# Utility maximization and expenditure minimization

The random utility maximization (RUM) is aiming to choose the least generalized travel cost in the process of the passenger route choice and choose the route with the maximum utility value. Under the rule of utility maximization, it is assumed that the passengers’ choice is completely rational; that is, the passengers always compare the ... Outline 1. Expenditure Minimization II 2. Expenditure Min: First order conditions 3. Slutzky equation 4. Complements and substitutes 5. Do utility functions exist? Practice Questions: Cost Minimization and Proﬁt Maximization. Problem 1. A ﬁrm has Cobb-Douglas production function y = KL. Input prices are as follows: rental rate on capital r = 4, wage is w = 1. a) Suppose in SR capital is ﬁxed at 5 units, ﬁnd short run TC function. Utility MaximizationConsumer BehaviorUtility MaximizationIndirect Utility FunctionThe Expenditure FunctionDualityComparative Statics (5) We will often need to assume that the solution to the Utility Maximization Problem (UMP) is unique. It turns out that strict convexity ensures uniqueness. 1 Proof. Assume not. x and x both solve the UMP. Then px m • Utility maximization problem (UMP) • Walrasian demand and indirect utility function • WARP and Walrasian demand • Income and substitution effects (Slutsky equation) • Duality between UMP and expenditure minimization problem (EMP) • Hicksian demand and expenditure function • Connections Advanced Microeconomic Theory 2 View Homework Help - Utility Max and Expenditure Min from ECON 11 at University of California, Los Angeles. TA Session Week 4: Utility Maximization and Expenditure Minimization Lei Zhang In this This random regret minimization (RRM) model, which has recently been introduced in the field of transport, forms a regret-based counterpart of the canonical random utility maximization (RUM) paradigm. In microeconomics, the expenditure minimization problem is the dual of the utility maximization problem: “how much money do I need to reach a certain level of happiness?”. This question comes in two parts. Given a consumer’s utility function, prices, and a utility target, how much money would the consumer need? Profit maximization and cost minimization Average and marginal costs 16 . Profit Maximization Joint choice of input and production levels Utility maximization: To maximize utility, given a fixed amount of income to spend, an individual will buy those quantities of goods that exhaust his or her total income and for which the psychic rate of trade-off between any two good (the MRS) is equal to the rate at which the goods can be traded one for the other in the marketplace utility-maximization problem with income as a variable while keeping the prices of the goods constant. expenditure-minimization problem with income as a variable while keeping the prices of the goods constant. expenditure-minimization problem with prices of the goods as variables while keeping income constant. Jan 22, 2016 · In microeconomics, the expenditure minimization problem is another perspective on the utility maximization problem: "how much money do I need to reach a certain level of happiness?".This question... Practice Questions: Cost Minimization and Proﬁt Maximization. Problem 1. A ﬁrm has Cobb-Douglas production function y = KL. Input prices are as follows: rental rate on capital r = 4, wage is w = 1. a) Suppose in SR capital is ﬁxed at 5 units, ﬁnd short run TC function. Jul 17, 2014 · The expenditure minimization function is the minimum money that is required to achieve a given level of utility and prices. This is very similar to the utility maximization question that you would be familiar dealing with in an intermediate microeconomics class. Normally, you are given a set of prices and some income and you are […] Theorem 10.5 Utility Maximization Implies Expenditure Minimization. Assume the conditions of Theorem 10.1 and ū ∈ [u (0), sup U). For every p ≫ 0: e * (p, ū) is continuous and unbounded on ū ∈ U if and only if u*(p, m) is strictly increasing in m ∈ R + n. If these conditions hold then x * (p, m) ⊂ h * (p, u * (p, m)) and m = e*(p, u*(p, m)). Exercise 10B.4 Verify that (10.11) and (10.12) are true for the functions that emerge from utility maximization and expenditure minimization when tastes can be modeled by the Cobb-Douglas function u(x1,x2) = xix41 –a) Tollowing logical relationsnip: x;(P1, P2, E(P1,P2, u)) = h;(P1,P2,u). Theorem 10.5 Utility Maximization Implies Expenditure Minimization. Assume the conditions of Theorem 10.1 and ū ∈ [u (0), sup U). For every p ≫ 0: e * (p, ū) is continuous and unbounded on ū ∈ U if and only if u*(p, m) is strictly increasing in m ∈ R + n. If these conditions hold then x * (p, m) ⊂ h * (p, u * (p, m)) and m = e*(p, u*(p, m)). View Homework Help - Utility Max and Expenditure Min from ECON 11 at University of California, Los Angeles. TA Session Week 4: Utility Maximization and Expenditure Minimization Lei Zhang In this Theorem 10.5 Utility Maximization Implies Expenditure Minimization. Assume the conditions of Theorem 10.1 and ū ∈ [u (0), sup U). For every p ≫ 0: e * (p, ū) is continuous and unbounded on ū ∈ U if and only if u*(p, m) is strictly increasing in m ∈ R + n. If these conditions hold then x * (p, m) ⊂ h * (p, u * (p, m)) and m = e*(p, u*(p, m)). Exercise 10B.4 Verify that (10.11) and (10.12) are true for the functions that emerge from utility maximization and expenditure minimization when tastes can be modeled by the Cobb-Douglas function u(x1,x2) = xix41 –a) Tollowing logical relationsnip: x;(P1, P2, E(P1,P2, u)) = h;(P1,P2,u). Production Maximization and Cost Minimization. Recall that in consumer choice we take budget constraint as fixed and move indifference curves to find the optimal point. The analogy of firm/producer/seller choice is a bit different, since a firm is not bounded by a fixed income. The optimization could go in two directions—either we maximize production for a given expenditure amount (cost), or we minimize cost for a given production quantity. Outline 1. Expenditure Minimization II 2. Expenditure Min: First order conditions 3. Slutzky equation 4. Complements and substitutes 5. Do utility functions exist? 