Exercise 1 A consumer's utility function is u(x, y) = x²y³ and the budget constraint is pax +Pyy ≤ w, where the parameters pa, py and w are all strictly positive. (a) Solve the consumer's utility maximization problem. (b) Use the envelope theorem to estimate the change in the indirect utility function (i.e., the problem's value function) when the price py is changed to py + €, with € > 0.

Exercise 1 A consumer's utility function is u(x, y) = x²y³ and the budget constraint is pax +Pyy ≤ w, where the parameters pa, py and w are all strictly positive. (a) Solve the consumer's utility maximization problem. (b) Use the envelope theorem to estimate the change in the indirect utility function (i.e., the problem's value function) when the price py is changed to py + €, with € > 0.

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

See similar textbooks

Similar questions

A consumer has the following utility function: U (x, y) = (x + a) (y + b) Prices of the two goods x and y respectively are px and py and the consumer has income m. We assume that all prices and income are strictly positive. Furthermore, throughout this question we assume that m is high enough so that both x and y are strictly positive in equilibrium. (a) Solve the consumer’s optimization problem and express the demand for the two goods in terms of prices and income. Note:- Do not provide handwritten solution. Maintain accuracy and quality in your answer. Take care of plagiarism. Answer completely. You will get up vote for sure.
1. An agent consumes quantity (X1,X2) of goods 1 and 2. Here is his utility function: U (x1,x2) = x13x2, his budget constraint is: p1x1+p2x2=m. (a) Calculate the agent's Marshallian demand (x*, x*2). (b) Derive the agent's expenditure function. (c) Roy's identity for good 1 states that: aV(pl,p2, m)/ х1* 3 дV (р1, р2, т) am Verify this equation (you need not verify it for good 2).
12. If indifference curves cross, then:(a) the assumption of a diminishing marginal rate of substitution is violated.(b) the assumption of transitivity is violated.(c) the assumption of completeness is violated.(d) consumers minimize their satisfaction.(e) all of the above
Upon graduating from UT this May, you take on a management position working at UtMax theater. You will consider the utility of seeing performance over 1 month, and suppose that at a regular price of $$$ per ticket (my assigned ticket price is 145), customers will see no performance, however with the price reduced by $5, customers will see one performance per month and when reduced by $10, customers will see two performances. As long as the number performances, x, is small, your demand function for performance can be modeled by p=D(x). Write down your demand function.
The utility function of a consumer is u = √x + 2y. (a) Look for the demand function of goods x and goods y. (b) Demonstrate mathematically that the demand function of goods x does not depend on income m. (C) Demonstrate mathematically that the demand function of goods y depends on m, Px, and Py.
A consumer has set of budget of $800 for the consumption of good X a d Y. The price of Good X is $10 and the price of good Y is $40.The consumer has a utility function given by U (X,Y) = X(0.80)Y(0.20).Find the optimal soloution using a methof that does not involves equations and illustrate the soloution?
A consumer has the following utility function U (X1, X2) = (x1 + 3) (x2+4) Prices of the two goods X₁ and x₂ respectively are p₁ and p2 and the consumer has income m. We assume that all prices and income are strictly positive. Furthermore, throughout this question we assume that m is high enough so that both goods are consumed in strictly positive amount in equilibrium. (a) Solve the consumer's optimization problem and express the demand for the two goods in terms of prices and income. (1) Compute the missing values of elasticities in the following table assuming M = 10, p₁ = 2, and p2 = 1. Show your work. Income elasticity Absolute value of own price elasticity Cross price elasticity X₁ T≤ X₂ (b) While x2 is produced locally, x₁ is transported from a different region. Assume P₂ = 1. Building a new railroad (that connects the regions) will reduce transportation cost which in turn will reduce the price of good 1 from p₁ = 2 to p₁ = 1. Railroad will be funded by taxes which will reduce…
Higher Indifference curve means: (a) Consumer has more income (b) Price of goods have reduced (c) Higher utility level (d) All of the above
Question 5 Solving Consumer's Choice Problem under CES Utility, Numerical (sympy solution not required) The consumer with income Y has a preference represented by CES utility function U(91,92) = (90.4 + 92.4) 04. Given prices for the two goods, denoted by P₁, P2, solve the consumer's optimal choice problem following the steps below. (a) Write down the consumer's maximization problem, i.e be clear about (1) the choice variables, (2) the objective function and (3) the constraint. (b) Write down the Lagrangian for this optimization problem. (c) Derive the three first-order conditions. (d) Solve for the optimal consumption bundles q₁,92 . (e) Now we analyze the properties of demand. How does q respond to changes in P₁, P2 and Y? Specifically, does q increase or decrease if 1. p₁ increases? 2. P2 increases? 3. Y increases? For simplicity, we only analyze q, results for q are similar. (Hint: comparative statics, calculate, etc.)
PROBLEM (5) Cars (x) is a normal good, and public transportation (y) is a substitute for cars. The demand for cars(x) is given by where Qp = 20 ± px ± py ± pfuture ±I where px is the price of cars, py is the price of public transportation, pfuture is the expected future price of cars and I is the average income level. (a) Decide whether a plus (+) or a minus (-) sign is appropriate for each "+" in the demand equation above. (b) Does the demand become more or less elastic (at a given px) as py increases? (c) Calculate the cross price elasticity of demand (for cars with respect to the price of public transportation py) at the point I = 10, px = 5, py=2, pfuture = 1 (d) If the point in (c) is the market equilibrium point where the supply is known to have constant price elasticity with E = 2 at any point on the curve, derive the supply function.
Joko is a university student, working part-time at copying service center for Rp. 8/hour with zero non-labor income. (a) Graph Joko’s budget constraint and label the utility-maximing outcome if Joko opts to work 40 hours per week. (b) Suppose Joko’s parents decide to send him Rp.100/week. Graph Joko’s new budget constraint. (c) How many hours will Joko now have to work to maintain the same weekly income as in (a)?
An individual has an income of 240 and finds his satisfaction by consuming goods x1 for 2$ and x2 for 3$. It is known that the individual utility function is as follows U (x1, x2) = x1²x2³ Determine the value of x1 and x2 for maximum individual satisfaction and calculate the value of utility.