One type of financial capital control is a rule that requires a foreign financial investment to deposit part of the capital in an account in the central bank that pays no interest. This is equivalent to a tax on the interest accrued from the capital. This was implemented in Chile during the 1990s. (See Macroeconomía Teoría y Políticas - Jose De Gregorio, page 198).

  1. If the proportion of the compulsory interest-free deposit is \(e\), what is the equivalent tax rate, \(t\)?
  2. What is the equilibrium domestic interest rate under the capital control, i.e. that which, given the interest-free proportion, \(e\), is equivalent to the international interest rate?

Question (i)

Let:
K = Capital
\(r^*\) = International interest rate
r = Domestic interest rate
e = Proportion of capital that must be deposited in central bank with no interest
T = Tax amount
\(\tau\) = Tax rate
R = Revenue under no capital controls and domestic interest rate
\(R^*\) = Revenue under capital controls

First, if there were no capital controls, the revenue under the domestic interest rate would be: \[R = rK\] And the revenue under capital controls is that which is accrued on the capital that is allowed to be deposited with the domestic interest rate: \(K(1-e)\): \[R^* = rK(1-e)\] And the tax amount is the difference between these two revenues: \[T = R - R^*\] \[\therefore T = rK - rK(1-e)\] \[\therefore T = rK(1-1+e)\] \[\therefore T = rKe\] \[\therefore T = Re\] \[\therefore \frac{T}{R} = e\] But \(\frac{T}{R}=\tau\): \[\therefore \tau = e\] QED

Note that \(\tau=e\) for any domestic interest rate r.

Question (ii)

Revenue under capital controls and domestic interest rate = Revenue under no capital controls and international interest rate \[\therefore rK(1-e) = r^*K\] \[\therefore r = \frac{r^*}{1-e}\]

Example:

K = 100
\(r^*\) = 5%
e = 10%

\[r = \frac{r^*}{1-e} = \frac{0.05}{1-0.1} = 0.05556\]

r0 = 0.05
e = 0.1
r = r0/(1-e)
r
## [1] 0.05555556

\[R = rK = 0.05556 \cdot 100 = 5.556\]

K = 100
R = r*K
R
## [1] 5.555556

\[R^* = r^*K = 0.05 \cdot 100 = 5\] \[T = R - R^* = 5.556 - 5 = 0.5556\] \[\tau = \frac{T}{R} = \frac{0.5556}{5.556} = 0.1\]

R0 = r0*K
TT = R - R0
t = TT/R
TT
## [1] 0.5555556
t
## [1] 0.1