Toulouse School of Economics, Macroeconomics II Franck Portier. Homework 1 Solutions. Problem I An AD-AS Model

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1 Toulouse School of Economics, Macroeconomics II Franck ortier Homework Solutions max Π = A FOC: d = ( A roblem I An AD-AS Model ) / ) 2 Equilibrium on the labor market: d = s = A and = = A Figure : Equilibrium of the labor market s d 3 See figure 2. Figure 2: Aggregate Supply s

2 4 IS: set of (, i) such that planned and actual expenditures are equal, i.e. = C + I + G, which gives = c( τ) γ c( τ) i (IS) 5 M: set of (, i) such that the money market clears; i.e. M d = M s i = β M + β (M) 6 IS-M : 7 D = e can check that D < 0 where B/ is public debt issuing. possible policies are Fiscal policy dg = db/, dg = dt, dg = dm/ Tax cut: dt = -d B/ Open market dm/ = db/ 8 See figure 3 γ β M ( τ)c + γ (G c T + C + I) + β ( τ)c + γ β G = T + τ + M + B Figure 3: Aggregate Demand- Aggregate Supply s d 9 Equilibrium real wage: = A labor market equilibrium price : = (A) This price needs not to coincide with the price that is given by aggregate demand there can be unemployment: if <, d < unemployment if >, d > underemployment e assume that transactions are given by the min of demand and supply (voluntary exchange assumption) see figure 4. 0 The AD curve is the same than before. The AS curve is now given by: { d if < = if { ( /( ) = A) /( ) if < if 2

3 Figure 4: Transactions on the labor market s represents transactions = min(demand,supply) d and therefore See figure 5 S = { ( /( ) A A) /( ) if < if Figure 5: AD-AS Model s Classical zone Intermediate zone IS-M zone d See Romer s book, chapter 6, part A roblem II A ucas 73 type model 3

4 roblem III Cagan s model log(m t / t ) = 0 + log y t + 2 R t + u t () - A priori, the signs of the coefficients are: > 0 and 2 < 0. A high activity (high y) implies a need for liquidity and then a high demand of money. The nominal interest rate represents the opportunity cost to hold money, thus if R is high, the demand of money is low. 2- R t = r t + π t, y t = y, r t = r then () is equivalent to m t p t = ( 0 + log y t + 2 r t ) + 2 π t + u t = γ + π t + u t (2) where γ = 0 + log y t + 2 r t and = 2 3- e can observe that when π is high, m p is low. This is in accordance with the money demand if < 0, if one accumulates expected and actual inflation. 4- The problem with equation (9) is that we do not observe expectations. e only observe actual inflation, and we don t know how much was expected. 5- Adaptive expectations (error correcting mechanism). The anticipations of inflation are proportional to the last period prediction errors. π t π t = λ( p t π t ) (3) 6- (3) implies π t = λ p t + ( λ)π t = λ p t + ( λ){λ p t + ( λ)π t 2 } = λ p t + λ( λ) p t + ( λ) 2 π t 2 = λ i=0 δp t i if we assume lim j ( λ) j π t j = 0. It is a weighted average of past observation. The parameter λ represents the weight of past observations in the expectations of inflation. 7- e have and Then m t p t = γ + π t + u t (5) π t = λ p t + ( λ)π t (4) lug in (5), we obtain: m t p t π t = λ p t + ( λ){ (m t p t ) γ u t } ( λ) ( λ)γ ( λ) = λ p t + (m t p t ) u t = γ + λ p t + ( λ)(m t p t ) ( λ)γ ( λ)u t + u t = λ + λ p t + ( λ)(m t p t ) ( λ)u t + u t (6) 8- e observe negative and λ lies between 0 and. The fit is good (R 2 are high). The λ s are small: the expectations are very sticky, persistent. 9- (6) m t p t = λ + λp t λp t + ( λ)m t ( λ)p t ( λ)u t + u t 4

