Question

Suppose that an individual consumes three goods, \(x_{1}, x_{2},\) and \(x_{3},\) and that \(x_{2}\) and \(x_{3}\) are similar commodities (i.e., cheap and expensive restaurant meals) with \(p_{2}=k p_{3},\) where \(k < 1-\) that is, the goods' prices have a constant relationship to one another. a. Show that \(x_{2}\) and \(x_{3}\) can be treated as a composite commodity. b. Suppose both \(x_{2}\) and \(x_{3}\) are subject to a transaction cost of \(t\) per unit (for some examples, see Problem 6.6 ). How will this transaction cost affect the price of \(x_{2}\) relative to that of \(x_{3}\) ? How will this effect vary with the value of \(t\) ? c. Can you predict how an income-compensated increase in \(t\) will affect expenditures on the composite commodity \(x_{2}\) and \(x_{3} ?\) Does the composite commodity theorem strictly apply to this case? d. How will an income-compensated increase in \(t\) affect how total spending on the composite commodity is allocated between \(x_{2}\) and \(x_{3} ?\)

Short answer

Short Answer: Yes, \(x_2\) and \(x_3\) can still be treated as a composite commodity, denoted as \(x_C\). However, the transaction cost affects their relative prices which could change the allocation of spending on these two goods within the composite commodity but might not affect the total expenditure on the composite commodity.

Step-by-step solution

Determine if \(x_2\) and \(x_3\) can be treated as a composite commodity

To demonstrate that \(x_2\) and \(x_3\) can be treated as a composite commodity, we need to show that they can be combined into a single good. Since the prices of \(x_2\) and \(x_3\) have a constant relationship as \(p_2 = kp_3\), we can form a single good consisting of a combination of \(x_2\) and \(x_3\). Denoting this composite commodity as \(x_C\), we can represent it as: \(x_C = x_2 + kx_3\) Since the relationship between \(x_2\) and \(x_3\) is constant, it is valid to treat \(x_C\) as a composite good.

Analyzing the impact of transaction cost on prices of \(x_2\) and \(x_3\)

Suppose both \(x_2\) and \(x_3\) are subject to a transaction cost of \(t\) per unit. The new prices, accounting for transaction costs, would be: \(p_2' = p_2 + t\) and \(p_3' = p_3 + t\) We are asked to analyze how the transaction cost would affect the price of \(x_2\) relative to that of \(x_3\). To do this, we can compute the new relationship between the prices: \(\frac{p_2'}{p_3'} = \frac{p_2 + t}{p_3 + t} = \frac{kp_3 + t}{p_3 + t}\) As \(t\) increases, the fraction converges to 1, weakening the relationship between the prices of \(x_2\) and \(x_3\). This implies that the effect of the transaction cost reduces the price ratio, making \(x_2\) relatively less cheap compared to \(x_3\).

Predicting the effect of an income-compensated increase in \(t\) on expenditures

If an income-compensated increase in \(t\) is implemented, the consumer's nominal income increases to compensate for the increased transaction cost, but their true purchasing power remains unchanged. This means that, theoretically, their consumption of the composite commodity should remain unchanged when considering the expenditures on both \(x_2\) and \(x_3\) together. However, because the relationship between \(x_2\) and \(x_3\) is not strictly constant anymore, it cannot be stated with certainty that the composite commodity theorem will strictly apply to this case. That is, the theorem's prediction about the unchanged consumption of the composite commodity might not hold exactly in this situation.

Assessing the allocation of total spending on \(x_2\) and \(x_3\)

Due to the income-compensated increase in \(t\), the consumer's income remains constant in real terms. However, because the relationship between the prices of \(x_2\) and \(x_3\) is no longer strictly constant, the allocation of spending on the composite good might change. As the transaction cost increases, the relative price of \(x_2\) becomes higher in comparison to \(x_3\). This price change can lead the consumer to allocate more spending on \(x_3\) rather than \(x_2\), shifting the allocation. In summary, while the total expenditure on the composite commodity may not be strictly influenced by the income-compensated transaction cost increase, the allocation of spending on \(x_2\) and \(x_3\) is likely to be affected by this change.

Key concepts

Transaction Costs

Transaction costs are the additional costs associated with buying or selling a good, beyond the actual price of the good. These costs can be monetary, such as fees and taxes, or non-monetary, like the time spent traveling to a store. In the context of our exercise, a transaction cost (\( t \)) is applied to each unit of goods \( x_2 \)and \( x_3 \).

These additional expenses can alter the way consumers perceive the affordability of products.
For instance, if a transaction cost of \(t\) is added per unit, the effective prices of the goods change:

• The new price of \(x_2\) becomes \(p_2 + t\).
• Similarly, the price for \(x_3\) changes to \(p_3 + t\).

As transaction costs increase, the comparative price difference between \(x_2\) and \(x_3\) diminishes, making \(x_2\) relatively more expensive. This impacts consumer choice by potentially reducing the attractiveness of \(x_2\).

Understanding these nuances helps us see why transaction costs can significantly impact market dynamics, altering demand and influencing how goods are consumed.

Consumption Analysis

Consumption analysis examines how consumers allocate their purchasing power among different goods and services to maximize satisfaction or utility. In the exercise, the notion of treating \( x_2 \) and \( x_3 \) as a composite commodity involves observing their price relationship, where \( p_2 = k \cdot \ p_3 \), and considering their overall consumption as a single unit.

The composite commodity \( x_C = x_2 + k \cdot \ x_3 \) reflects the combined utility derived from these similar commodities. By analyzing how consumers interact with this composite good, we can deduce patterns in their consumption behavior given different economic circumstances, such as changes in prices or transaction costs.

Such analysis reveals the flexibility in a consumer's budget allocation. It demonstrates how consumers might adjust their consumption when faced with varying prices and costs, highlighting the potential shift in expenditure proportions between \( x_2 \)and \( x_3 \) due to transaction costs.

Income Compensation

Income compensation refers to adjustments in consumer income to neutralize effects caused by changes in prices or costs, ensuring the consumer’s purchasing power remains unchanged.

In this scenario, when a transaction cost increase is matched with an income compensation, the consumer ideally should be able to maintain the same level of consumption for a composite commodity consisting of \( x_2 \) and \( x_3 \).

However, because the price ratio between \( x_2 \) and \( x_3 \) is impacted by transaction costs, the detailed distribution of the consumer's spending might shift.

• As \(t\) causes a rise in \(p_2\) relative to \(p_3\), it affects decisions on how much of each to purchase.
• Although overall expenditures on the composite could stabilize, the balance within this expenditure between \(x_2\) and \(x_3\) might change.

The composite commodity theorem assumes constant relative prices, so deviations in price ratios challenge its application. Understanding these dynamics in income-compensated scenarios is key to grasping consumer responsiveness to economic changes.