Econ 304: Problem Set 4

PROBLEM SET 4 - INTERNATIONAL - ANSWERS

1. a. The formula for the real exchange rate is e = EP*/P. In this problem: P* = 1.2; E = 1/5 = .2, and P = 1.5, so

= (1.2)(.2)/1.5 = .16

b. Now, E = 1/8 = .125, so e = (1.2)(.125)/1.5 = .10

2. The interest parity condition tells us that

i_t

i*_t + (E^e_t+1 - E_t)/E_t

With i_t = 4%, and i*_t = 6%, then (E^e_t+1 - E_t)/E_t has to be -2%. This means that the dollar price of the German mark is expected to decline by 2%. This is an appreciation of the dollar.

3. The interest parity condition tells us that

i_t

i*_t + (E^e_t+1 - E_t)/E_t

With i*_t = 20% and (E^e_t+1 - E_t)/E_t = 15%, US interest rates should be 5%. Since you think that the return to holding Brazilian bonds is 5% higher than everyone else thinks it is (because you think that there will be less depreciation of the Brazilian currency, you should invest in Brazil.

4. a. To find equilibrium income set Z (total demand for goods and services) equal to Y and solve for Y

Y = Z = c₀ + c₁ (Y - T) + d₀ + d₁ Y - d₂ r + G₀ - q₁ Y + x₁ Y*

now collect all the terms involving Y on the left-hand side

Y - c₁ Y - d₁ Y + q₁ Y = c₀ - c₁ T + d₀ - d₂ r + G₀ + x₁ Y*

solving for Y yields:

Y = 1/(1 - c₁ - d₁ + q₁) [c₀ - c₁ T + d₀ - d₂ r + G₀ + x₁ Y*]

b. The multiplier for a D G is 1/(1 - c₁ - d₁ + q₁).

c. The multiplier for a D Y* is x₁/(1 - c₁ - d₁ + q₁).

5. The formula for net exports (exports minus imports) is:

NX = x₁ Y* - q₁ Y.

If there is a change in taxes, the resulting change in income would be

Y = [-c₁/(1 - c₁ - d₁ + q₁)]

The effect of this change in income on net exports would all come through the imports side, yielding

NX = [q₁ c₁/(1 - c₁ - d₁ + q₁)]

which represents an increase in net imports.

6. To get an IS curve in this instance take the equation for equilibrium income derived in question 4 part a, and solve it for r, the variable on the vertical axis in the r,Y (IS-LM) plane. To do this first isolate the term for r:

Y = 1/(1 - c₁ - d₁ + q₁) [c₀ - c₁ T + d₀ + G₀ + x₁ Y*]

- [d₂ /(1 - c₁ - d₁ + q₁)]r

Now solve for r yielding

r = 1/d₂ [c₀ - c₁ T + d₀ + G₀ + x₁ Y*] - [(1 - c₁ - d₁ + q₁)/d₂ ] Y

Now we can say that

a. changes in x₁ have no effect on the slope of the IS curve,

b. an increase in q₁ would cause the IS curve to be steeper (the slope becomes larger in absolute value, therefore the D r for a D Y will be larger, i.e. the curve becomes steeper.)

7. The diagram shows the effects of a tax increase under both flexible and fixed exchange rates.

The increase in taxes shifts the IS curve to the left.

a. Under flexible exchange rates this shift in the IS curve leads to a new equilibrium at point a, and income contracts, interest rates fall, and the exchange rate rises. Under fixed

b. Under fixed exchange rates, the monetary authority has to respond with a cut in the money supply shifting the LM curve up until the change in interest rates and exchange rates disappears. This leads to an equilibrium at point b, and a large fall in income.

8. The diagram shows the effects of a decrease in the money supply under both flexible and fixed exchange rates.

The decrease in the money supply shifts the LM curve up.

a. Under flexible exchange rates this shift in the LM curve leads to a new equilibrium at point a, and income contracts, interest rates rise, and the exchange rate falls.

b. Under fixed exchange rates, the monetary authority has to undo the change in the money supply, because it has to undo the change in the exchange rate. This means that equilibrium position of the economy is unchanged, i.e. it returns to point 0.

9. Start out by setting up the three equilibrium conditions:

Equilibrium in the Goods Market

(1) Y = 100 +.80 (1-.25) Y + 300 -1,000 i + 200 + 195 - .1 Y + 100 E

[note: This is just Y = C + I + G + NX]

Equilibrium in the Money Market

(2) 800 = .8 Y - 2,000 i

[note: This is money supply equals money demand.]

Interest Parity Condition

(3) E = .75 - 5 i

next substitute (3) for E in (1), and simplify slightly, yielding

(4) Y = 100 + .6 Y + 300 - 1,000 i + 200 + 45 - .1 Y + 100 (.75 - 5 i)

and simplify

(5) Y = 720 +.5 Y - 1,500 i.

now solve equation (2) for i, yielding

(6) i = -.4 + .0004 Y.

now substitute (6) into (5), yielding

(7) Y = 720 + .5 Y - 1,500 (-.4 + .0004 Y)

simplifying yields

(8) Y = 720 + .5 Y + 600 - .6 Y

Y = 1320 - .1 Y

which solves for Y as

(9) Y = 1200

Next substitute equation (9) into (6) to find the equilibrium interest rate

(10) i = -.4 + .0004 (1200) = .08

Finally substitute equation (10) into equation (3) yields

(11) E = .75 - 5 (.08) = .35

10. To calculate the effect of the change in I₀ (the autonomous portion of investment) it is easiest to break into the derivation of equilibrium income at equation (8) in the answer to question 9. Equation (8) would become

(8)' Y = 1320 + 110 - .1 Y

which solves for Y as

(9)' Y = 1300

This shows us that the D Y = 100.

Now using the results from question 4, the relevant multiplier for a D I₀ is the multiplier derived for a D G, i.e.

1/(1 - c₁ - d₁ + q₁)

Using the numbers from the model in problem 9, we find that

c₁ = .8(1-.25) (note we have to add the tax rate here), d₁ = 0, and q₁ = .1.

The multiplier would be:

1/(1 - .6 + .1) = 1/(.5) = 2.0

The reason that the actual change in Y which we calculated is lower than this multiplier suggests it should be is that we have a full IS-LM model in question 9, and only an IS curve in question 4's model. The multiplier of 2.0 would tell us how far the IS curve shifted to the right, but (since this does not take into consideration either the effects in the money market or the effects of exchange rate changes on net exports, it overestimates the change in income.