up the firm’s profit maximization problem and find the FOCs. the constraint optimization problem is max x 1;x 2 x 1 x 1 2 subject to p 1x 1 + p 2x 2 = I. %%EOF Set out the basic consumer optimisation problem • the primal problem 2. (2) In (b)(2), several people said that M = U if P/R= 1 (should be M = PU= RU). Utility maximization. Access the answers to hundreds of Utility maximization problem questions that are explained in a way that's easy for you to understand. 1. For Q 5 : Utility maximization problem (with free disposal) of the consumer is : 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? Yu\$��wȀj !=\$� \$��f`bd�I00��� �� 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# ��� h�bbd``b`f�@�i���s��9 ��bi��%�� A consumer has preferences over consump- tion bundles that are strongly monotone, strictly convex, and represented by the following (differentiable) utility function: utility the consumer can achieve when facing a given set of prices with income I? 285 0 obj <>stream There two goods, X and Y , available in arbitrary non-negative quantities (so the consumption set is R2+). Problem 1: Utility maximization. (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. 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 COMMON ERRORS: (1) Some of you solved a utility maximization problem instead of the expenditure-minimization problem that is needed. Problem Set . Then Lx 1 and qx 2. Choose variables to represent the quantities involved. Write down the solution • copy directly from the solution to the firm’s problem 5. This is OK provided you then invert the indirect utility function to get the expenditure function, and some did not do this. ) is a global maximum strictly concave ⋅ unique global maximum Sufficient condition: ∗ is optimal if 10.2.Utility maximization implies expenditure minimization. Notice that production set is linear over some range and then starts to exhibit increasing returns to scale. 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. h�b```f``����� ��π �@V�8ǃ��F�� 5��iA �Lb�唜|�����J��3Y*�i`���V���1j.+Cf �fb`� �Y;�4'C+H #� 5`� In fact, in this sort of problem, λ has the interpretation of being the marginal utility of income. And the more classes he takes, the easier each one gets, making him enjoy each additional class more than the one before. Write an expression for the objective function using the variables. (c) Given Y, utility is maximized at (x1;x2) = (0;Y). L = labor q = consumption. endstream endobj startxref This means that the demands for goods 1 and 2 are x1 = 0 and x2 = Y. Let t represent the number of tetras and h represent the number of headstanders. 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 … endstream endobj 242 0 obj <. endstream endobj startxref Consider the utility maximization problem max U (x, y) = √ x + y s.t. The price of good xis pxand the price of good yis py.We denote income by M,as usual, with M>0.This 1. Maximization of a function with a constraint is common in economic situations. Get help with your Utility maximization problem homework. 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. 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. 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. The more economics classes Al takes, the more he enjoys the subject. 241 0 obj <> endobj d. Set this slope equal to the slope of the budget line and solve for the consumption in period 1 and 2. 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. It is focused on preferences, utility functions, and utility maximization. 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. 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 Maximization . 0 "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. Set up the Lagrangian 2. 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. 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. a. a) Solve the utility maximization problem for a representative consumer. 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 The more economics classes Al takes, the more he enjoys the subject. 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. 3.2 Utility-maximizing worker Convert to a problem with positive variables. The utility function is u(x,y)= √ x+ √ y. Will Mainy be better or worse off? (Or, after losing one unit of x Answers to Problem Set 3 0. an interior solution to a consumer's utility maximization problem implies. the constraint optimization problem is max x 1;x 2 x 1 x 1 2 subject to p 1x 1 + p 2x 2 = I. %%EOF 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 . unconstrained, univariate optimization problem by eliminating the constraint. 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. Utility Maximization . 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. Utility maximization. Utility Units 0 1 2 3 4 5 6 7 Total Utility 0 20 35 45 50 50 45 35 Problem Set . 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) unconstrained, univariate optimization problem by eliminating the constraint. (2) In (b)(2), several people said that M = U if P/R= 1 (should be M = PU= RU). 1. To solve this problem, you set up a linear programming problem, following these steps. %PDF-1.5 %���� Choose variables to represent the quantities involved. 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. h�bbd``b`:\$��X[��C ��H�I�X�@�9 D�/A+�`] Preview this quiz on Quizizz. The Engel curve for good 2 is the graph of Y = x2, which is the 45-degree line. 1.1 Commodity and Price Utility MaximizationConsumer BehaviorUtility MaximizationIndirect Utility FunctionThe Expenditure FunctionDualityComparative Statics This is OK provided you then invert the indirect utility function to get the expenditure function, and some did not do this. In microeconomics, the utility maximization problem is the problem consumers face: "how should I spend my money in order to maximize my utility? His optimal consumption bundle is \$(x_1, x_2) = (1,1)\$. Show that this problem is identical to that of the firm 4. 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. We consider three levels of generality in this treatment. Problem set 1 ECON 4330 Part 1 We are looking at an open economy that exists for two periods. For 0 x 1 20, the problem is max x 1;x 2 logx 1 + logx 2; s.th. 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 Output in each period Y 1 and Y 2 respectively, is given exogenously. 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 . 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 Show that the solution is equivalent to another problem • the dual problem 3. x + 4 y = 100 (a) Using the Lagrange multiplier method, find the quantities demanded of the two goods. (b) Suppose income increases from 100 to 101. Get help with your Utility maximization problem homework. Will she borrow or save in the first period. Let t represent the number of tetras and h represent the number of headstanders. 3. The robust utility maximization problem for this set Q was studied by Baudoin , 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 Fig. For each of the following situations, decide whether Al has increasing, constant or diminishing marginal utility. 825 0 obj <>stream %PDF-1.5 %���� Uncertainty Jonas Thern maximises expected utility: U(π 1, π 2,c 1,c 2) = π 1 c 1 + π 2 c 2 This Problem set tests the knowledge that you accumulated in the lectures 5 to 8. Be very careful in writing the budget constraint as the consumer has many sources of income in this model. It is the increase in the level of utility that would be achieved if income were to increase by one unit. Problem 1. The price of good xis pxand the price of good yis py.We denote income by M,as usual, with M>0.This (b) Suppose income increases from 100 to 101. (52 points) In this exercise, we consider a standard maximization problem with an unusual utility function. Problem 1. 0 Utility Maximization . 10.2.Utility maximization implies expenditure minimization. COMMON ERRORS: (1) Some of you solved a utility maximization problem instead of the expenditure-minimization problem that is needed. Use the table below to answer questions 1-2.
2020 utility maximization problem set