CLASS 13: FIXED EXCHANGE RATES AND 'CRAWLING PEG'

• Implications for monetary policy of fixed exchange rate regime
• ... and of so called 'crawling peg': in a 'crawling peg' system the Central Bank has a 'moving target' for the exchange rate.
• This was the case of Mexico before 1994.

The Equilibrium under 'Crawling Peg'

• The case of fixed exchange rate is a special case of 'crawling peg' because the target for the exchange rate is fixed rather than moving. So we may as well study crawling peg first.
  
• In a 'crawling peg' regime the the Central Bank has a moving target for the exchange rate: the depreciation of the exchange rate has to be equal to a prespecified amount $\gamma$:
  
  $$\frac{e_t}{e_{t-1}}=(1+\gamma)\mbox{, all }t$$
  
  
  and conducts monetary policy consistently with this target.
• As you may anticipate, for the crawling peg to be successful fiscal policy has to follow suit.
• Note in the first place that the Law of One Price implies that prices have to grow at the same rate as the planned depreciation: since
  
  $$p_t=e_t p^w\mbox{, all }t,$$
  
  
  and since p^w is constant, then:
  
  $$\frac{p_t}{p_{t-1}}=\frac{e_t}{e_{t-1}}=(1+\gamma)$$
  
  
  therefore in equilibrium inflation will be constant.
• Note that the no-arbitrage condition:
  
  $$(1+r^w)\frac{e_t}{e_{t-1}}-1= \frac{u_m(c_t,\frac{M_t}{p_t})}{u_c(c_t,\frac{M_t}{p_t})}$$
  
  
  becomes:
  
  $$(1+r^w)(1+\gamma)-1= \frac{u_m(c_t,\frac{M_t}{p_t})}{u_c(c_t,\frac{M_t}{p_t})}\mbox{, all }t$$
  
  

• We can substitute this condition in the first order condition with respect to money holdings, namely:
  
  $$\frac{1}{p_t} u_c(c_t,\frac{M_t}{p_t})=$$
  
  
  
  
  $$\beta \frac{1}{p_{t+1}}[u_m(c_{t+1},\frac{M_{t+1}}{p_{t+1}})+ u_c(c_{t+1},\frac{M_{t+1}}{p_{t+1}})]$$
  
  
  to obtain:
  
  $$u_c(c_t,\frac{M_t}{p_t})= \beta \frac1{(1+\gamma)}(1+r^w)(1+\gamma)u_c(c_{t+1},\frac{M_{t+1}}{p_{t+1}})$$
  
  

• Since in the foreign country inflation is 0, we know that this implies that:
  
  $$(1+r^w)=\frac1\beta$$
  
  
  So we obtain that:
  
  
  $$u_c(c_t,\frac{M_t}{p_t})=u_c(c_{t+1},\frac{M_{t+1}}{p_{t+1}})$$
  
  

• From the no-arbitrage condition we see that this implies:
  
  $$u_m(c_t,\frac{M_t}{p_t})=u_m(c_{t+1},\frac{M_{t+1}}{p_{t+1}})$$
  
  

• Let us focus -for the time being- only on stationary equilibria, that is, equilibria where the total amount of foreign activities of the country do not change over time, that is:
  
  $$A^p_t+A^g_t=A^p_0+A^g_0\mbox{, all}t.$$
  
  
  If the total amount of foreign activities of the country do not change over time, then the current account,
  
  $$\frac1{p^w}[(A^p_t+A^g_t) (1+r^w)-(A^g_{t+1}+A^p_{t+1})]$$
  
  
  is constant. But this implies that consumption is constant because:
  
  $$c^*-y=\frac1{p^w}[(A^p_t+A^g_t) (1+r^w)-(A^g_{t+1}+A^p_{t+1})]$$
  
  
  and in particular:
  
  $$c_t=y+\frac1{p^w} (A^p_0+A^g_0) r^w=c^*.$$
  
  

• Since both equation have to hold at the same time, it is clear that, if c_t=c_t+1=c^*, then:
  
  $$\frac{M_t}{p_t}=\frac{M_{t+1}}{p_{t+1}}=m^*$$
  
  

• If real money balances are constant, and prices are growing at the rate $\gamma$, it must be that money supply is also growing at a rate $\gamma$:
  
  $$\frac{M^s_{t+1}}{M^s_t}=(1+\gamma)$$
  
  

• if money supply is growing at a rate $\gamma$, and if real money balances and consumption are constant, we know from the equilibrium relationship:
  $u_c(c^*,m)-\frac{\beta}{1+\gamma-\beta}u_m(c^*,m)=0$
  that real money balances are equal to $m^*(\gamma)$

The Impossible Trinity (Mundell?)

