Lecture Notes Exchange Rates and the ‘New Open Economy Macroeconomics’ 1. So far we have focused only on the closed economy. Here we take the dynamic optimizing model of a two country open economy model, and introduce sticky prices. This extends the framework of the one-consumer closed economy model in natural ways. An important feature of any multi-country model is the degree of risk sharing (the nature of financial markets). We will make a particular assumption of complete markets. 2. Let there be two countries, Home and Foreign. There is a measure 1 of goods and people in the world, with n goods and people in the home country and 1 − n goods and people in the foreign country.Let agents in the home country have preferences given by

Us =

Cs1−σ χ Ms 1−² 1 + ( ) −η H 1+v (1 − σ) (1 − ²) Ps 1+v s

They value both consumption real money balances

Ms Ps

in utility and (negatively) work

effort. Labor is immobile across countries. Full utility is then given by U = Et

∞ X

β s−t Us

s=t

Agents face the following budget constraint:

Pt Ct + Mt +

X

q(zt+1 , zt )B(zt+1 ) = Ht Wt + Πt + B(zt ) + Mt−1 + T Rt

zt+1 ∈Z

There are complete markets, so that the agents can buy state contingent nominal bonds B(zt+1 ) at price q(zt+1 ), where zt is the particular ‘state of the world’ at time t, which must be contained in the set of all states Z. 3. We assume that consumption is differentiated. That means that the composite consumption term Ct is really an aggregate over the two country’s goods, where

C = (n

1 ρ

1− 1 Ch ρ

+n 1

1 ρ

1

1− 1 1 Cf ρ ) 1− ρ

This says that consumption depends on consumption of a home and foreign composite consumption good; Ch and Cf respectively. In turn, these composites are functions of individual home and foreign differentiated goods. There is a measure n of home goods and measure 1 − n of foreign goods. Thus, we have: Z n 1 −1 1 1 Ch = (n ψ Ch (i)1− ψ di) (1− ψ 0

The price index is then defined as 1

P = (nPh1−ρ + nPf1−ρ ) 1−ρ Note that it turns out that the optimal consumption of the home good and the foreign good for the home consumer will always be

Ch = n(

Ph −ρ ) C P

Cf = (1 − n)(

Pf −ρ ) C P

It is also true that the optimal consumption of an individual home good may be written as:

Ch (i) =

1 Ph (i) −ψ ( ) Ch n Ph

and similarly for the consumption of individual foreign goods. 4. The home consumer will choose bond holding, money holdings, and labor supply, to maximize utility given the budget constraint described above. Imagine that there is a finite number of possible states of the world Z, such that z ∈ Z and we may describe a conditional probability distribution f (zt+1 , zt ) described as the probability of state zt+1 , given state zt . We may show that the first order conditions for the home consumer are q(zt+1 , zt )

Ct−σ C(zt+1 )−σ = f (zt+1 , zt )β Pt Pt+1

(1)

σ

1 Mt Ct² = χ( ² ) 1 Pt (1 − Dt ) ²

2

(2)

Wt = ηPt Ctσ Htψ

(3)

The first condition pertains to the optimal choice of state contingent bonds. The second condition describes the equilibrium money demand schedule. The third condition gives the implicit labour supply curve. Here Dt is defined as the nominal discount factor, or 1 over 1 plus the nominal interest rate. That is

Dt =

X

q(zt+1 , zt ) = βEt

zt+1

C(zt+1 )−σ Pt Pt+1 Ct−σ

5. Firms in the economy maximize profits. Assume first that firms can continuously adjust prices. In addition, assume firms have technologies given by y(i) = h(i) i.e. one unit of labour produces one unit of each good i. Then each firm is a monopolistic competitor, and faces the demand curve given by those above (from both home and foreign consumers). The elasticity of demand is ψ, so they will set the price cost markup given by p(i) =

ψ W ψ−1

(4)

6. We may easily describe a symmetric general equilibrium of the world economy with flexible prices. The home economy price index is as given above. The foreign economy price index is ∗(1−ρ)

P ∗ = (nPh

∗(1−ρ)

+ (1 − n)Pf

1

) 1−ρ

In addition, there is full free trade in all goods, so that Pf = SPf∗ and Ph = SPh∗ where S is the exchange rate (price of foreign currency). That is, the ‘law of one price’ holds at all times. This must therefore mean that PPP holds, i.e. 3

