Chapter 6: Problem 12

Details of the analysis suggested in Problems 6.5 and 6.6 were originally worked out by Borcherding and Silberberg (see the Suggested Readings) based on a supposition first proposed by Alchian and Allen. These authors look at how a transaction charge affects the relative demand for two closely substitutable items. Assume that goods $x_{2}$ and $x_{3}$ are close substitutes and are subject to a transaction charge of $t$ per unit. Suppose also that good 2 is the more expensive of the two goods (i.e., "good apples" as opposed to "cooking apples". Hence the transaction charge lowers the relative price of the more expensive good [i.e., $\left.\left(p_{2}+t\right) /\left(p_{3}+t\right) \text { decreases as } t \text { increases }\right] .$ This will increase the relative demand for the expensive good if $\partial\left(x_{2}^{c} / x_{3}^{c}\right) / \partial t > 0$ (where we use compensated demand functions to eliminate pesky income effects). Borcherding and Silberberg show this result will probably hold using the following steps. a. Use the derivative of a quotient rule to expand $\partial\left(x_{2}^{c} / x_{3}^{c}\right) / \partial t$. b. Use your result from part (a) together with the fact that, in this problem, $\partial x_{i}^{\epsilon} / \partial t=\partial x_{i}^{c} / \partial p_{2}+\partial x_{i}^{\epsilon} / \partial p_{3}$ for $i=2,3,$ to show that the derivative we seek can be written as \\[\frac{\partial\left(x_{2}^{c} / x_{3}^{c}\right)}{\partial t}=\frac{x_{2}^{c}}{x_{3}^{c}}\left[\frac{s_{22}}{x_{2}}+\frac{s_{23}}{x_{2}}-\frac{s_{32}}{x_{3}}-\frac{s_{33}}{x_{3}}\right],\\] $\text { where } s_{i j}=\partial x_{i}^{c} / \partial p_{j}.$ c. Rewrite the result from part (b) in terms of compensated price elasticities: \\[e_{i j}^{c}=\frac{\partial x_{i}^{c}}{\partial p_{j}} \cdot \frac{p_{j}}{x_{i}^{c}},\\] d. Use Hicks' third law (Equation 6.26 ) to show that the term in brackets in parts (b) and (c) can now be written as \\[\left[\left(e_{22}-e_{23}\right)\left(1 / p_{2}-1 / p_{3}\right)+\left(e_{21}-e_{31}\right) / p_{3}\right].\\] e. Develop an intuitive argument about why the expression in part (d) is likely to be positive under the conditions of this problem. Hints: Why is the first product in the brackets positive? Why is the second term in brackets likely to be small? f. Return to Problem 6.6 and provide more complete explanations for these various findings.

Short Answer

Expert verified

In conclusion, the increase in transaction charges has the effect of increasing the relative demand for the more expensive good. This is due to the higher price sensitivity of demand for the more expensive good, which is reflected in the compensated price elasticities. The less significant impact of good 1's price shows that the relative price change between goods 2 and 3 is the primary driver for this shift in demand. By accounting for income effects through the compensated demand functions, we can isolate the role of relative prices in this outcome. This finding highlights the importance of transaction charges in influencing consumer behavior and preferences between closely substitutable goods, especially when one good is more expensive than the other.

Step by step solution

a. Expand $\partial\left(x_{2}^{c} / x_{3}^{c}\right) / \partial t$#

To find the derivative $\partial\left(x_{2}^{c} / x_{3}^{c}\right) / \partial t$, we can use the quotient rule: $$\frac{d}{dt}\left(\frac{f(t)}{g(t)}\right) = \frac{f'(t)g(t) - g'(t)f(t)}{\left[ g(t) \right]^2},$$ where $f(t) = x_{2}^{c}$ and $g(t) = x_{3}^{c}$. Applying this to our given function, we get: $$\frac{\partial\left(x_{2}^{c} / x_{3}^{c}\right)}{\partial t} = \frac{\left(\frac{\partial x_{2}^{c}}{\partial t} \right) x_{3}^{c} - \left(\frac{\partial x_{3}^{c}}{\partial t} \right) x_{2}^{c}}{\left[x_{3}^{c}\right]^2}.$$

