CLASS 12: MONETARY POLICY IN AN OPEN ECONOMY

• Implications for monetary policy of crawling-peg (Mexico before 1994) and flexible (Mexico after 1995) exchange rate regimes.
• The role of the exchange rate in a the stabilization policy (Mexico 1986, Real-Brazil, Argentina).
• The constraints for fiscal policy of a fixed exchange rate regime.
• How a fiscal policy that does not respect such constraints may induce speculative attacks against the currency.

The Equilibrium in an open economy

• Defining the exchange rate:
  The exchange rate e_t is defined as the price of a foreign currency in terms of the domestic currency (for instant 9.6 Pesos for 1 US Dollar).
• In this model we assume that the government issues debt only in foreign currency (Tesobonos), and that no debt in domestic currency is issued.
  This is without loss of generality.
• The reason being that, if r^w_t is the return on $ denominated securities, the arbitrage relationship:
  
  $$(1+R_t)=(1+r^w_t)\frac{e_t}{e_{t-1}}$$
  
  
  insures that domestic and foreign denominated securities have to earn the same return.
  If this is the case, however, domestic and foreign denominated securities are perfect substitutes in the portfolio of investors: as long as the condition is satisfied, investors do not care whether they are putting the money in domestic or foreign denominated securities: all they care about is the total amount of lending/borrowing they are doing.
  So for simplicity we assume that only foreign denominated securities are issued.
• Under this assumption, the budget constraint of the representative household is:
  p_t c_t+M_t+1+e_tA^p_t+1=p_t y_t+M_t+e_tA^p_t(1+r^w_t)+T_t
  where A^p_t represents the amount of financial activities ($ denominated) held by the household at the beginning of period t
• The budget constraint of the government is now:
  M_t+1-M_t-e_t(A^g_t+1-A^g_t)+e_tA^g_tr^w_t=T_t
  where A^g_t represents the amount of financial activities ($ denominated) held by the government at the beginning of period t.
• Three equilibrium conditions have to hold.
• The first one is an arbitrage conditions in the goods market (the only good in the model - the consumption good- is assumed to be traded), called the ``Law of One Price":
  
  $$p_t=e_t p^w_t \mbox{, all } t$$
  
  

• The second is the usual equilibrium condition in the money market: money supply equal money demand:
  
  $$M^s_t=M_t \mbox{, all} t$$
  
  

• The third condition is obtained from adding up the budget constraints of the household and of the government and is:
  
  $$c_t-y_t=\frac{e_t}{p_t}[(A^p_t+A^g_t) (1+r^w_t)-(A^g_{t+1}+A^p_{t+1})]$$
  
  

• In an endowment economy, like he one we are studying, the Current Account of a country is equal to its Savings:
  
  CA=S.
  
  
  Furthermore we know that the Capital Account and the Current account have to balance out:
  
  CA+KA=0,
  
  
  or
  
  KA=-CA.
  
  

• The Capital Account is equal to net foreign investment: the change in the amount of Mexican assets owned by foreigner minus the change in the amount of foreign assets owned by Mexicans.
• In this model the quantity of -net- foreign assets owned by Mexican at the end of each period -expressed in real terms- is:
  
  $$\frac{e_t}{p_t}[A^g_{t+1}+A^p_{t+1}]$$
  
  
  (if negative, the amount of Mexican assets owned by foreigner is greater then the amount of foreign assets owned by Mexicans).
  The change in the quantity of -net- foreign assets owned by Mexican during each period is:
  
  $$\frac{e_t}{p_t}[(A^g_{t+1}+A^p_{t+1})-(A^p_t+A^g_t) (1+r^w_t)],$$
  
  
  and therefore the Capital Account balance is:
  
  $$KA=-\frac{e_t}{p_t}[(A^g_{t+1}+A^p_{t+1})-(A^p_t+A^g_t)]=$$
  
  
  
  
  $$=\frac{e_t}{p_t}[(A^p_t+A^g_t) -(A^g_{t+1}+A^p_{t+1})].$$
  
  