P = SP ∗ . Now we assume that Ph (i) = Ph , for all i. That is, all home firms (and foreign firms) have the same price, so this means that consumption by home consumers of each home good will be the same. While the real exchange rate is constant, the terms of trade

Ph SPf

,

will be endogenous. Note also that because of complete markets, we will get the full risk sharing condition: Ct−σ (

St Pt∗ ) = Ct∗−σ Pt

(5)

In addition, using the price cost markup rules in the labour supply curves, we get ψ Pt σ ψ C H ψ − 1 Pht t t

(6)

ψ Pt∗ ∗σ ∗ψ C Ht ψ − 1 Pf∗t t

(7)

1=η

1=η

Then goods market clearing for the home and foreign country implies that Pht −ρ ) (nCt + (1 − n)Ct∗ ) Pt Pf∗t Ht∗ = ( ∗ )−ρ (nCt + (1 − n)Ct∗ ) Pt Ht = (

This gives 5 equations in Ct , Ct∗ , Ht , Ht∗ , and the terms of trade

(8) (9) Ph SPf∗ .

Because of the symmetry of the model, the solutions have a particularly simple representation. Note that the presence of PPP and the full risk sharing condition implies that Ct = Ct∗ , so that there is full international consumption risk-sharing. Combining this with the two goods market clearing conditions, and using PPP again, we have ¡ Ht ¢ −1 Pht ρ = St Pf t Ht∗ 4

From the labour market clearing conditions, using PPP, we get (

Pht Ht v ) = ∗ Ht St Pf t

Combining the last two equations, it is clear that an equilibrium must have Ht Pht = =1 ∗ Ht St Pf t Then, using this in the labour market clearing conditions, we deduce that, in a flexible price economy, C=H=

1 ¡ ηρ ¢(− 1+v ) ρ−1

Thus we have an equilibrium in which consumption and output are constant, equal across countries, and independent of the monetary regime. The equilibrium displays complete monetary neutrality. We can obtain the solution for price levels and the exchange rate from the money market clearing conditions. Say that each country’s money growth rate is constant, so that Mt = (1 + µ)Mt−1 and Mt∗ = (1 + µ∗ )Mt∗ Then from the home country money market clearing condition, we get Pt = M t φ where φ =

β (1− (1+µ) )

H

σ ²

. Thus, using PPP, we have that the exchange rate is St =

Mt φ Mt∗ φ∗

Obviously, this system displays neutrality of the exchange rate regime with respect to real variables. It is therefore irrelevant whether the exchange rate is fixed or flexible. While the exchange rate regime does determine the volatility of the real and nominal exchange rate, it doesn’t affect the real exchange rate or any other real variables. 5

7. Now lets look at the case of fixed prices. We will begin with the simple assumption that prices are set one period in advance, and can only be adjusted at the end of the period. Therefore, in response to an unanticipated shock to the money supply, the exchange rate can adjust, but not prices. There are two different assumptions with respect to price setting we could make. On the one had, we could assume that prices are set in the home currency of the producer or seller. This means that in face of an exchange rate change, the price of an imported good must respond immediately. Thus, even in the sticky price environment, this assumption implies 100 percent ‘pass-through’from exchange rates to import goods prices. We call this ‘producer currency pricing’. An alternative assumption however, and one more in accord with reality, is that prices are preset in the currency of the buyer. This means that the price of an import good does not respond to an exchange rate change. In the short run (i.e. within a period), there is zero ‘pass-through’ from exchange rate changes to imported goods prices. We will look separately at each of these assumptions. 8. Producer Currency Pricing (PCP). Note that when prices are pre-set, the markup condition given by (4) no longer applies. Essentially, firms set prices so that the price is equal to a constant markup over the expected marginal cost. But they don’t know the actual marginal cost, because there may be shocks that have not been perfectly forecast. Note that with PCP, we still have the law of one price holding at the level of each good, so therefore we have PPP holding at the aggregate level; i.e. Pt = St Pt∗ . Thus, the risk sharing condition implies that Ct = Ct∗ still. The solution of the model with sticky prices is more difficult, because the price index is a non-linear function of the exchange rate. We have to take a linear approximation to the equilibrium to derive the affect of money shocks on the exchange rate and other real magnitudes. Say we start with the equilibrium given above, i.e. the steady state with the terms of trade equal to zero. Then let the approximation be such that xt =

