How to Price Rigidity

Question 1

How to Price Rigidity

Answer 1

1. Write the maximization problem up, everything is given:
   - Utility function
   - s.t. 4 different equations normally
   - Plug the condition inside into the utility function
   - Take the FOC of the utility function wrt. Y_i
   - plug one more of the condition in, because we want to have something with p_i and p.
2. Now linearizing the FOC taking logs. Isolate y_i

(2). CHECK for effect of some variable. eg. increase in lambda, will decrease price rigidity, thus output will vary less.

3. Decide a symmetric equilibrium, where prices adjust without friction.
   - The price is equal in both periods, no friction, so we will just se the two p go out in the optimal output.

(3) CHECK for the effect of a change in psi on optimal y_i:
- Take the FOC of y_i wrt. psi.
- comment, when sub stability between the goods traded in the monopolistic market increases, the deadweight loss due to imperfect information decreases, giving higher equilibrium output.

4. If prices have friction, we will assume some prices are decided on expection of the price and other on the real price.

x = 1/2 ( real price + expected price)

Show that x relies on money supply. Recall that y = m - p (got it from the last constraint taking log).
- substitute m =y + p in p
- Comment that higher money supply increases prices, thus higher x.

5. Decide the aggregate price inflation
   - pi = p_t + p_t+1, which can be written as pi = 1/2 ( x_t + x_t-1) + 1/2 ( x_t-1 + x_t-2)
   - insert the x
   - a lot of manipulation of m_t = m_t-1 + epsilon_t

Question 2

Math

Answer 2

Product rule: g(x)h(x) = g´(x)h(x)+g(x)*h´(x)

Chain rule: F´(x)=f´(g(x))*g(x) , den ydre og den indre

Kan godt være kombineret, set flere gange.

Ln rules

Ln(xy)=ln(x)+ln(y)
Ln(x/y)=ln(x)-ln(y)
ln(1/y)=ln(y^-1)=-1ln(y)
ln(k)=ln(k)
ln(x^a)=aln(x)