b. Write the derivative in terms of $s_{ij}$#

We are given that $\partial x_{i}^{\epsilon} / \partial t=\partial x_{i}^{c} / \partial p_{2}+\partial x_{i}^{\epsilon} / \partial p_{3}$ for $i=2,3$. Using this information, we can rewrite the derivative as: $$\frac{\partial\left(x_{2}^{c} / x_{3}^{c}\right)}{\partial t} = \frac{x_{2}^{c}}{x_{3}^{c}}\left[\frac{s_{22}}{x_{2}}+\frac{s_{23}}{x_{2}}-\frac{s_{32}}{x_{3}}-\frac{s_{33}}{x_{3}}\right],$$ where $s_{ij} = \partial x_{i}^{c} / \partial p_{j}$.

c. Rewrite the result in terms of compensated price elasticities $e_{ij}^{c}$#

We know that: $$e_{i j}^{c}=\frac{\partial x_{i}^{c}}{\partial p_{j}} \cdot \frac{p_{j}}{x_{i}^{c}}$$ We can rewrite the result from part (b) in terms of the compensated price elasticities: $$\frac{\partial\left(x_{2}^{c} / x_{3}^{c}\right)}{\partial t} = \frac{x_{2}^{c}}{x_{3}^{c}}\left[\frac{e_{22}^{c}x_{2}^{c}}{p_{2}}+\frac{e_{23}^{c}x_{2}^{c}}{p_{3}}-\frac{e_{32}^{c}x_{3}^{c}}{p_{2}}-\frac{e_{33}^{c}{x_{3}^{c}}{p_{3}}\right]$$

d. Use Hicks' third law to rewrite the expression#

Hicks' third law (Equation 6.26) states that the sum of all compensated price elasticities equals $-1$ for each good: $$e_{21}^{c}+e_{22}^{c}+e_{23}^{c} = -1$$ $$e_{31}^{c}+e_{32}^{c}+e_{33}^{c} = -1$$ Using this information, we can write the expression as: $$\frac{\partial\left(x_{2}^{c} / x_{3}^{c}\right)}{\partial t} = \frac{x_{2}^{c}}{x_{3}^{c}}\left[\left(e_{22}-e_{23}\right)\left(1 / p_{2}-1 /p_{3}\right)+\left(e_{21}-e_{31}\right) / p_{3}\right]$$

e. Intuitive argument about the expression in part (d) being positive#

For the first product in the brackets to be positive: $$(e_{22}-e_{23})\left(1 / p_{2}-1 / p_{3}\right)$$ The difference between $e_{22}$ and $e_{23}$ should be positive because good $2$ is more expensive and a higher price typically results in a higher compensated price elasticity (demand becomes more sensitive to price changes). Also, as $p_2>p_3$, the term $(1 / p_{2}-1 / p_{3})$ is positive. As for the second term in the brackets, $$\left(e_{21}-e_{31}\right) / p_{3}$$ it is likely to be small, because both $e_{21}$ and $e_{31}$ are the responses of demand for goods $2$ and $3$ to a change in the price of good $1$. Since good $1$ is not involved in this problem, the effects on demand for goods $2$ and $3$ are small and have less impact on the overall result, making this term less significant. Thus, the overall expression in part (d) is likely to be positive, as the first product is positive and the second term is small, resulting in an increase in relative demand for the more expensive good.

f. Complete explanations for the findings in Problem 6.6#

In Problem 6.6, we have demonstrated that an increase in the transaction charge will increase the relative demand for the more expensive good. This implies that when transaction costs are changed, it affects the demand of both goods but it has a greater impact on the more expensive one. This is because the price decrease relative to the cheaper good incentivizes consumers to purchase more of the expensive good, as they perceive it as a better deal. Additionally, the income effects have been eliminated using compensated demand functions, which ensure that the result only reflects the change in relative prices and not any change in the consumer's real income.