Simple comparative statics. (a) Suppose the price p of b increases. (In this part, it’s fine to consider just direct (su

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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
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can assume that the increase in p is small enough and/or the share
of b in total
expenditure is small enough 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
expect
db∗
dp < 0? You don’t need to use 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 that equation): Go through the equation
implicitly defining
b
∗ and explain your answer in part i.
A. suppose we are in an equilibrium with p = p0 and the household
has
optimally chosen b
∗
0
given the price p0. Now suppose p increases to p1 > p0.
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 ∂
2U/∂h2 and
∂
2U/∂c2
. What will happen to ∂U/∂h and ∂U/∂c as the household
re-optimizes?
iii. Extra credit: formally derive db∗
dp and show that db∗
dp < 0. Hint 1: you can
rearrange Equation (1) to obtain an implicit function g (b, p) = 0
defining b
∗
:
b
∗
:
∂U
∂h
∂h
∂b /p −
∂U
∂c = 0. (2)
Then, remember how to do implicit differentiation: if you have an
implicit
function defining the relationship between y and x, as in f (x, y)
= 0, then to
calculate the derivative dy
dx, you do not need to solve for y as a function of x,
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you can use the implicit function theorem, which says that
dy
dx = −
∂f/∂x
∂f/∂y .
In this case, since we have an implicit function g (b, p), the
implicit function
theorem tells us that
db
dp = −
∂g/∂p
∂g/∂b .
Hint 2: it’s helpful to write out the elements of Equation 2 a bit
more completely, as in
∂U (c, h (b, w))
∂h
∂h (b, w)
∂b /p −
∂U (c, h (b, w))
∂c .
This will help you remember where to use the chain rule. Hint 3:
you can
assume hbb = ∂
2h/∂b2 or hww = ∂
2h/∂w2 are negative or zero.
(b) Suppose exogenous income I increases. How do we expect b
∗
to change? Follow 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.