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Chapter 6: Problem 5

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

Expert verified
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

01

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.
02

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 \(x_3\).
03

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.
04

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.

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Most popular questions from this chapter

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?

Heidi receives utility from two goods, goat's milk ( \(m\) ) and strudel (s), according to the utility function \\[U(m, s)=m \cdot s.\\] a. Show that increases in the price of goat's milk will not affect the quantity of strudel Heidi buys; that is, show that \(\partial s / \partial p_{m}=0\). b. Show also that \(\partial m / \partial p_{s}=0\). c. Use the Slutsky equation and the symmetry of net substitution effects to prove that the income effects involved with the derivatives in parts (a) and (b) are identical. d. Prove part (c) explicitly using the Marshallian demand functions for \(m\) and \(s\).

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.

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 \(6.1 .)\)

Donald, a frugal graduate student, consumes only coffee ( \(c\) ) and buttered toast (bt). He buys these items at the university cafeteria and always uses two pats of butter for each piece of toast. Donald spends exactly half of his meager stipend on coffee and the other half on buttered toast. a. In this problem, buttered toast can be treated as a composite commodity. What is its price in terms of the prices of butter \(\left(p_{b}\right)\) and toast \(\left(p_{t}\right) ?\) b. Explain why \(\partial c / \partial p_{b t}=0\). c. Is it also true here that \(\partial c / \partial p_{b}\) and \(\partial c / \partial p_{t}\) are equal to \(0 ?\)
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