Answer Happy

ah (d) (Extra credit) Show that (3) and an assumption <0 together imply that 262 ah db i. If < 0, then <0 abəw dw h db ii. If 0, then Әьд <0 but “only a little dw h db iii. If > 0, then it's likely but not certain that abaw du 6
4. Simple comparative statics. (a) Suppose the price p of b increases. (In this part, it's fine to consider just direct (substitution) effects. That is, you can ignore income effects. In other words, you 3 can me that the increase in pis small enough and/or the share of bin total expenditure is small ettongh that we can ignore changes in the marginal utility of
c.) i. Intuitively, why do we expect b* to fall when p increases? That is, why do we p db expect <0? You don't need to use any equations. dp ii. Semi-formal reasoning (you don't need to do a full derivation, but you should refer to specific equations, especially the condition for optimal c and b and specific terms in that equation): Go through the equation implicitly defining C b' and explain your answer in part i. A. suppose we are in an equilibrium with p = Po and the household has optimally chosen bó given the price po. Now suppose p increases to pı > po. What will happen in Equation (1)? How do we know that this is no longer optimal for the household?
B. How can the household adjust its optimal choice to bring Equation (1) back into balance? What will happen in Equation (1) to ensure a new equilibrium is found? Hint: remember our assumptions on ?u/ah2 and 8PU/8c? What will happen to au/ah and au/ac as the household re-optimizes? db db iii. Extra credit: formally derive and show that <0. Hint 1: you can dp dp rearrange Equation (1) to obtain an implicit function g (b.p) = 0 defining b*: au ah /p ah ab au ac 0. (2) Then, remember how to do implicit differentiation if you have an implicit function defining the relationship between y and w, as in gr. y) = 0. then to dy calculate the derivative you do not need to solve forty as a function of dir
you can use the implicit function theorem, which says that dy dr afla aflay In this case, since we have an implicit function g (b.p), the implicit function theorem tells us that db dp əg/p მq/მს Hint 2: it's helpful to write out the elements of Equation 2 a bit more com- pletely, as in OU (ch (w) au (chu (b.) an. (6.) Oh -/ ᎧᏂ Ос This will help you remember where to use the chain rule. Hint 3: you can sume hahab? or = a/Owne negative or zero
au (ch(b, w)) əh (b, w) ah ab /p- au (c, hb, w)) Әс This will help you remember where to use the chain rule. Hint 3: you can assume ho = əh/ab? or hww = 22h/awa are negative or zero. (b) Suppose exogenous income I increases. How do we expect b* to change? Fol- low the same sequence as above: intuition; semi-formal reasoning. The formal derivation is a bit too difficult, so I'm leaving it out, i. Intuition. You don't need any equations. ii. Semi-formal reasoning. You don't need to do a full derivation, but you should refer to specific equations (especially the condition for optimal c and b) and specific terms in those equations,