Question

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

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

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.

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.

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.

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#

Create an initial representation

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

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.

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#

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.

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#

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.

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.

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.

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.

Key concepts

Compensating Variation

Compensating Variation (CV) is a measure used in welfare economics to assess the change in wellbeing due to price changes. It's a way to calculate how much money a consumer would need to reach their original utility level after a price increase. Think of it as the extra money needed to "compensate" a person for higher prices.

To calculate CV, you compare expenditures before and after prices change. If two prices increase, compute the initial expenditure at the old prices, and then the new expenditure at the increased prices, keeping utility constant. The CV equals the difference between these two computations:

\[ CV = e(p_1', p_2', ..., p_n, u_0) - e(p_1, p_2, ..., p_n, u_0) \]

This calculation reveals the financial impact of price hikes, aiding policy makers in understanding economic shifts' effects on consumers.

Expenditure Function

The expenditure function is a crucial concept in welfare economics. It refers to the minimum amount of money required to maintain a given level of utility, given a set of prices. Mathematically, it's denoted as \( e(p, u) \), where \( p \) is the price vector and \( u \) the utility level.

When analyzing price changes, the expenditure function helps determine how much additional expenditure is necessary to achieve the same utility. Initially, you calculate it with the initial prices, and after any price changes, you recalculate it with new prices.

For example, if prices of two goods \( p_1 \) and \( p_2 \) go up, the expenditure function aids in computing how much more a person needs to spend to maintain their utility level. This step is foundational in calculating the compensating variation.

Compensated Demand Curves

Compensated demand curves are an illustration of how demand for goods changes when consumers are compensated to maintain their utility after price changes. These curves differ from regular demand curves, which show how demand varies with price, holding income and preferences constant.

To create a compensated demand curve, start by illustrating the initial demand at the original prices and utility. Next, change one price while keeping utility constant, showing how demand adjusts. Repeat for additional price changes, visually explaining how compensation affects demand.

This graphical representation helps demonstrate welfare losses from price changes. It's an effective method to illustrate the need for additional compensation due to price increases.

Price Changes

Price changes are fundamental in understanding how they affect consumers' welfare. When prices increase, especially for goods essential to a consumer, it can significantly impact their ability to maintain the same level of satisfaction.

In welfare economics, we assess how these changes affect expenditure through the lens of CV and expenditure functions. Increasing prices of goods \( p_1 \) and \( p_2 \) require recalculating expenditures to see how much more needs to be paid just to keep the same utility. This understanding aids in designing policies to mitigate adverse effects on consumers.

When graphing compensated demand curves, we observe how demand shifts with each price change, giving visual insight into welfare costs.

Net Substitutes and Complements

The relationship between goods as net substitutes or complements can drastically affect compensating variations after price changes. Net substitutes are goods where an increase in the price of one results in an increased demand for the other. Conversely, net complements are goods where a price increase in one can cause the demand for both to drop.

When goods are net substitutes, a price hike in one good may lead to less of a welfare cost. Consumers can switch to the other good, mitigating some effects. On the other hand, if goods are net complements, both suffer when one price rises, often resulting in a larger welfare cost due to reduced consumption overall.

Understanding if goods are substitutes or complements helps predict the impact of price changes on consumer welfare and guides more informed economic decisions.