2 Expenditure Minimization Problem The consumer problem can be approached in a diﬀerent way which produces some useful tools. Instead of maximizing utility given a certain income, imagine how much income it would take to achieve a certain level of utility. In other words consider the following expenditure minimization problem (EMP for short), This random regret minimization (RRM) model, which has recently been introduced in the field of transport, forms a regret-based counterpart of the canonical random utility maximization (RUM) paradigm. 2. Utility Maximization 1 Budget Constraint Two standard assumptions on utility: Œ Non-satiation: @U(Cx;Cy) @Cx > 0 for all values of Cx;Cy > 0 Œ Convexity: Let C1;C2 and C3 be commodity bundles such that C1 C3 and C2 C3: Then any convex combination of C1 and C2 is also weakly preferred to C3: tC1+(1 t)C2 C3 for all t 2 [0;1]: Jul 17, 2014 · The expenditure minimization function is the minimum money that is required to achieve a given level of utility and prices. This is very similar to the utility maximization question that you would be familiar dealing with in an intermediate microeconomics class. Normally, you are given a set of prices and some income and you are […] Because of the duality between the utility maximization and the expenditure minimization problems, we know that the optimal value of the expenditure minimization problem is given by the function E (p x, p y, V) where V is the indirect utility function (U (x, y) evaluated at the optimum). The 10 Cost minimization: find point at which the isocost line is tangent to the isoquant curve Utility maximization: find point at which budget line is tangent to the indifference curve Both problems use (1) an optimality/tangency condition and (2) a budget constraint or output constraint Theorem 10.5 Utility Maximization Implies Expenditure Minimization. Assume the conditions of Theorem 10.1 and ū ∈ [u (0), sup U). For every p ≫ 0: e * (p, ū) is continuous and unbounded on ū ∈ U if and only if u*(p, m) is strictly increasing in m ∈ R + n. If these conditions hold then x * (p, m) ⊂ h * (p, u * (p, m)) and m = e*(p, u*(p, m)). Outline 1. Expenditure Minimization II 2. Expenditure Min: First order conditions 3. Slutzky equation 4. Complements and substitutes 5. Do utility functions exist? Nov 15, 2010 · Just like the utility maximization subject to a given budget constraint, the expenditure minimization is solved as it is subject to a utility constraint. In essence, it is gives the minimal expenditure budget ( p1x1+p2x2 ) that one requires to reach the certain level of utility (uo) . Expenditure Minimization It is going to be very useful to de–ne Expenditure minimization problem This is the dual of the utility maximization problem Prime problem (utility maximization) choose x 2 RN + in order to maximize u(x) subject to N å i=1 p ix i w Dual problem (cost minimization) choose x 2 RN + in order to minimize N å i=1 p ix i ... Profit maximization and cost minimization Average and marginal costs 16 . Profit Maximization Joint choice of input and production levels Theorem 10.5 Utility Maximization Implies Expenditure Minimization. Assume the conditions of Theorem 10.1 and ū ∈ [u (0), sup U). For every p ≫ 0: e * (p, ū) is continuous and unbounded on ū ∈ U if and only if u*(p, m) is strictly increasing in m ∈ R + n. If these conditions hold then x * (p, m) ⊂ h * (p, u * (p, m)) and m = e*(p, u*(p, m)). utility-maximization problem with income as a variable while keeping the prices of the goods constant. expenditure-minimization problem with income as a variable while keeping the prices of the goods constant. expenditure-minimization problem with prices of the goods as variables while keeping income constant. View Homework Help - Utility Max and Expenditure Min from ECON 11 at University of California, Los Angeles. TA Session Week 4: Utility Maximization and Expenditure Minimization Lei Zhang In this utility-maximization problem with income as a variable while keeping the prices of the goods constant. expenditure-minimization problem with income as a variable while keeping the prices of the goods constant. expenditure-minimization problem with prices of the goods as variables while keeping income constant. 2 Expenditure Minimization Problem The consumer problem can be approached in a diﬀerent way which produces some useful tools. Instead of maximizing utility given a certain income, imagine how much income it would take to achieve a certain level of utility. In other words consider the following expenditure minimization problem (EMP for short), The expenditure minimisation problem (EMP) looks at the reverse side of the utility maximisa- tion problem (UMP). The UMP considers an agent who wishes to attain the maximum utility from a limited income. The EMP considers an agent who wishes to ﬂnd the cheapest way to attain a target utility.