• The Impossible Trinity is the following:
  • Fixed (or predetermined) exchange rates
  • Open capital markets
  • Independent monetary policy
• Our model says that the Impossible Trinity is indeed impossible:
  If the no-arbitrage condition holds, once the government has chosen the path for the exchange rate the growth in the money supply (monetary policy) is determined by the equilibrium of the model - there are no more degrees of freedom.
• Note that fixed exchange rate means $\gamma=0$, so that money supply and the price level must be constant.
• This is a lesson Asian countries, and Mexico, learned the hard way: some chose to have the exchange rate floating (Mexico, most Asian countries), some chose to impose capital controls (Malaysia, Chile).
• Let us try to understand why.
  If the Central Bank wants to have a predetermined level of the exchange rate, it is fixing the price of one currency against another.
  Therefore it must be ready to exchange Peso denominated assets for $ denominated assets at the given exchange rate: at the given exchange rate the supply of foreign currency must be flat.
• With perfect capital mobility, the demand of real money balances is given by the no arbitrage condition:
  
  $$(1+r^w)\frac{e_t}{e_{t-1}}-1= \frac{u_m(c_t,\frac{M_t}{p_t})}{u_c(c_t,\frac{M_t}{p_t})}$$
  
  
  which says that the return from holding domestic currency has to be equal to the return from holding foreign denominated assets.
• For given consumption, the above equation determines real money balances. But the law of one price determines the evolution of prices! So the demand for nominal money balances is determined.
• Since in equilibrium demand must meet supply, the supply of money is also given - no longer a choice of the Central Bank.

The Implications for Fiscal Policy

• Fiscal policy as a fourth element of the impossible trinity.
• Let us look at the intertemporal budget constraint of the government:
  
  $$\frac1p^w A^g_t(1+r^w) +\sum_{s=0}^{\infty}\frac{M^s_{t+s}-M... ...1+r^w)^s}= \sum_{s=0}^{\infty}\frac{T_{t+s}}{p_{t+s}(1+r^w)^s}$$
  
  

• Given that real money balances are equal to $m^*(\gamma)$, seignorage in real terms is constant and equal to $\gamma m^*(\gamma)$
• The intertemporal budget constraint of the government becomes:
  
  $$\frac1p^w A^g_t(1+r^w) +\sum_{s=0}^{\infty}\frac{\gamma m^*(... ...= \sum_{s=0}^{\infty}\frac{\frac{T_{t+s}}{p_{t+s}}}{(1+r^w)^s}$$
  
  
  or
  
  $$\frac1p^w A^g_t(1+r^w) +\gamma m^*(\gamma)\frac{(1+r^w)}{r^w}= \sum_{s=0}^{\infty}\frac{\frac{T_{t+s}}{p_{t+s}}}{(1+r^w)^s}.$$
  
  

• This condition pins down the discounted present value of real transfers from the government to the private sector.
• The implication is that once the rate of devaluation is chosen, fiscal policy is also constrained.
  For instance, with fixed exchange rates, at any period the present discounted value of future real transfers is a function of the current amount of activities owned by the government:
  
  $$\frac1p^w A^g_t(1+r^w)=\sum_{s=0}^{\infty}\frac{\frac{T_{t+s}}{p_{t+s}}}{(1+r^w)^s}.$$
  
  

• If real transfers are constant over time at a level $\tau^*$, then the real amount of activities owned by the government also has to be constant:
  
  $$\frac1{p^w} A^g_t(1+r^w) +\gamma m^*(\gamma)\frac{(1+r^w)}{r^w}=\tau^*\frac{(1+r^w)}{r^w}$$
  
  
  so that the equilibrium level of transfers is equal to:
  
  $$\tau^*=\gamma m^*(\gamma)+\frac1{p^w} A^g_0 r^w,$$
  
  
  where A^g₀ is the initial amount of activities owned by the government.
• Of course real transfers do not have to be constant over time. In general, the path for real transfers will determine the path for the reserves of the Central Bank:
  
  $$\frac1{p^w}(A^g_t-A^g_{t+1})=\tau_t-\gamma m^*(\gamma)-\frac1{p^w}A^g_t r^w$$
  
  
  If real transfers are above seignorage plus the interest from reserves, then reserves will decrease.
• Remember that A^p_t+A^g_t=A^p₀+A^g₀, all t.
  The total amount of foreign assets in the country is constant, so if the government is decumulating reserves, the private sector must be accumulating them.
• This simply reflects the identity:
  
  CA=S_p+S_g
  
  
  since CA is constant, if S_g is going down, then S_p must be increasing.
• The government cannot be decumulating reserves forever, because of the no-Ponzi game condition: since the present value of real transfers is given, if the government chooses to increase transfers today, it must increase them tomorrow.
• However, if fiscal policy is not consistent with the equilibrium, at some point the fixed exchange rate must be abandoned.