10.07.2023 Tutorial 6#

Model:

\[ y = \underbrace{\bar{y}}_{trend GDP} + \underbrace{I(i)}_{investment} + \underbrace{d}_{demand\ shock} \]

  • I = „real/ physical“ Investment as Function of Interest Rate (negative):

    • \(I = \phi_1-\phi_2i\): negative sloped line

  • Function describes GDP as part of Investment and Trend

\[ \pi = \underbrace{\pi^e}_{expected} + \alpha \underbrace{(y-\bar{y})}_{output\ gap} + \underbrace{s}_{supply \ shock} \]

  • Inflation is influenced by price shocks (e.g. oil), output gap and expectations

  • output gap is unfleunced by aggergate demand

\[ L = \beta (y- \tilde{y})^2+ \underbrace{(\pi - \pi^*)^2}_{inflation \ target} \]

  • Loss Function for central bank with two targets (inflation and GDP / employment)

  • \(\tilde{y}= k \bar{y}\) = estimate of trend with k=1 is perfect estimate

Questions:

a) price and wage setters are forward looking, incorporate exceptions into todays prices

b) explicti channel = Investment and interest rate, implicit = asset prices, exchange rate etc.

c) Barro Gordon: optimal i = minimal of loss function

\[\begin{split} L = \beta (y- \tilde{y})^2+ (\pi - \pi^*)^2 \\ = \beta [\bar{y}(1-k)+ I(i)+d]^2 + [\pi^e + a (I(i)+d)+s-\pi^*]^2 \\ \\ \frac{ dL }{di} = 2 \beta [\bar{y}(1-k)+I(i)+d] I'(i) + 2 [\pi^e+\alpha * (I(i)+d)+s-\pi^e]\alpha I'(i) = 0 \\ \\ \implies i = \frac{ \phi_1 }{\phi_2}+...+...+ \frac{ 1 }{\phi}d+ ... \end{split}\]

d) demand shock and interest rate

  • Assumptions: no supply shock, no expectations

  • then interest rate only influenced by d

\[ i = \frac{ \phi_1 }{\phi_2}+ \frac{ 1 }{\phi}d \]

  • every demand shock should be counteracted by interest rate

  • when central bank reacts accordingly, output gap = 0, \(\Delta\) unemployment = 0

Umstellen von Equation 1: und i einsetzen

\[ y-\bar{y}= \phi_1-\phi_2i+d \]

e) supply shock

\[ i = \frac{ \phi_1 }{\phi_2}+ \frac{ a }{\phi_2 (a^2+b)}s \]

  • supply shock such as oil shock

  • but here central bank reaction produces negative output gap

\[\begin{split} y-\bar{y}= \phi_1-\phi_2i \\ = \phi_1-\phi_2i \big(\frac{ \phi_1 }{\phi_2}+ \frac{ a }{\phi_2 (a^2+b)}s\big) \\ = - \frac{ a }{a^2+b}s < 0 \end{split}\]

=> lowering interest rates in supply shock = instrument with side effects

f) central bank willingness

  • if \(\beta\) high, then central bank more focused on output gap than interest rate

  • b = 0, only inflation relevant, every (supply) shock will be absorbed by CB with rising interest rates

  • b = \(\infty\) , only unemployment relevant, CB will not raise i in response to shocks

=> b > 0 , then CB not as aggressive

g) 1970s

  • oil price shock, but not enough reaction from CB

  • inflation was getting permanent, expectations cement

  • CB reaction: get kore hawkish, lower b

h) Taylor rule

\[ i-\pi = \underbrace{r}_{\phi_1/ \phi_2} + \underbrace{a(y-y)}_{d}+ \underbrace{b(\pi-\pi)}_{s} \]

j) CB Miscalculates

interest rate with no shocks and anchored inflation:

\[ i = \frac{ \phi_1 }{\phi_2}+ \frac{ b}{\phi_2 (a^2+b)}(1-k)\bar{y} \]

misplaced guess: k < 1

output gap:

\[\begin{split} y-\bar{y}= \phi_1-\phi_2 \overbrace{\big(\frac{ \phi_1 }{\phi_2}+ \frac{ b}{\phi_2 (a^2+b)}(1-k)\bar{y} \big)} \\ = \frac{ b }{a^2+b}(k-1)\bar{y} < 0 \end{split}\]

  • CB overrreaction, interest rate is too high

  • output gap is negative, inflation below target

k) presidents influence

  • short term: k>1, GDP above potential, inflation above target

  • medium term: expected inflation rises, inflationn remains high

  • long term: inflation higher than target level