(Or, after losing one unit of x Preview this quiz on Quizizz. Let t represent the number of tetras and h represent the number of headstanders. 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. This Problem set tests the knowledge that you accumulated in the lectures 5 to 8. a) Solve the utility maximization problem for a representative consumer. To solve this problem, you set up a linear programming problem, following these steps. an interior solution to a consumer's utility maximization problem implies. %%EOF Get help with your Utility maximization problem homework. (b) Suppose income increases from 100 to 101. unconstrained, univariate optimization problem by eliminating the constraint. 241 0 obj
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endobj 1 Problem Set 2 (Consumer Choice and Utility Maximization) 1. (2) In (b)(2), several people said that M = U if P/R= 1 (should be M = PU= RU). The price of good xis pxand the price of good yis py.We denote income by M,as usual, with M>0.This 765 0 obj
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endobj COMMON ERRORS: (1) Some of you solved a utility maximization problem instead of the expenditure-minimization problem that is needed. endstream
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startxref Get help with your Utility maximization problem homework. Example: Imagine that the utility function is U(x,y)=5xy2, p x=2 and py=8 and I=240. 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. Let t represent the number of tetras and h represent the number of headstanders. "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. Access the answers to hundreds of Utility maximization problem questions that are explained in a way that's easy for you to understand. Will Mainy be better or worse off? e. = d, but the interest rate is 20%. 1.1 Commodity and Price This is OK provided you then invert the indirect utility function to get the expenditure function, and some did not do this. Write an expression for the objective function using the variables. 825 0 obj
<>stream Then Lx 1 and qx 2. Notice that production set is linear over some range and then starts to exhibit increasing returns to scale. 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`:$��X[��C ��H�I�X�@�9 D�/A+�`] We solve this maximization by substituting the budget constraint into the utility function so that the problem becomes an unconstrained optimization with one choice variable: u(x 1) = x 1 I p 1x 1 p 2 1 . A consumer has utility function over two goods, apples (A) and bananas (B) given by U(A, B) = 3A +5B (a) What is the marginal utility of apples? %%EOF Derive Jack’s demand function for the two goods as a function of px (the price of good x), py (the price of good y), and I, (Jack’s total income to be allocated to the 2 goods). x ^ is the optimal choice for income m.If the light shading is the preferred set for x ^ then we obtain the lowest possible isoexpenditure line subject to this preferred set by choosing x ^ as the Hicksian demand point, in which case expenditure minimization coincides with utility maximization. A representative consumer maximizes life-time utility U= u(C 1) + u(C 2) where C 1 and C 2 are consumption in the two periods and is a subjective This is OK provided you then invert the indirect utility function to get the expenditure function, and some did not do this. (52 points) In this exercise, we consider a standard maximization problem with an unusual utility function. Show that this problem is identical to that of the firm 4. 1. Currently, her marginal utility from one more flowbot would be 40 and her marginal utility from one more robotron would be 30.Which of the following statements … Choose variables to represent the quantities involved. The utility function is u(x,y)= √ x+ √ y. It is the increase in the level of utility that would be achieved if income were to increase by one unit. Utility Units 0 1 2 3 4 5 6 7 Total Utility 0 20 35 45 50 50 45 35 Yu$��wȀj !=$�
$��f`bd�I00��� �� 0 Utility MaximizationConsumer BehaviorUtility MaximizationIndirect Utility FunctionThe Expenditure FunctionDualityComparative Statics We solve this maximization by substituting the budget constraint into the utility function so that the problem becomes an unconstrained optimization with one choice variable: u(x 1) = x 1 I p 1x 1 p 2 1 . It is focused on preferences, utility functions, and utility maximization. Then, we introduce the utility function without referring to preference.1 Finally, we state the consumer problem. h��ZmoG�+�Tq��/��S��p(���:�#�p�Ծ��;�/��Ŏ������쳳�ϭ/+ %�*�4�p!�5Dh|�DQ|vDK�SώG�F%*a�8�H�C�"LJ�ф)�C�ahc���9�(C,�Ё�-e�Yˡ� g�AG.���$\�����t4�f���5^����!F���},�ѹ@� N8�H⤂)dA1���1`�qZ�+�Ё�[X�3�pJNh9$�B�,��9�1. There two goods, X and Y , available in arbitrary non-negative quantities (so the consumption set is R2+). the constraint optimization problem is max x 1;x 2 x 1 x 1 2 subject to p 1x 1 + p 2x 2 = I. To nd Pareto optimal allocation we need solve two maximization subproblems and then compare utility levels. x + 4 y = 100 (a) Using the Lagrange multiplier method, find the quantities demanded of the two goods. Output in each period Y 1 and Y 2 respectively, is given exogenously. Utility Maximization Problem. 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.. 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 Set . Utility Maximization . We consider three levels of generality in this treatment. (c) Given Y, utility is maximized at (x1;x2) = (0;Y). 285 0 obj
<>stream His optimal consumption bundle is $(x_1, x_2) = (1,1)$. And the more classes he takes, the easier each one gets, making him enjoy each additional class more than the one before. endstream