• The amount of savings of the country during period t is given by income minus consumption:
  
  S=y_t-c_t+NFP
  
  
  where
  
  $$NFP=\frac{e_t}{p_t}(A^p_t+A^g_t) r^w_t$$
  
  

• since -CA=-S and KA=-CA, then -S=KA, which is exactly the equilibrium condition:
  
  
  $$c_t-y_t-\frac{e_t}{p_t}(A^p_t+A^g_t) r^w_t=\frac{e_t}{p_t}[(A^p_t+A^g_t) -(A^g_{t+1}+A^p_{t+1})]$$
  
  
  or
  
  $$c_t-y_t=\frac{e_t}{p_t}[(A^p_t+A^g_t) (1+r^w_t)-(A^g_{t+1}+A^p_{t+1})]$$
  
  

• In an open economy the country can consume more than its income by borrowing from abroad (decumulating the amount of foreign assets held by residents)...
  ...provided that the country is able to pay its debt in the future.
• For simplicity from now on I will assume that foreign prices p^w and interest rates r^w are constant.
  This assumption does not affect the results that we will obtain.
  From the Law of One Price this implies that $\frac{e_t}{p_t}=\frac1p^w$ is also constant.
• The Current Account identity can be rewritten as:
  
  $$c_t-y_t=\frac1{p^w}[(A^p_t+A^g_t) (1+r^w)-(A^g_{t+1}+A^p_{t+1})]$$
  
  

• What are the intertemporal implications of the Current Account identity?
• Let us write the identity in period t=0 as:
  
  $$\frac1{p^w}(A^g_{1}+A^p_{1})=y_0-c_0+\frac1p^w(A^p_0+A^g_0) (1+r^w).$$
  
  
  This means that the amount of foreign debt held by Mexicans in period 2 is:
  
  $$\frac1{p^w}(A^g_{2}+A^p_{2})=(y_1-c_1)+(y_0-c_0)(1+r^w)$$
  
  
  
  
  $$+\frac1{p^w}(A^p_0+A^g_0)(1+r^w)^2$$
  
  
  and the amount of foreign debt held by Mexicans in period T is:
  
  $$\frac1{p^w}(A^g_{T}+A^p_{T})=\frac1{p^w}(A^p_0+A^g_0)(1+r^w)^T$$
  
  
  
  
  $$+\sum_{s=0}^{T-1}(1+r^w)^{T-1-s}(y_s-c_s)$$
  
  

• If we divide both sides by (1+r^w)^T we get:
  
  $$\frac1{p^w}\frac{(A^g_{T}+A^p_{T})}{(1+r^w)^T} =\frac{A^p_0+A^g_0}{p^w}+ \sum_{s=0}^{T-1}(1+r^w)^{-1-s}(y_s-c_s)$$
  
  

• We impose the no Ponzi game condition:
  
  $$\frac1{p^w}\frac{(A^g_{T}+A^p_{T})}{(1+r^w)^T} \rightarrow 0 \mbox{ for} T \rightarrow \infty$$
  
  
  which says that Mexicans are not willing to accumulate indefinitely foreign debt, or, if A^g_T+A^p_T is negative, that foreigners are not willing to accumulate indefinitely Mexican debt.
• We obtain:
  
  $$\frac1{p^w}(A^p_t+A^g_t)(1+r^w)= \sum_{s=0}^{\infty}\frac{c_{t+s}-y_{t+s}}{(1+r^w)^s}$$
  
  

