Mccallum Chapter 14

Mccallum Chapter 14

McCallum Chapter 14

Open Economy Macroeconomics

  1. Introduction
  1. Our objective is to adapt a simplified macroeconomic model to an open-economy setting.
  1. Exchange Rate Conventions:
  1. The home country currency is dollars
  2. The exchange rate for the home country is expressed in terms of dollars/unit of foreign currency; i.e., dollars/mark.
  3. Under this convention a decline in the exchange rate represents an appreciation of the dollar.
  1. The Basic Open Economy Model
  1. National Income Identity:

(1)

where X is net exports.

  1. The Net Exports function is:

(2)

where is foreign income and

where Q is the price of exports relative to imports.

  1. Substituting into (1)

(3)

  1. Solve to get the IS curve

(4)

  1. Conventional assumptions are:
  1. Letting and collapsing Y*and G in the error term we have:

where (5)

  1. Recall the real interest rate is given by:

(6)

  1. Q, the price of domestic (export) goods, can be written as:
  1. In logs:

(7)

  1. Hereafter e refers to the exchange rate and E refers to the expectations operator.
  1. The LM curve is conventional:

, where .(8)

  1. We ignore complications

a)We deflate the money stock by the domestic price level rather than a weighted average of domestic and foreign price levels.

b)It is possible that the transactions measure should exclude net exports from yt.

  1. Interest rate parity condition:

(9)

  1. The domestic interest rate equals the foreign interest rate plus the expected depreciation of the exchange rate. The foreign interest rate is taken to be exogenous (a small country assumption).
  1. Example:
  1. US interest rate is 5%
  2. Mexican interest rate is 15%
  3. The for the condition to hold, the expected depreciation of the dollar versus the peso is -10% (i.e., the dollar is expected to appreciate 10%).
  4. If the dollar appreciates and we have the appropriate negative sign for .
  1. Rearranging the Model
  1. We assume that the model is classical, so that .
  2. The model consists of 5 equations (5)-(9), and must explain the paths of 5 variables, .

where (5)

(6)

(7)

, where .(8)

(9)

  1. Insert (6) and (7) into IS (5) to get

(10)

  1. Further, substituting the interest rate parity condition into (10) and LM (8) we get:

(11)

(12)

  1. Taking as constants, we now have 2 equations in and two endogenous variables, .
  1. Analysis: Steady States
  1. We will initially suppress random disturbances and analyze steady-state properties of the model.
  1. Assume:
  1. Error terms = 0.
  2. .
  3. .
  4. .
  5. .
  6. Expectations are correct: and .
  1. Write (11) and (12) in first differences:
  1. Consequences:
  1. Domestic inflation equals the rate of growth of domestic money.
  2. We can get , so the change in the exchange rate is equal to the difference in the money growth rates at home and abroad.
  3. The real price of exports is constant:

where

  1. The latter result implies that purchasing power parity prevails. An increase in a country’s money stock will ultimately increase the price level and the exchange rate, leaving real variables unchanged.
  1. Analysis: Random Shocks
  1. Assume:
  1. are constant.
  2. The shocks are independent white noise errors with variances
  1. Rewrite (11) and (12) combining constants into B and C:

(11’)

(12’)

  1. Now we employ a slight generalization of our earlier procedure for solving rational expectations models. We have two endogenous variables in the two equations above, . We have just two exogenous variables, . We conjecture solutions of the form:

(15)

(16)

  1. Substituting in (11’) and (12’) yields

(17)

(18)

  1. Note the typo in McCallum in (17).
  1. Now equate coefficients on the LHS and RHS of (170 and (18). This yields 6 equations to be solved for six parameters .
  1. Ignoring the conditions related to the intercepts, the remaining 4 equations are:

(19a-d)

  1. Solutions are:
  1. Signs are:
  1. Interpretation of results:
  1. A positive shock to the money demand function () lowers pt and et. When people wish to hold more money this increases the value of money relative to goods (so the price level falls) and relative to the foreign currency (so the exchange rate falls).

a)Similar results would apply for a negative money supply shock (if our model had and error term in the money supply process)

  1. A positive shock to the IS curve () lowers but raises . Such a shock increases demand for home country output leading to a higher price level and a lower value of foreign currency (the dollar appreciates).

a)The effect of the positive IS shock on can also be found. Since the per unit response to and IS shock is:

b)This implies that a positive IS shock will raise the real price of domestic goods.

c)Similar reasoning shows that qt is unaffected by a money demand shock.

