Open Economy: Exercises and Solutions
Exercise 1 (Topic 6)
Consider an open economy, where the goods market is characterized by:
- C = 5 + 0.1 * (Y – T)
- I = 4 + 0.3 * Y
- IM = 0.3 * Y * ε
- X = 0.2 * Y* – 2 * ε
We also know the government sets a level of public spending and of taxes equal to 8, the real exchange rate is 1.4, and foreign income is 1. Given this information, what is the value of autonomous spending in this economy?
The demand for domestic goods in this economy is:
Z = C + I + G + NX = 5 + 0.1 * (Y – T) + 4 + 0.3 * Y + 8 – 0.3 * Y *ε+ 0.2 * Y* – 2 * ε = 5 + 0.1Y – 0.8 + 4 + 0.3 * Y + 8 -0.42Y + 0.2 – 2.8 = 13.6 -0.02Y
Thus, in this economy autonomous spending is 13.60 and the slope of the demand is -0.02.
Exercise 2 (Topic 6)
Consider the standard model of the goods market in an open economy, as the one we’ve seen in class. Imagine that the domestic government decides to increase taxes. Represent the effects of this policy on equilibrium output. Explain the effects on net exports. You can support your answer with a diagram.
If the (domestic) government increases taxes, households’ disposable income decreases, inducing a fall in both consumption and (private) savings. When consumption decreases, demand in the economy also goes down, decreasing output and income. This lower income makes households further decrease their consumption (& savings), and firms reduce their investments. Because now domestic agents demand less (because of the lower consumption) our imports fall (we don’t demand as many foreign goods as before). So in the end, net exports increases. Graphically, an increase in taxes generates a fall in the demand for any level of income, so we represent it as a downwards shift in the demand curve.
Exercise 3 (Topic 6)
Consider an open economy, where the goods market is characterized by:
- C = 120 + 0.3 * (Y – T)
- I = 60
- IM = 0.1 * Y * ε – ε
- X = 0.8 * Y* – 1.3 * ε
The government sets a level of taxes equal to 31, the real exchange rate is 1, and foreign income is 80. We don’t know the level of public spending. Given this information, find the value of net exports as a function of public spending. Explain. Note: trade balance means that net exports are equal to 0.
First, the function for net exports is:
NX = X – IM/ε = 0.8Y* – 1.3ε – (0.1 * Y * ε – ε)/ε = 0.8Y* – 1.3 * ε – 0.1Y + 1 = 63.7 – 0.1Y
Equilibrium output in this economy is:
Z = C + I + G + NX = 120 + 0.3(Y – T) + 60 + G + 63.7 – 0.1Y = 120 + 0.3Y – 0.3 * 31 + 60 + G + 63.7 – 0.1Y = 234.4 + G + 0.2Y
Z = Y → Y = 234.4 + G + 0.2Y and solving for Y: Y = 293 + 1.25G
Finally, using this expression for Y in the definition of net exports we’ve derived previously:
NX = 63.7 – 0.1Y = 63.7 – 0.1 * (293 + 1.25G) = 34.4 – 0.125G
This is the expression for net exports as a function of public spending (“as a function of X” means that we should treat X as an unknown). This equation tells us that net exports are decreasing in public spending: if the government increases public spending, the economy will import more, lowering net exports. In a more elaborated way, we could also say that this equation shows that public spending is partly financed by foreigners (as increasing G means our position with respect to the rest of the world is smaller, or even negative).
Extra: For those who interpreted that the question was about finding the level of public spending that makes the economy to be in trade balance. Trade balance means that net exports are zero. Thus:
NX = 63.7 – 0.1Y = 0 → Y = 637
This is: the economy will be in trade balance whenever income is equal to 637. We’ve already seen that output is given by Y = 293 + 1.25G
So, we can just set Y = 637 and solve for G:
Y = 234.4 + G + 0.2Y → 637 = 234.4 + G + 0.2 * 637 → G = 637 – 361.8= 275.2
Then, public spending should be equal to 275.2 in order for the economy to be in trade balance.
Exercise 4 (Topic 6)
Consider an open economy, where the goods market is characterized by:
- C = 20 + 0.3 * (Y – T)
- I = 15 + 0.4 * Y
- IM = 0.4 * Y * ε
- X = 0.2 * Y*
The government sets a level of public spending and of taxes equal to 10, the real exchange rate is 1, and foreign income is 100. Suppose the government reduces taxes to 6. Given this information, answer the following questions:
a) What is the effect on equilibrium output?
