Question

Suppose that an individual consumes three goods, \(X_{u} X_{2},\) and \(X_{3},\) and that \(X_{2}\) and \(X_{3}\) are similar commodities (i.e., cheap and expensive restaurant meals) with \(P_{2}=\mathrm{KP}_{3}\) where \(K<1-\) that is, the goods' prices have a constant relationship to one another. a. Show that \(\mathrm{X}_{2}\) and \(\mathrm{X}_{3}\) can be treated as a composite commodity. b. Suppose both \(\mathrm{X}_{2}\) and \(\mathrm{X}_{3}\) are subject to a transaction cost of \(t\) per unit (for some exam ples, see Problem 6.6 ). How will this transaction cost affect the price of \(X_{2}\) relative to that of \(\mathrm{X}_{3} ?\) 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 \(\mathrm{X}_{2}\) and \(\mathrm{X}_{3} ?\) 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 com posite commodity is allocated between \(X_{2}\) and \(X_{3} ?\) (For a further discussion of the complications involved in this problem, see \(T . E\) Borcherding and E. Silberberg, "Shipping the Good Apples Out: The Alchian- Allen Theorem Reconsidered," Journal ofPolitical Economy [February 1978]: 131-138.)

Short answer

Answer: The allocation of spending between X2 and X3 in response to an income-compensated increase in transaction costs depends on the consumer's preferences. If the consumer tends to substitute more expensive goods with cheaper alternatives when transaction costs increase, they may spend more on the cheaper good, X2. Conversely, if the consumer's preference is biased towards the more expensive good, spending on X3 may remain higher. The precise allocation of spending cannot be predicted without information on the consumer's preferences.

Step-by-step solution

Consider the Budget Constraint

The consumer's budget constraint can be written as follows: \(M = P_1X_1 + P_2 X_2 + P_3X_3\) Given that \(P_2 = KP_3\), we can rewrite the budget constraint as: \(M = P_1X_1 + KP_3X_2 + P_3X_3\) Now, we can factor out the common price factor \(P_3\): \(M = P_1X_1 + P_3(KX_2 + X_3)\) By defining \(Y = KX_2 + X_3\), we have the new budget constraint: \(M = P_1X_1 + P_3Y\) We can treat \(Y\) as a composite commodity, as it represents the sum of the consumption of goods \(X_2\) and \(X_3\) weighted by their constant relationship of prices (\(K\)). b.

Add the transaction cost

With the transaction cost \(t\) per unit for both \(X_2\) and \(X_3\), the new budget constraint is: \(M = P_1X_1 + (P_2+t)X_2 + (P_3+t)X_3\) As \(P_2 = KP_3\), we can write this as: \(M = P_1X_1 + (KP_3+t)X_2 + (P_3+t)X_3\) Now, the ratio of effective prices becomes: \(\frac{(KP_3+t)}{(P_3+t)}\) Comparing this ratio with \(K\), we notice that the effect of the transaction cost on the relative price will depend on the value of \(t\). If \(t\) is large compared to \(P_2\) and \(P_3\), the transaction cost will dominate and the relative price will be close to 1. On the other hand, if \(t\) is small, the relative price will be close to \(K\). c.

Effects of Income-compensated increase in t

For this part, we need to predict how an income-compensated increase in \(t\) will affect expenditures on the composite commodity \(Y = KX_2 + X_3\). The income compensation means that the consumer's income increases so as to offset the increase in the transaction cost for a fixed consumption level. If the increase in \(t\) is absorbed by the compensation, the expenditure would remain unchanged. However, since we have a composite commodity where both goods have a transaction cost, the composite commodity theorem does not strictly apply to this case, as the income effect may not be completely compensated for both \(X_2\) and \(X_3\). Therefore, it is difficult to predict the exact outcome of the income-compensated increase in \(t\) without information on the consumer's preferences. d.

Allocate Spending between X2 and X3

An income-compensated increase in \(t\) will affect how total spending on the composite commodity is allocated between \(X_2\) and \(X_3\), depending on the consumer's preferences. If the consumer tends to substitute more expensive goods with cheaper alternatives when transaction costs increase, then we can expect a larger share of spending on the cheaper good \(X_2\). Conversely, if the consumer's preference is biased towards the more expensive good, the share of spending on \(X_3\) may remain higher. As preferences are not given in the problem, we cannot predict the precise allocation of spending between \(X_2\) and \(X_3\).

Key concepts

Budget Constraint

The idea behind a budget constraint is to explain how people decide to allocate their limited income across different goods. Picture this constraint as an invisible line that tells us the limits within which they can spend money. When it comes to this problem, imagine a consumer who has funds amounting to M. This consumer can spend this money on three goods: \(X_1, X_2, \) and \(X_3\). Each one of these comes with its own price, being \(P_1, P_2, \) and \(P_3\), respectively. Initially, the budget constraint can be written as:

• \(M = P_1X_1 + P_2 X_2 + P_3X_3\)

However, when we know \(P_2 = KP_3\), we can modify this pesky equation to simplify things:

• \(M = P_1X_1 + KP_3X_2 + P_3X_3\)

By organizing terms, you can see that we define a new variable \(Y\), which represents a combination of \(X_2\) and \(X_3\):

• \(M = P_1X_1 + P_3Y\)

This tells us that the consumer can also think about how much they spend on composite goods. They get to choose between buying more of \(X_1\) or combining \(X_2\) and \(X_3\) as one commodity. This simplified model helps consumers and economists alike understand how a fixed income is spread over different goods or services.

Transaction Cost

Transaction costs arise from hurdles like taxes, fees, or even time that influence the actual cost of purchasing goods. Think of transaction costs as these additional charges that play a role in the purchasing process. Now, both \(X_2\) and \(X_3\) have a transaction cost \(t\), which affects how effective the price is.Now, redefine your budget constraint as:

• \(M = P_1X_1 + (P_2+t)X_2 + (P_3+t)X_3\)

Given \(P_2 = KP_3\), substitute once more and the constraint adjusts to:

• \(M = P_1X_1 + (KP_3+t)X_2 + (P_3+t)X_3\)

Here’s the twist: calculate the new ratio for effective prices:

• \(\frac{(KP_3+t)}{(P_3+t)}\)

When \(t\) is significant compared to \(P_2\) and \(P_3\), it can push the relative price closer to 1, which leads to more equal pricing between \(X_2\) and \(X_3\). With a smaller \(t\), it hangs around the proportional price relationship \(K\). So, transaction costs have a vital influence, reshaping how prices stack up relative to each other. This indicates the need to justifiably adjust purchases based on not just price, but the accompanying transaction cost.

Income-Compensated Price Change

An income-compensated price change is a concept that focuses on balancing a consumer's income in response to price changes. Here, the compensation reflects increases in transaction costs through an adjustment in income, ensuring the consumer's ability to afford goods remains unchanged at a certain consumption level.The main question is: how does this affect consumer choices between the composite commodity \(Y = KX_2 + X_3\)? If income compensates for an increase in \(t\), then, in principle, the expenditure should stay the same. However, two complexities arise:

• The composite commodity theorem doesn't quite hold due to transaction costs; treating these goods as one might not completely represent real choices.
• Consumer preferences play a significant role. If more expensive items see a substitution effect towards cheaper options due to price hikes, spending can shift dramatically according to personal preferences.

Because the theorem doesn't apply strictly, one cannot predict decisively without diving deep into consumer preferences or observing real-world reactions. Hence, even though expenditures might seem traditionally stable per compensatory adjustments, other forces drive spending between \(X_2\) and \(X_3\), making the forecast tricky without further insights into consumer behavior.