¯t X t −X ¯t , X

small case letter for a variable denotes a deviation from its initial steady state level. Then from the price index (with full symmetry) we have 6

i.e. a

(1−ρ)

Pt = (nPht

1

+ (1 − n)(St Pf∗t )(1−ρ) ) 1−ρ

Differentiating this, we have pt = (1 − n)st . From the foreign country price index, we also have p∗t = −nst . Now take the money market clearing conditions, and differentiate to get mt − pt = where λ =

β (1+µ) β 1+ (1+µ)

σ λ ct + dt ² ²

(10)

and dt = −Et (σ(ct+1 − ct ) + pt+1 − pt )).

Now derive the foreign equivalent to (10), and use the presence of PPP and full risk sharing. We obtain mt − m∗t = st −

λ (Et st+1 − st ) ²

. This is a single linear difference equation in the (response of) the exchange rate. If relative money supplies are given by a random walk, then the solution is st = mt − m∗t i.e. the solution is the same as in the flexible price model. The presence of sticky prices does not increase exchange rate variability. But the sticky prices do have implications for real variables. Take the two money market clearing conditions again. Add them up, weighting by n and 1 − n. We get

nmt + (1 − n)m∗t =

λ σ ct − Et (σ(ct+1 − ct ) + Et (npt+1 + (1 − n)p∗t+1 ) − npt − (1 − n)p∗t )) ² ² 7

Now conjecture that Et ct+1 = 0 and Et pt+1 = mt , Et p∗t+1 = m∗t . This will be in fact true when money shocks are driven by random walks. Then we also know that npt + (1 − n)p∗t = n(1 − n)st − (1 − n)nst = 0. Then we get that ct = (nmt + (1 − n)m∗t )

(² + λ) σ(1 + λ)

From this, and the market clearing conditions, we may derive the response of home output to the money shock.

ht = ρ(1 − n)st + ct = ρ(1 − n)(mt − m∗t ) + (nmt + (1 − n)m∗t )

(² + λ) σ(1 + λ)

Take the case where σ = ² = 1. Then the response of home output is ht = ρ(1 − n)(mt − m∗t ) + nmt + (1 − n)m∗t ) Home output rises in response to a home money shock, but will fall in response to a foreign money shock (when ρ > 1). Home output will clearly rise more than foreign output for a home money shock. The rationale for these results is straightforward. An unanticipated rise in the money stock raises consumption, and leads to a nominal exchange rate depreciation. The nominal exchange rate depreciation reduces the foreign price level and therefore raises consumption abroad too. The exchange rate depreciation causes an ‘expenditure switching’ of world demand towards the home good. Thus, at given prices, the home country will experience a larger rise in its output. Note then the three implications: The exchange rate rises in proportion to the rise in the money stock, but the real exchange rate is unchanged, consumption increases equally in both home and foreign country, and there is a rise in relative home country output (all in response to an unanticipated increase in the money supply of the home country). How does the trade balance respond? Imagine we begin in a steady state with a zero trade balance. Then the response of the home country trade balance to a money shock depends on the difference between the response of Pht Ht and Pt Ct , i.e. the difference 8

being the home country trade balance. From above we have the movement in the first expression being ht , and the movement in the second expression is pt + ct . Substitute the above equations, and we obtain the movement in the trade balance being (ρ − 1)(1 − n)(mt − m∗t ) Thus, the trade balance will rise in response to a home country money shock, when ρ > 1, and fall in response to a foreign country money shock. 9. Local Currency Pricing (LCP). Now lets look at the opposite pricing assumption - all goods prices are set in local currency. This means that the price indices P and P ∗ do not respond at all to exchange rate shocks. Thus, from the risk sharing condition, we get ct = c∗t +

1 st σ

Thus, there is a deviation from full risk-sharing, in the presence of local currency pricing. This occurs because an unanticipated movement in the exchange rate leads to a real depreciation. What this says is that an unanticipated real depreciation generates a rise in home relative to foreign consumption. Now use the money market equilibrium conditions again. From mt − pt =

σ λ ct + dt ² ²

m∗t − p∗t =

σ ∗ λ ∗ c + dt ² t ²

and

, noting the definition of dt and d∗t , and using the fact that Et (pt+1 − p∗t+1 ) = Et st+1 , i.e. PPP holds in an expected sense, we obtain: mt − m∗t =