• This condition says that if Mexico has positive net holdings of assets with respect to the rest of the world it can run Current Account deficits in the future, as long as the present value of such deficits is equal to the value of current net holdings.
• Likewise, if Mexico has negative net holdings of assets with respect to the rest of the world (negative A^p₀+A^g₀) the present discounted value of all future Current Account surpluses has to be enough cover the current debt.
• A similar condition holds for the government alone:
  
  $$\frac{A^g_t(1+r^w)}{p^w} +\sum_{s=0}^{\infty}\frac{M^s_{t+s... ...1+r^w)^s}= \sum_{s=0}^{\infty}\frac{T_{t+s}}{p_{t+s}(1+r^w)^s}$$
  
  
  which says that the present discounted value of future transfers has to equal to present discounted value of seignorage revenues, plus the net amount of assets held by the government.
• The first order condition with respect to money holdings is the usual one:
  
  $$\frac{1}{p_t} u_c(c_t,\frac{M_t}{p_t})= \beta \frac{1}{p_{t+... ...frac{M_{t+1}}{p_{t+1}})+ u_c(c_{t+1},\frac{M_{t+1}}{p_{t+1}})]$$
  
  
  from which, using transversality, we obtain:
  
  $$\frac{1}{p_t} = \frac{1}{u_c(c_t,\frac{M_t}{p_t})} \sum_{i=1... ...beta^i \frac{1}{p_{t+i}} u_m(c_{t+i},\frac{M_{t+i}}{p_{t+i}})$$
  
  

• The first order condition with respect to -foreign- assets yields:
  
  $$(1+r^w)\frac{e_t}{e_{t-1}}-1= \frac{u_m(c_t,\frac{M_t}{p_t})}{u_c(c_t,\frac{M_t}{p_t})}$$
  
  
  which is the same as the usual condition after considering the no-arbitrage relationship:
  
  $$(1+R_t)=(1+r^w_t)\frac{e_t}{e_{t-1}}$$
  
  

The Equilibrium with Flexible Exchange Rates

• Let us assume money supply grows at a constant rate $\mu$, and let us find the equilibrium inflation, consumption, etc.
• We will consider only stationary equilibria, that is, in which the assets held by households and government are constant over time:
  
  $$A^g_{t}=A^g \mbox{ and }A^p_{t}=A^p \mbox{ all }t$$
  
  
  Let us also assume that income y_t is constant.
• From the budget constraint of the household we obtain:
  
  $$c_t=y+\frac1p^w (A^p+A^g) r^w.$$
  
  

• Given that we know consumption, we can use the usual argument to determine real money demand and inflation.
• In particular inflation is equal to $\mu$, and real money balances are determined by the condition:
  
  $$u_c(c,m)-\frac{\beta}{1+\mu-\beta}u_m(c,m)=0$$
  
  
  where c is given by the expression above.
• Equilibrium real transfers are obtained from the budget constraint of the government, given that seignorage in real terms is $\mu m^*(\mu)$:
  
  $$\frac{T_t}{p_t}=\mu m^*(\mu)+\frac1p^w A^g r^w$$
  
  

• What about the equilibrium exchange rate?
• The condition
  
  $$(1+r^w)\frac{e_t}{e_{t-1}}-1=\frac{u_m(c,m^*(\mu))}{u_c(c,m^*(\mu))}$$
  
  
  determines the rate of depreciation of the currency, which is:
  
  $$\frac{e_t}{e_{t-1}}=\frac{1+\frac{u_m(c,m^*(\mu))}{u_c(c,m^*(\mu))}} {1+r^w}=\frac{\frac{1+\mu}{\beta}}{1+r^w}$$
  
  
  in other words, the rate of depreciation is equal to the ratio of the nominal interest rate at home and abroad. Notice that:
  
  $$\frac{e_t}{e_{t-1}}=1+\mu$$
  
  
  if
  
  $$1+r^w=\frac1{\beta}$$
  
  

• The faster money supply grows, the higher the rate of depreciation of the currency.
• Finally, the level of the exchange rate is determined by the Law of One Price, and by the fact that the price level is:
  
  $$p_t=\frac{M^s_t}{m^*(\mu)}$$
  
  
  obtaining:
  
  $$e_t=\frac{M^s_t}{p^w m^*(\mu)}$$
  
  

• With flexible exchange rates the equilibrium is just the same as in a closed economy. All we get is an additional set of restrictions that pin down the exchange rate.