The effect of a fall in taxes on output is:
ΔY = Multiplier × (-c1ΔT)
Then, to find the effect of a 4-unit decrease in taxes in our economy, we need to know the value of the multiplier. We (should) know that the multiplier is always equal to:
Multiplier = 1 / (1 – Slope of Z)
In our economy, the slope of the demand function is just given by 0.3 + 0.4 – 0.4 = 0.3 (note that the slope of the demand function is just the sum of all the coefficients that multiply income; in our example these are 0.3 in the consumption function, 0.4 in the investment function, and -0.4 in the imports function). Thus:
ΔY = 1 / (1 – 0.3) × (0.3 * 4) = 1.71
That is: a 4-unit decrease in taxes increases output by 1.71 units. This is just one way of computing the effects. One can always recompute the demand function with the new level of taxes and get the value of output.
b) What would have been the effect on equilibrium output in a closed economy? Briefly explain the differences.
To find the effect of the fall in taxes in the closed economy we can just do the same as before, but with the proper value of the multiplier. In the closed economy there are no imports so the marginal propensity to import is zero: m = 0. Then, the slope of the demand is 0.7 (0.3 from the consumption function, and 0.4 from the investment function). Thus:
ΔY = 1 / (1 – 0.7) × (0.3 * 4) = 4
That is: the effect of a 4-unit decrease in taxes is much higher in a closed economy. Again, this is just one way of computing the effects. One can always recompute the demand function with the new level of taxes and get the value of output.
The reason why the effect in a closed economy is larger than in an open economy is because, in an open economy, part of the multiplier effect (the increase in demand that follows the initial increase in income) is partly used to buy foreign goods that do not increase the demand. Thus, the presence of imports lowers the multiplier in the open economy. This can be seen by comparing the slopes of the demand function in the open and in the closed economy: 0.3 versus 0.7 (the difference is the marginal propensity to import, m).
Exercise 5 (Topic 7)
Consider an open economy described by the following system of equations:
- C = 10 + 0.5 * (Y – T)
- I = 10 + 0.2 * Y – 100 * i
- IM = 0.2 * Y * E
- X = 0.3 * Y* – 60 * E
The government sets a level of public spending of 30 and a level of taxes of 20, expected nominal exchange rate is equal to 1, and foreign income is 200:
G = 30, T = 20, Ee = 1, Y* = 200
Assume also that the foreign interest rate is 20%. Given this information, answer the following questions:
a) Obtain the IS curve. What would the IS curve if the economy were closed?
The demand in this economy is:
Z = C + I + G + NX = 10 + 0.5(Y – 20) + 10 + 0.2Y – 100i + 30 + 0.3Y* – 60E – 0.2Y = 10 + 0.5Y – 10 + 10 + 0.2Y – 100i + 30 + 60 – 60E – 0.2Y = 100 – 100i – 60E + 0.5Y
where the exchange rate, E, is equal to:
E = (1 + i) / (1 + i*) *Ee = (1 + i) / 1.20
and therefore:
Z = 100 – 100i – 60 * ((1 + i) / 1.20) + 0.5Y = 100 – 100i – 50(1 + i) + 0.5Y = 50 – 150i + 0.5Y
Applying the equilibrium condition and solving for Y yields:
Y = (1 / (1 – 0.5)) × (50 – 150i) → Y = 100 – 300i
Finally, if the economy were closed, we can just repeat the same process but, in this case, without including net exports in the definition of demand. This is:
Z = 40 – 100i + 0.7Y → Y = 40 – 100i + 0.7Y → Y = 133.3 – 333.3i
b) The central bank decides to set an interest rate of 20%. What is the level of equilibrium output? What is the value of net exports?
The value of equilibrium output is:
Y = 100 – 300i = 100 – 300 * 0.2 = 40
and the value of net exports is:
NX = 0.3Y* – 60E – 0.2Y = 0.3 * 200 – 60 * (1.20 / 1.20) – 0.2 * 40 = -8
This economy exhibits a trade deficit of 20% of GDP.
c) Find the value of output, net exports, and exchange rate in the following two scenarios:
(i) The interest rate rises to 10%
To compute the effect on income, we just need to set the interest rate equal to 10% in the IS curve we derived in part (a):
Y‘ = 100 – 300i‘ = 100 – 300 * 0.1 = 70
and the new value of net exports is:
NX‘ = 0.3Y* – 60 * E‘ – 0.2 * Y‘ = 0.3 * 200 – 60 * (1.10 / 1.20) – 0.2 * 70 = -9
(ii) The interest rate rises to 10% and taxes increase to 30.
As we’ve seen in class, the effects of a combination of fiscal and monetary policy is simply the sum of the effects of the two policies separately. The effect of the fall in the interest rate is computed in the previous part: output increases by 30 units, from 40 to 70. The effect of the increase in taxes is:
ΔY = Multiplier × (-c1ΔT)
The multiplier in this economy is 2 (the slope of the demand function is 0.5). Thus:
ΔY = 2 × (-0.5 * 10) = -10
So the final effect on income is:
ΔY = +30 (monetary policy) – 10 (fiscal policy) = +20
and therefore:
Y” = Y + 20 = 40 + 20 = 60
Finally, the effect on net exports is:
NX” = 0.3Y* – 60 * E” – 0.2 * Y” = 0.3 * 200 – 60 * (1.10 / 1.20) – 0.2 * 60 = -7