3 A Fixed Exchange Rate Regime: Argentina After the Stabilization
5.3 Technical Notes on (Steady-State) Comparative Statics in Section 3
It is also useful to notice that
) . Thus, the following must be true:
a) If r(1)(1), then well defined. In addition, as I mentioned before, <0.
b) If r(1)(1), then well defined. Also, notice that, as I mentioned before, >0.
Thus, from (A.33), it follows that
5.3 Technical Notes on (Steady-State) Comparative Statics in Section 3
5.3.1 Steady-State Equilibria in a Walrasian Regime
Comparative Statics with respect to the Policy Parameters and Differentiating equation (65) with respect to and , respectively, I obtain
(1 *)( 1)((1 ) )( (11)(1 ) ) *(1 )
2 0,given 1,In addition, differentiating equation (63) also with respect to and , respectively, one gets
Finally, differentiating equation (64) with respect to and , respectively, yields
,
Effects of Increases in *, r, d or f on the capital-labor ratio (kˆ)
Given equation (62), it is obvious that there is a negative relationship between kˆ and .
Then, it will be sufficient to show the effects of *, r, d and f on . Differentiating (56) Differentiating equation (65) with respect to * yields
(1 *)( 1)( (1) (1 ) ) *(1 )
2 0. expression.
Notice that ( 1)0 for lending to be positive.
Comparative Statics under a Currency Board
In a currency board, the subsidy rate on capital can be written as
(1 *)( (11)( )( )1)((1 ) ) *(1 )
.Differentiating (A.46) with respect to r, one obtains
(1 *)( (11)( ) () (1 )(1 ) ) *(1 )
2 .Thus, the subsidy rate on capital is increasing in r.
On the other hand, differentiating equation (64) with respect to r, we obtain the following expression
. Then, the share of net investment abroad in total savings is increasing in the world interest rate r.
Regarding the effects of changes in domestic reserve requirements, I start with the analysis of increases in d. Differentiating (A.46) with respect to d, one gets
.
This condition always holds if
.
Obviously, (A.51) always holds under a currency board and, therefore, 0
Similarly, differentiating (46) with respect to f yields
This condition always holds if
.
Again, it is obvious that (A.55) always holds under a currency board and therefore,
straightforward to show that ˆ is increasing in f: Thus, differentiating (64), one obtains ˆ 0
Comparative Statics under a Pure Fixed Exchange Rate
In a pure fixed exchange rate regime, the subsidy rate on capital can be written as
(1 ) *(1 )
.By inspection of (A.57), it is obvious that 0
Thus, the share of net investment abroad in total savings is always increasing in r.
Next, differentiate (A.57) with respect to d to obtain
(1 (1 ) )(1*(1 ) )
2 .In addition, differentiating equation (64) with respect to d, yields
Using (A.43), (A.59), and (A.60) I am able to determine that the share of net investment abroad is increasing in d for all but very small values of *.
In similar way, differentiating (A.57) with respect to f I obtain
Differentiating equation (64) with respect to f, I obtain
. investment abroad is increasing in f for all but very small values of *.
5.3.2 Steady-State Equilibria in a Credit Rationing Regime
Comparative Statics with respect to the Policy Parameters and
Differentiating equation (68) with respect to and with respect to , and using (A.35) as well as (A.36) I am able to determine, respectively, that
obtains. 2 and using (A.35), (A.36), (A.63), and (A.64), I am able to determine that
obtains,
Finally, differentiating equation (69) with respect to and with respect to , and using (A.35), (A.36), (A.63) and (A.64), I am able to determine, respectively, that
obtains, 2
obtains. 2
Comparative Statics under a Currency Board
Proof of Proposition 8 In this section I proceed to prove Proposition 8 as well as the corresponding results for a Case 2 economy arises for the case when a currency board is in place.
I start with the analysis of the effects of increases in * on the capital-labor ratio k~ . Differentiating equation (68) with respect to * one obtains
In addition, using equations (A.58) and (A.65), gives
Obviously, under a currency board =1 and [0,(1-d-f)), and therefore (A.70) becomes
Notice that the first term in the numerator of (A.71) is positive if ).
In addition, recall that 0(1-d-f) by assumption. Thus, (1-) is unambiguously increasing in * r in situations where [c,(1d f)) holds and a currency board is in place. As a consequence, k~ is decreasing (increasing) in * if Case 1 (Case 2) obtains or, equivalently if r(1)(1)().
On the other hand, notice that if (0,c), the first term in the numerator of (A.71) is negative and, obviously,
*
is decreasing in r. Also, while the denominator in (A.71) is always positive, we can rewrite the numerator as
( )( )
,) 1
)(
(d f d f rd d f d (A.73) and, therefore
parameter values similar to the ones found in Latin American countries and, therefore, it seems that r<rc always holds. Therefore, under plausible conditions, (1-) is also an determine that
Using (A.41) and (A.44), (A.75) can be rewritten as
Obviously, under a currency board =1 and [0,(1-d-f)), and therefore (A.76) can be rewritten as
obtains
if ,
while when Case 2 obtains
if . respectively.
Notice that, under a currency board
). Also notice that if Case 1 (Case 2) obtains
) case, it is straightforward to show that
Notice that when Case 1 obtains) . increasing function of *. On the other hand, notice that when Case 2 obtains and when both [0,c) and r<rc hold, on the case where r<rc. It is straightforward to show that
) , accordingly. On the other hand, using also (A.81) and (A.85), it is possible to determine that x>xc when Case 2 obtains and both [c,(1d f)) and r<rc hold. Thus, under these circumstances, ~ is always increasing in *. Q.E.D.
Effects of Increases in r on k~
and ~ I start with the analysis of the effects of increases in r on k~
. Differentiating equation (68) with respect to r, I obtain
Using (A.42) and (A.47) I obtain
r , unambiguously. If )
r , then (1-) is always increasing in r. I next proceed
to analyze when the last condition holds.
) . 1
(
) (
) 1
( r
1
) 1
(
) 1
)(
1 1 (
1
f d
f d f
d d
f d
d f
d
r
(A.88)
Obviously, if [c,(1d f)), the right-hand-side of the second inequality in (A.88) is negative, and the inequality always holds, resulting in (1-) being always increasing in r. On the other hand, if [0,c), the right-hand-side of the second inequality is positive. In this case, it is helpful to rewrite the second inequality in (A.88) as
d(1 (1d f ) (d) f)
.f
r d
(A.89)
Notice that, given that both r>1 and
d(1(1df)(d)f)
1f
d , (A.89) might
hold (or not), especially if is very close to c. Thus, when [0,c), (1-) is increasing in r for all values of * but maybe those arbitrarily close to
1 1
r . I can then say that (1-) is typically increasing in r, [0,(1d f)),
* 1 1 , *
Max r .
Now, I return to the analysis of equation (A.86). When Case 1 (Case 2) obtains, both r(1)(1)() and
(1)x(1)
()0 hold, and, therefore, k~is decreasing (increasing) in r, given that, as we have previously proved, (1-) is typically increasing in r.
Next, I analyze the effects of increases in r on the share of net investment abroad in total savings. Differentiating equation (69) with respect to r, I obtain
) . obtains and a currency board is in place, 0
~
r
. However, when Case 2 obtains and a
currency board is in place, both k~ and are increasing in r, resulting in two forces that act in opposite directions. In this case, it is helpful to rewrite (A.90) as