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Chapter 6: Problem 3

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 \(0 ?\)

Short Answer

Expert verified
Answer: The partial derivative of coffee consumption with respect to the price of buttered toast is equal to zero because Donald spends half of his stipend on coffee regardless of the price of buttered toast. Any change in the price of buttered toast does not affect the amount he spends on coffee, as he will still spend half of his stipend on it.

Step by step solution

01

Find the price of buttered toast

Considering that Donald always uses two pats of butter for each piece of toast, the price of a buttered toast (bt) can be written as the sum of the price for the toast (\(p_t\)) and twice the price for butter (\(p_b\)). So, the price of buttered toast, \(p_{bt}\), is given by: \[p_{bt}=p_t+2p_b\]
02

Explain why \(\partial c / \partial p_{b t}=0\)

Since Donald spends exactly half of his stipend on coffee and the other half on buttered toast, any change in the price of buttered toast (\(p_{bt}\)) will not affect the amount he spends on coffee (\(c\)), as he will still spend half of his stipend on it regardless of the prices. This implies that the partial derivative of coffee consumption with respect to the price of buttered toast is zero, i.e., \(\partial c / \partial p_{b t}=0\).
03

Check if \(\partial c / \partial p_{b}\) and \(\partial c / \partial p_{t}\) are equal to \(0\)

We know that Donald spends half of his stipend on coffee and the other half on buttered toast. Let's denote Donald's stipend as \(s\). Therefore, we can write: \[c = \frac{s}{2p_c}\] \[bt = \frac{s}{2p_{bt}}\] where \(p_c\) is the price of coffee, and we already found \(p_{bt}\) in Step 1. Now, we need to find the partial derivatives of coffee consumption (\(c\)) with respect to the prices of butter (\(p_b\)) and toast (\(p_t\)): \[\frac{\partial c}{\partial p_{b}} = \frac{\partial}{\partial p_{b}}\left(\frac{s}{2p_c}\right) = 0\] since the price of butter doesn't affect the amount spent on coffee. \[\frac{\partial c}{\partial p_{t}} = \frac{\partial}{\partial p_{t}}\left(\frac{s}{2p_c}\right) = 0\] since the price of toast doesn't affect the amount spent on coffee. Therefore, it is true that \(\partial c / \partial p_{b}=0\) and \(\partial c / \partial p_{t}=0\).

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Most popular questions from this chapter

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.

Ms. Sarah Traveler does not own a car and travels only by bus, train, or plane. Her utility function is given by \\[\text { utility }=b \cdot t \cdot p,\\] where each letter stands for miles traveled by a specific mode. Suppose that the ratio of the price of train travel to that of bus travel \(\left(p_{t} / p_{b}\right)\) never changes a. How might one define a composite commodity for ground transportation? b. Phrase Sarah's optimization problem as one of choosing between ground \((g)\) and air \((p)\) transportation. c. What are Sarah's demand functions for \(g\) and \(p ?\) d. Once Sarah decides how much to spend on \(g\), how will she allocate those expenditures between \(b\) and \(t\) ?

Example 6.3 computes the demand functions implied by the three-good CES utility function \\[U(x, y, z)=-\frac{1}{x}-\frac{1}{y}-\frac{1}{z}.\\] a. Use the demand function for \(x\) in Equation 6.32 to determine whether \(x\) and \(y\) or \(x\) and \(z\) are gross substitutes or gross complements. b. How would you determine whether \(x\) and \(y\) or \(x\) and \(z\) are net substitutes or net complements?

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 \(6.1 .)\)

Graphing complements is complicated because a complementary relationship between goods (under Hicks' definition) cannot occur with only two goods. Rather, complementarity necessarily involves the demand relationships among three (or more) goods. In his review of complementarity, Samuelson provides a way of illustrating the concept with a two-dimensional indifference curve diagram (see the Suggested Readings). To examine this construction, assume there are three goods that a consumer might choose. The quantities of these are denoted by \(x_{1}, x_{2},\) and \(x_{3} .\) Now proceed as follows. a. Draw an indifference curve for \(x_{2}\) and \(x_{3},\) holding the quantity of \(x_{1}\) constant at \(x_{1}^{0} .\) This indifference curve will have the customary convex shape. b. Now draw a second (higher) indifference curve for \(x_{2}, x_{3},\) holding \(x_{1}\) constant at \(x_{1}^{0}-h .\) For this new indifference curve, show the amount of extra \(x_{2}\) that would compensate this person for the loss of \(x_{1} ;\) call this amount \(j .\) Similarly, show that amount of extra \(x_{3}\) that would compensate for the loss of \(x_{1}\) and call this amount \(k\) c. Suppose now that an individual is given both amounts \(j\) and \(k\), thereby permitting him or her to move to an even higher \(x_{2}, x_{3}\) indifference curve. Show this move on your graph, and draw this new indifference curve. d. Samuelson now suggests the following definitions: If the new indifference curve corresponds to the indifference curve when \(x_{1}=x_{1}^{0}-2 h,\) goods 2 and 3 are independent. If the new indifference curve provides more utility than when \(x_{1}=x_{1}^{0}-2 h,\) goods 2 and 3 are complements. If the new indifference curve provides less utility than when \(x_{1}=x_{1}^{0}-2 h,\) goods 2 and 3 are substitutes. Show that these graphical definitions are symmetric. e. Discuss how these graphical definitions correspond to Hicks' more mathematical definitions given in the text. f. Looking at your final graph, do you think that this approach fully explains the types of relationships that might exist between \(x_{2}\) and \(x_{3} ?\)
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