  1. Extensions
  1. Facts from post-71 (Flexible exchange rate regime) that do not fit the theory:
  1. The purely random component of exchange rate fluctuations is much greater than for national price levels.

a)In the theory, equations (15) and (16) imply that the random component of pt will exceed that of et unless (which is not considered to be the case).

  1. Exchange rates respond more promptly to shocks than do national price levels.

a)In the theory, both variables respond fully to shocks in the period in which the shocks occur.

  1. Relative prices of imported goods exhibit a great deal of persistence in their fluctuations.

a)In the theory q fluctuates randomly around its mean from period to period (with no persistence).

  1. Modifying the Model
  1. Alter the stochastic process assumed for the errors . Assume random walks:

(21 & 22)

  1. We still use (15) and (16) to represent solutions, but now we have:

a)These results follow directly from (15) and (16) and the random walk assumptions about the error terms.

  1. Recall equations (11’) and (12’):

(11’)

(12’)

  1. Rewrite (11’) and (12’) using the results noted above:

(23)

(24)

  1. Equating coefficients and solving, we get solutions for :

(25)

(26)

  1. Implications:

a)The variance of exceeds that of .

b)Since , is also a random walk.

  1. Now, in addition, assume that there is price stickiness:
  1. Assume 1-period-ahead price setting:

(28)

where is the market clearing price.

  1. Again rewrite (11) and (12), now allowing to be variable:

(11”)

(12”)

  1. Define as the market-clearing value of that would be determined under a flexible price system in which . So is given by (25) and:
  1. From the equation above, it follows that:
  1. Now, substitute (11”) into (12”) to eliminate , and also substitute the results above:

=

(29)

  1. This equation now has just one endogenous variable, the exchange rate, and we can solve using our usual method. Conjecture:
  1. Solve to find:
  1. Recall that is negative. If is close to 1 and if the absolute value of is much smaller than 1, then . Comparing to (26) we see that the current impact of a monetary shock on the exchange rate is now bigger than it was in the flexible price model. This is exchange rate over-shooting.

a)Intuition: A positive money demand shock puts upward pressure on the interest rate (when prices are sticky).To satisfy interest rate parity, a higher interest rate must imply that the currency is expected to depreciate. Other things equal, this means that the current value of the dollar must be higher (i.e., et must be lower).

b)In succeeding periods the exchange rate moves back to the level that would prevail in the flexible price model.

  1. Fixed Exchange Rates
  1. Again consider the classical version of our model as given by equations (11) and (12), but now assume that et is fixed at e.

(11)

(12)

  1. Now we have two equations in two unknowns: .
  1. Implication: With a fixed exchange rate, it is not possible for monetary policy to be set exogenously. Holding the exchange rate fixed requires the monetary authority to surrender control of the domestic money stock.
  1. In practice, governments have some limited ability to target both exchange rates and the money supply (because they have multiple instruments, i.e. fiscal policy). But this is only possible temporarily.
  1. For example, faster money growth would lead to a depreciation of the currency. This could be offset in a period by an increase in government spending. But fiscal policy would have to be increasingly expansive in successive years to maintain the exchange rate, which would not be possible forever.

a)Note error in McCallum p. 288. to prevent et from falling would require contractionary fiscal policy.

  1. Friedman on Fixed versus Flexible Exchange Rates
  1. When a shock hits the economy, one would normally expect adjustments in . But with e fixed and pt inflexible, output will have to adjust. Thus output will be more unstable under fixed exchange rates.
  1. Temporarily fixing the exchange rate will be even worse, because it leads to speculative pressures.
  1. One possible case favoring a fixed exchange rate is that it represents adherence to a rule (i.e. we limit our inflation rate to that prevailing elsewhere in the world).
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