1 λ st − Et (st+1 − st ) ² ²

Using again the random walk assumption for the money supply, we get st = (mt − m∗t ) 9

(² + λ) (1 + λ)

For ² > 1, the exchange rate will respond by more than the money supply, in response to an unanticipated shock. That is, there will be exchange rate overshooting. We may derive the solution for consumption from the home country money market equilibrium relationship (on its own). We get

mt =

σ λ ct − Et (σ(ct+1 − ct ) + Et pt+1 ) ² ²

Again, conjecture that Et ct+1 = 0 and Et pt+1 = mt (it is easily verified to be true later) . So we get

ct = mt χ , where χ =

(²+λ) σ(1+λ)

Home country consumption is affected only by the home money

supply, and unaffected by shocks to the foreign money supply. The opposite holds for foreign consumption. Home and foreign output however, will respond by the identical amounts, since the relative prices are unchanged in equations (8) − (9), and therefore we have ht = h∗t = nct + (1 − n)c∗t = (nmt + (1 − n)m∗t )χ Note the different results under LCP. We have that the exchange rate may respond by more than fundamentals, that the real and nominal exchange rate response is identical (with the period of the shock), that consumption is not equated across countries, but output responds by an identical amount. Finally we may derive the trade balance response response to a money shock, in the LCP pricing regime. The response of Pht Ht is ht = (nmt + (1 − n)m∗t )χ . The response of Pt Ct is ct = mt χ A home country money shock will then lead to a trade balance deterioration. 10. Staggered Pricing in the Open Economy: PCP. 10

While this one-period ahead pricing captures the essence of the difference between PCP and LCP, it is not a realistic description of price adjustment in the open economy. Now lets extend the model to allow for persistence in price setting using the basic Calvo model. We will not need to develop this from first principles because we know how the Calvo model works. Note now that prices are not predetermined, but rather a fraction of firms may adjust in any given period, governed by the parameter κ. We may write the model in the following way. In the home economy the firms set prices in producers currency for sale to home or foreign consumers. Their price setting equations are governed by; p˜ht = βκwt + (1 − βκ)Et p˜ht+1 pht = (1 − κ)˜ pht + κpht−1 These equations are explained in a straightforward fashion as the result of the Calvo pricing model. They lead to the domestic goods price inflation equation governed by

πt = λ(wt − pht ) + βκEt πt+1 That is, πt = pht − pht−1 . A similar condition applies to the foreign economy. Note that from the labour supply equation, we have

wt = pt + ρct + vht in log terms, where pt = npht + (1 − n)(p∗f t + st ). We focus on the case v = 0, which represents an assumption that preferences are linear in the disutility of labour. Substituting this into the inflation equation you get:

πt = λ(ρct + (1 − n)τt ) + βκEt πt+1 where τt = (p∗f t + st ) − pht is the terms of trade, or the relative price of the foreign good to the home good. The foreign economy has an analogous condition. 11

∗ πt∗ = λ(ρc∗t − nτt ) + βκEt πt+1

In addition, we have the condition determining consumption risk sharing, which is, for the case of PCP, just ρct = ρc∗t This gives us three equations in 5 variables; πt , πt∗ , τt , ct and c∗t . We need the monetary rules and the inter-temporal Euler equations to solve the system. If consumers in the home economy are choosing an optimal consumption path, we must have: ∗ it = ρEt (ct+1 − ct ) + Et (nπt+1 + (1 − n)(πt+1 + st+1 − st ))

This just says that the nominal interest rate equals the real interest rate plus anticipated inflation in the CPI. We may re-write this as:

it = ρEt (ct+1 − ct ) + Et πt+1 + (1 − n)(τt+1 − τt )) That is, we have a system in the terms of trade and inflation. Finally, let the monetary policy rule be:

it = σπt + vt where vt = µvt−1 + ut represents a monetary policy shock. Putting the last two equations together gives us:

σπt + vt = ρEt (ct+1 − ct ) + Et πt+1 + (1 − n)(τt+1 − τt )) Likewise, for the foreign economy, we have: ∗ σπt∗ + vt∗ = ρEt (c∗t+1 − c∗t ) + Et πt+1 − n(τt+1 − τt ))