5 If m t = m, u t = 0 for all t Stable if +λ λ +λ < p t ( + λ) = ( + λ λ)p t + m t ( λ)m t + ( λ)u t λγ p t = + λ λ + λ p t + λm λγ 0- In Germany and Russia, one can have hyper inflation with constant money growth. If this model is correct, hyperinflation is not always a monetary phenomenon, caused by too expansionist monetary policy. It can be driven by expectations. - The problem with adaptive expectations is that agents make forecastable prediction errors. 2- π t = p t+ Now the model solution is the solution of: with m t = m, u t = 0 t. m t p t = γ + (p t p t ) + u t (7) where = > if < 0. et s assume < 0 for now on. p t+ p t+ = ( )p t + m γ = p t + m γ (8) 3- ong run value p And: p = ( )p + m γ p = m γ p = (m γ) p t+ (p t+ p) (p t+ p) = = = p t + m γ (p t p) + p + m γ (p t p) (9) (0) 4- Therefore, (p t+ p) = ( )t+ (p 0 p) p t+ = ( )t+ (p 0 p) + p () 5- if p 0 p, lim t p t+ = ± or if p 0 = p, p t+ = p t. see figure 6 6- et p = ( m γ) and p = ( m γ) see figure 7 Then π t = E t [p t+ ] p t (2) m t p t = γ + (E t [p t+ ] p t ) + u t ( )p t = E t [p t+ ] γ + m t u t (3) 7-8- Take E t implies: ( )p t+ = E t+ p t+2 γ + m t+ u t+ (4) ( )E t p t+ = E t p t+2 γ + E t m t+ E t (p t+ ) = E t[p t+2 ] γ + Et[mt+] (5) 5

6 Figure 6: The Equilibrium rice Jumps on its Stationary Value p p t p Figure 7: The Effect of a ermanent Shock p p T t 6

7 9- (3) p t = E t[p t+ ] γ + m t u t using (5) and solving forward: (6) p t = { E t[p t+2 ] =... = {m t γ{ γ + Et[mt+] } γ + mt ut + ( )2 +...} u t + E tm t+ + ( )2 E t m t } (7) 20- The current price level is a function of all expected future. 2- persistence in money supply. 22- m t = µ 0 + µ m t + e t E t (m t+ ) = E t (µ 0 + µ m t + e t+ ) = µ 0 + µ m t (8) E t (m t+2 ) = E t (µ 0 + µ m t+ + e t+2 ) = µ 0 + µ µ 0 + µ 2 m t p t 23- Money demand shock: u t. u t = < 0. A positive shock on real money demand for a r and m t given, we need real money supply to increase to reach equilibrium t decreases. 24- pt m t = µ = ( µ ) > 0 if µ =, p t m t = 25-Quantitative theory of money py = mv, if y = v = cste, affected (if it corresponds to a change in v). p m =. If µ < p m < because the money demand is roblem IV Identification in VARs Consider VAR with includes output, prices, nominal interest rates and money, t = [GD t, t, i t, M t ]. There are four shocks in the economy. Suppose that a class of models suggests that output contemporaneously reacts only to its own shocks; that prices respond contemporaneously to output and money shocks; that interest rates respond contemporaneously only to money shocks, while money contemporaneously responds to all shocks. Are the four structural shocks identifiable? Solution : Using the notations of the course, ν = A(0)ε, where ν is the vector of structural shocks, ε the vector of non structural (and non orthogonal) shocks. A(0) represents the impact effect of shocks on the macro variables. et s denote a ij the line i column j element of A(0). For example, a 2 is the impact effect on output of the money shock. Knowing Ω the variance-covariance matrix of ε, we have V (A(0)ε) = V (ν) A(0)A(0) = Ω This give us 0 independent equations and we need to find the 6 coefficients of A(0). The restrictions stated in the question imply some zeros in the A(0) matrix: a A(0) = a 2 a 22 0 a a 33 a 34 a 4 a 42 a 43 a 44 Note that we are left with 0 coefficients, which is the number of equations that we have. The structural shocks can then be identified. 2 Suppose we have extraneous information which allows us to pin down some of the parameters of the matrix A(0) (this notation is the one of chapter ). For example, suppose in a trivariate system with output, hours and taxes, we can obtain estimates of the elasticity of hours with respect to taxes. How many restrictions do you need to identify the shocks? Does it make a difference if zero or constant restriction is used? 7

8 Solution : In this case, the matrix A(0) is 3 3 so that 9 coefficients are to be found. 6 equations are given by the equality of variance of ν and A(0)ε. e then need 3 more restrictions. Assume that we want to identify an output shock, an hour shock and a tax shock (in that order). Knowing the elasticity of hours to a tax shock (say it is k) implies that a 23 = k, which already give one restriction. Two other will needed to identify the 3 shocks. Note that for identification, it does not matter whether the restriction is a zero or a real non zero number (i.e. whether k is null or not). 8