Chapter 6

Hard Times Burt buys only rotgut whiskey and jelly donuts to sustain him. For Burt, rotgut whiskey is an inferior good that exhibits Giffen's paradox, although rotgut whiskey and jelly donuts are Hicksian substitutes in the customary sense. Develop an intuitive explanation to suggest why a rise in the price of rotgut must cause fewer jelly donuts to be bought. That is, the goods must also be gross complements.

Ms. Sarah Traveler does not own a car and travels only by bus, train, or plane. Her utility function is given by $$\text{ utility }=\mathit{B}-{\mathit{T}}^{\prime }\mathit{P}$$ 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 $(P,/P_B)$ 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 Gand P? d. Once Sarah decides how much to spend on $G$, how will she allocate those expenditures between $B$ and $7?$

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.)

Hicks's "second law" of demand states that the predominant relationship among goods is net substitutability (see footnote 3 of Chapter 6 ). To prove this result: a. Show why compensated demand functions $$X;=h,(P_U\dots ,P_n,V)$$ are homogeneous of degree zero in $P_x\dots P_n$ for a given level of $V$ b. Use Euler's theorem for homogeneous functions (for a statement of this theorem, see footnote 5 of Chapter 7 ) to show that $=0\text{ (for all }i=1,n)$ c. Use the "first law of demand" to conclude that is, net substitution must prevail, on average.

A utility function is termed separable if it can be written as \[\left.U(X, Y)=U^{\wedge} X\right)+U_{2}(Y)\] where $U\mathrm{\setminus }>0,U^{\prime \prime }<0,$ and $U_u{U}_2$ need not be the same function. a. What does separability assume about the cross partial derivative $U_{XY}?$ 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, 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.

Example 6.3 computes the demand functions implied by the three-good CES utility function a. Use the demand function for Xin Equation 6.28 to determine whether Xand For Xand 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?