We now have a system of 5 equations in the 5 variables ct , c∗t , τt , πt and πt∗ . 12

Lets look at the impact of temporary money shocks in the home country. Using an undetermined coefficients method, we have:

πt = −

λ ut (σλ + 1)

πt∗ = 0

ct = −

τt =

n ut (σλ + 1)ρ

−1 ut (σλ + 1)

A monetary shock generates an appreciation of the terms of trade, and a fall in domestic consumption. These combine to push down domestic inflation. The depreciation of the foreign terms of trade (the reciprocal of the home terms of trade) combines with the fall in consumption to leave foreign inflation unchanged. The home nominal interest rate rises - this is represented by a rise in the real interest rate, and a rise in the anticipated rate of inflation, which comes from the anticipated terms of trade depreciation. What happens the nominal exchange rate? We can compute this by just adding domestic inflation and the terms of trade:

st = −

(1 + λ) ut (σλ + 1)

The nominal exchange rate appreciates by more than the terms of trade. Note that a very tight monetary policy can prevent any real impacts of the money shock at all - in this case this is equivalent to a fixed exchange rate. 11. Staggered Pricing in the Open Economy: LCP. Now make the opposite assumption - assume that there is local currency pricing. How does this change the system? We may write the system now in terms of four separate prices; the price for the home good in the home market; phh , the foreign good in the home 13

market, pf h , the home good in the foreign market, p∗hf , and the foreign good in the foreign market, p∗f f . The pricing equations may be written as: p˜hht = βκwt + (1 − βκ)Et p˜hht+1 phht = (1 − κ)˜ phht + κphht−1

p˜f ht = βκ(wt∗ + st ) + (1 − βκ)Et p˜f ht+1 pf ht = (1 − κ)˜ pf ht + κpf ht−1

p˜∗hf t = βκ(wt − st ) + (1 − βκ)Et p˜∗hf t+1 p∗hf t = (1 − κ)˜ p∗hf t + κp∗hf t−1

p˜∗f f t = βκwt∗ + (1 − βκ)Et p˜∗f f t+1 p∗f f t + κp∗hht−1 p∗f f t = (1 − κ)˜ In principle this involves four separate prices and four inflation equations, instead of two. But note that the dynamics of prices are set so that producers are targeting an equal price in both markets, adjusted for the exchange rate. Thus, while sticky prices lead to a deviation from the law of one price and therefore deviation from PPP, over time we will see a convergence to PPP, as prices are adjusted. Although the system can get a bit complicated, there is an easy way to solve it in terms of CPI inflation rates. Let Πt = nπhht + (1 − n)πf ht , that is, the domestic CPI inflation rate, which is obviously in terms of home currency. Then we may combine these equations, along with the definitions of marginal cost, to get:

Πt = λρct + βκEt Πt+1 14

Likewise, for the foreign economy, we get: Π∗t = λρc∗t + βκEt Π∗t+1 The risk sharing condition gives: ρct = ρc∗t + qt where qt is the real exchange rate: qt = st + p∗t − pt where the prices here are now CPI prices. We can write the interest rate equations again as:

σΠt + vt = σEt (ct+1 − ct ) + Et Πt+1 We assume now that monetary policy targets the CPI inflation rate. σΠ∗t + vt∗ = ρEt (c∗t+1 − c∗t ) + Et Π∗t+1 How does a money shock affect the real exchange rate? We can calculate this easily from this model. Combine all the equations to get: ˜ t+1 − Π ˜ ∗t+1 ) Πt − Π∗t = λqt + βκEt (Π

σ(Πt − Π∗t ) + vt − vt∗ = Et (qt+1 − qt ) + Et (Πt+1 − Π∗t+1 ) This is just a two equation system in inflation differentials and the real exchange rate change. A real exchange rate depreciation will generate a higher home country inflation rate, temporarily. Thus, in terms of inflation differentials, the real exchange rate acts like the output gap in the domestic economy. The steady state real exchange rate is zero, but a real depreciation is a signal of inflation pressures. A temporary money shock has the following effects on the domestic inflation differential, and the real exchange rate: 15

Πt − Π∗t = −

qt = −

λ ut (σλ + 1)

1 ut (σλ + 1)

Again, a tight monetary policy can eliminate the real effects of the money shock on the real exchange rate.

16