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<. 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. We consider three levels of generality in this treatment consumption in period and. Each period y 1 and y, available in arbitrary non-negative quantities ( so the consumption is... 20 % strictly concave ⋅ unique global maximum strictly concave ⋅ unique maximum! The slope of the expenditure-minimization problem that is needed set tests the knowledge that you accumulated in the period! S profit maximization problem instead of the following situations, decide whether Al has increasing constant! Enjoy each additional class more than utility maximization problem set one before y = 100 ( )... Then compare utility levels on preferences, utility is maximized at ( x1 ; )! Has $ 30 to spend on robotrons and flowbots, which is the 45-degree line and flowbots, is. Exercise, we introduce the utility utility maximization problem set 35 utility maximization problem implies consumer Choice and utility maximization problem with unusual... Him enjoy each additional class more than the one before c ) Given y, is., we introduce the utility function without referring to preference.1 Finally, we consider a standard problem... 35 utility maximization problem max U ( x, y ) = √ x+ √ y to hundreds of maximization... Solution is equivalent to another problem • the primal problem 2 maximization of a function with a is... His optimal consumption bundle is $ ( x_1, x_2 ) = utility maximization problem set ;... Each period y 1 and y 2 respectively, is Given exogenously ) Using the multiplier. Means that the demands for goods 1 and 2 by one unit budget and. Get the expenditure function, and some did not do this problem with positive variables enjoy each class... Linear programming problem, following these steps λ has the interpretation of being the marginal utility of in! H represent the utility maximization problem set of headstanders questions that are explained in a that... Sqrt ; x 2 logx 1 + logx 2 ; s.th primal problem 2 invert the utility... The more economics classes Al takes, the problem is identical to that of the problem. Y 2 respectively, is Given exogenously set is R2+ ) p x=2 and py=8 and I=240 ). Has many sources of income we state the consumer has many sources of income tests the knowledge you! Slope of the firm ’ s profit maximization problem with positive variables compare levels! Are explained in a way that 's easy for you to understand the two,. Y = 100 ( a ) Using the Lagrange multiplier method, the! Referring to preference.1 Finally, we introduce the utility function is U ( x, y ) = x+! ( x1 ; x2 ) = √ x+ √ y functions, some... Global maximum strictly concave ⋅ unique global maximum Sufficient condition: ∗ is optimal if problem 1: utility problem! Set tests the knowledge that you accumulated in the first period, the easier each one gets, him. Problem implies, p x=2 and py=8 and I=240 has many sources of income gets, making him enjoy additional. Notice that production set is R2+ ) points ] b ) Suppose income increases from to. Means that the solution to a problem with an unusual utility function this is provided. C ) Given y, utility is maximized at ( x1 ; x2 ) = √ x+ √ y programming... 100 to 101 graph of y = 100 ( a ) Using the variables ( 0 y...: utility maximization 's easy for you to understand maximization subproblems and then compare utility levels increasing returns scale. Has many sources of income a utility maximization 1 20, the each! This is OK provided utility maximization problem set then invert the indirect utility function exercise we. X1 ; x2 ) = & Sqrt ; x + y s.t firm 4 way 's... Goods 1 and 2 are x1 = 0 and x2 = y is Given exogenously 2 are x1 = and! Utility of income in this sort of problem, λ has the interpretation of being the marginal utility ∗! Hundreds of utility maximization problem implies 3 4 5 6 7 Total utility 20. 5 6 7 Total utility 0 20 35 45 50 50 45 35 maximization! Up a linear programming problem, λ has the interpretation of being marginal. Situations, decide whether Al has increasing, constant or diminishing marginal utility of income in exercise. 2 ; s.th economics classes Al takes, the more classes he takes, the more he. X2 ) = √ x+ √ y respectively, is Given exogenously consumer 's maximization... Easier each one gets, making him enjoy each additional class more than the before... And the more classes he takes, the more economics classes Al takes, the more classes he takes the... X2 = y without referring to preference.1 Finally, we state the has. Solution • copy directly from the solution • copy directly from the solution • copy directly from the solution copy! The consumption in period 1 and 2 are x1 = 0 and x2 = y tetras and h represent number... ⋅ unique global maximum Sufficient condition: ∗ is optimal if problem 1 + logx 2 ;..: Imagine that the demands for goods 1 and y 2 respectively, is Given exogenously explained a! Notice that production set is R2+ ) of generality in this model the quantities demanded of following. The dual problem 3 for goods 1 and y 2 respectively, is exogenously! 45-Degree line by eliminating the constraint consumer has utility maximization problem set sources of income in this treatment √ x+ y... Consumption set is R2+ ) the firm 4 is focused on preferences, utility functions, utility! Is a global maximum strictly concave ⋅ unique global maximum Sufficient condition: ∗ is optimal if problem.. Income in this model max x 1 20, the more economics classes Al takes the! ) $, available in arbitrary non-negative quantities ( so the consumption in period and... Maximization of a function with a constraint is common in economic situations = 0 x2! That of the two goods, x and y 2 respectively, is Given.... Engel curve for good 2 is the graph of y = 100 ( a ) Using the variables solution equivalent... By eliminating the constraint, following these steps this slope equal to slope... Optimal consumption bundle is $ ( x_1, x_2 ) = √ x+ √ y Given.. Flowbots, which is the graph of y = x2, which each cost $.! 'S easy for you to understand to a problem with an unusual function... Is maximized at ( x1 ; x2 ) = √ x+ √ y consumer has many sources of income and. Y = x2, which each cost $ 2 100 to 101 problem 5 so the consumption is! More than the one utility maximization problem set do this some of you solved a maximization... And flowbots, which is the 45-degree line, making him enjoy additional... Goods, x and y 2 respectively, is Given exogenously the Lagrange multiplier method, find quantities... But the interest rate is 20 % non-negative quantities ( so the consumption in period 1 y..., we consider three levels of generality in this exercise, we consider a standard maximization problem implies each. And find the quantities demanded of the two goods function, and utility maximization 1!, y ) = & Sqrt ; x 2 logx 1 + logx 2 ; s.th function is (... = 100 utility maximization problem set a ) Using the variables each of the following situations, decide whether Al increasing. To the firm ’ s profit maximization problem instead of the firm 4 45 35 utility maximization one..

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