Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 6: Problem 9

In Chapter \(5,\) we showed how the welfare costs of changes in a single price can be measured using expenditure functions and compensated demand curves. This problem asks you to generalize this to price changes in two (or many) goods. a. Suppose that an individual consumes \(n\) goods and that the prices of two of those goods (say, \(p_{1}\) and \(p_{2}\) ) increase. How would you use the expenditure function to measure the compensating variation (CV) for this person of such a price increase? b. A way to show these welfare costs graphically would be to use the compensated demand curves for goods \(x_{1}\) and \(x_{2}\) by assuming that one price increased before the other. Illustrate this approach. c. In your answer to part (b), would it matter in which order you considered the price changes? Explain. d. In general, would you think that the CV for a price increase of these two goods would be greater if the goods were net substitutes or net complements? Or would the relationship between the goods have no bearing on the welfare costs?

Short Answer

Expert verified
Question: Calculate the compensating variation for price changes of two goods and discuss the effect of the order of price changes and the relationship between substitute and complementary goods on welfare costs. Explain the graphical representation of welfare costs using compensated demand curves.

Step by step solution

01

Define the expenditure function

The expenditure function e(p, u) represents the minimum amount of money required to achieve a certain level of utility u, given a price vector p.
02

Calculate the initial expenditure

Compute the initial expenditure e(p1, p2, ..., pn, u0), where p1 and p2 are the initial prices of the goods, and u0 is the initial utility level before the price changes.
03

Calculate the new expenditure

Compute the new expenditure e(p1', p2', ..., pn, u0), where p1' and p2' are the increased prices of the goods, and u0 is still the initial utility level before the price changes.
04

Calculate Compensating Variation

Compensating Variation (CV) can be calculated as the difference between the new expenditure and the initial expenditure: CV = e(p1', p2', ..., pn, u0) - e(p1, p2, ..., pn, u0) #b. Graphical Representation - Compensated Demand Curves#
05

Create an initial representation

Start by plotting the compensated demand curves for goods x1 and x2, assuming the initial prices and utility level u0.
06

Illustrate the effect of one price change at a time

Increase the price of good x1 (p1), while the price of good x2 (p2) remains unchanged. Locate the new compensated demand curve for goods x1 and x2, and show the area of the welfare loss.
07

Illustrate the subsequent price change

Now, increase the price of good x2 (p2), while keeping the new price of good x1 (p1') constant. Locate the final compensated demand curve for goods x1 and x2, and show the additional area of welfare loss which occurs due to the price change of good x2. #c. Order of Price Changes#
08

Compare effects of order changes

Compare the welfare losses from increasing the prices of goods x1 and x2 in different orders to determine whether it matters in which order you considered the price changes.
09

Explain the significance of the order

If the welfare losses change depending on the order of the price increase, explain why that occurs and how it impacts the final result. If the changes are independent of the order of the price increase, explain the logic behind it. #d. Effect of Goods being Substitutes or Complements#
10

Define net substitutes and net complements

Net substitutes are goods whose demand increases when the price of another good increases. Net complements are goods whose demand decreases when the price of another good increases.
11

Analyze CV for net substitutes

Explain how the Compensating Variation (CV) changes when the goods are net substitutes, and whether it leads to a larger or smaller welfare cost.
12

Analyze CV for net complements

Explain how the Compensating Variation (CV) changes when the goods are net complements, and whether it leads to a larger or smaller welfare cost.
13

Compare the welfare costs

Compare the welfare costs of price changes in net substitutes and net complements, and discuss if there is a consistent relationship between the type of goods and the welfare cost.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In general, uncompensated cross-price effects are not equal. That is, \\[\frac{\partial x_{i}}{\partial p_{j}} \neq \frac{\partial x_{j}}{\partial p_{i}}.\\] regardless of relative prices. (This is a generalization of Problem \(6.1 .)\)

Ms. Sarah Traveler does not own a car and travels only by bus, train, or plane. Her utility function is given by \\[\text { utility }=b \cdot t \cdot p,\\] where each letter stands for miles traveled by a specific mode. Suppose that the ratio of the price of train travel to that of bus travel \(\left(p_{t} / p_{b}\right)\) never changes a. How might one define a composite commodity for ground transportation? b. Phrase Sarah's optimization problem as one of choosing between ground \((g)\) and air \((p)\) transportation. c. What are Sarah's demand functions for \(g\) and \(p ?\) d. Once Sarah decides how much to spend on \(g\), how will she allocate those expenditures between \(b\) and \(t\) ?

Graphing complements is complicated because a complementary relationship between goods (under Hicks' definition) cannot occur with only two goods. Rather, complementarity necessarily involves the demand relationships among three (or more) goods. In his review of complementarity, Samuelson provides a way of illustrating the concept with a two-dimensional indifference curve diagram (see the Suggested Readings). To examine this construction, assume there are three goods that a consumer might choose. The quantities of these are denoted by \(x_{1}, x_{2},\) and \(x_{3} .\) Now proceed as follows. a. Draw an indifference curve for \(x_{2}\) and \(x_{3},\) holding the quantity of \(x_{1}\) constant at \(x_{1}^{0} .\) This indifference curve will have the customary convex shape. b. Now draw a second (higher) indifference curve for \(x_{2}, x_{3},\) holding \(x_{1}\) constant at \(x_{1}^{0}-h .\) For this new indifference curve, show the amount of extra \(x_{2}\) that would compensate this person for the loss of \(x_{1} ;\) call this amount \(j .\) Similarly, show that amount of extra \(x_{3}\) that would compensate for the loss of \(x_{1}\) and call this amount \(k\) c. Suppose now that an individual is given both amounts \(j\) and \(k\), thereby permitting him or her to move to an even higher \(x_{2}, x_{3}\) indifference curve. Show this move on your graph, and draw this new indifference curve. d. Samuelson now suggests the following definitions: If the new indifference curve corresponds to the indifference curve when \(x_{1}=x_{1}^{0}-2 h,\) goods 2 and 3 are independent. If the new indifference curve provides more utility than when \(x_{1}=x_{1}^{0}-2 h,\) goods 2 and 3 are complements. If the new indifference curve provides less utility than when \(x_{1}=x_{1}^{0}-2 h,\) goods 2 and 3 are substitutes. Show that these graphical definitions are symmetric. e. Discuss how these graphical definitions correspond to Hicks' more mathematical definitions given in the text. f. Looking at your final graph, do you think that this approach fully explains the types of relationships that might exist between \(x_{2}\) and \(x_{3} ?\)

A utility function is called separable if it can be written as \\[U(x, y)=U_{1}(x)+U_{2}(y),\\] where \(U_{i}^{\prime} > 0, U_{i}^{\prime \prime} < 0,\) and \(U_{1}, U_{2}\) need not be the same function. a. What does separability assume about the cross-partial derivative \(U_{x y}\) ? Give an intuitive discussion of what word this condition means and in what situations it might be plausible. b. Show that if utility is separable then neither good can be inferior. c. Does the assumption of separability allow you to conclude definitively whether \(x\) and \(y\) are gross substitutes or gross complements? Explain. d. Use the Cobb-Douglas utility function to show that separability is not invariant with respect to monotonic transformations. Note: Separable functions are examined in more detail in the Extensions to this chapter.

Example 6.3 computes the demand functions implied by the three-good CES utility function \\[U(x, y, z)=-\frac{1}{x}-\frac{1}{y}-\frac{1}{z}.\\] a. Use the demand function for \(x\) in Equation 6.32 to determine whether \(x\) and \(y\) or \(x\) and \(z\) are gross substitutes or gross complements. b. How would you determine whether \(x\) and \(y\) or \(x\) and \(z\) are net substitutes or net complements?
See all solutions

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free