Practice problems on TOPIC 3
Macroeconomics 1
PRACTICE PROBLEMS – TOPIC 3 – THE GOODS MARKET
Step-by-step Solution to Keynesian Cross (Expenditure-Output) Model
In the expenditure-output or Keynesian cross model, the equilibrium occurs where the
aggregate expenditure line (AE line) crosses the 45-degree line. Given algebraic
equations for two lines, the point where they cross can be readily calculated. Imagine
an economy with the following characteristics.
•
Y = Real GDP - national income
•
Tx = Taxes = 1.000
•
Tr = 300
•
T = Net Taxes = Tx – Tr = 700
•
Yd = Disposable income = (Y – T)
•
C = Consumption = 140 + 0’9(Y – T)
•
I = Investment = 400
•
G = Government spending = 800
•
EX = Exports = 0
•
IM = Imports = 0
Step 1. Determine the aggregate expenditure function. In this case, it is:
𝐸 = 𝐶 + 𝐼 + 𝐺 = 140 + 0′9(𝑌 – 𝑇) + 400 + 800
Step 2. The equation for the 45-degree line is the set of points where GDP or
domestic/national income on the horizontal axis is equal to aggregate expenditure on
the vertical axis. Thus, the equation for the 45-degree line is: AE = Y:
𝑌 = 𝐶 + 𝐼 + 𝐺 = 140 + 0′9(𝑌 – 𝑇) + 400 + 800
Step 3. The next step is to solve these two equations for Y (or AE, since they will be
equal to each other). Substitute Y for AE:
𝑌 = 140 + 0′9(𝑌 – 𝑇) + 400 + 800
Step 4. Insert the term T=700. This produces an equation with only Y.
𝑌 = 140 + 0′9(𝑌 − 700) + 400 + 800
Let’s then work through the algebra and solve for 𝑌 ∗ and for 𝐶 ∗
𝑌[1 − 0,9] = 140 − 630 + 400 + 800 → 𝑌 ∗ = 7.100
𝐶 ∗ = 140 + 0′ 9(𝑌 – 𝑇) = 140 + 0′ 9 (7.100 − 700) = 5.900
Practice problems on TOPIC 3
Macroeconomics 1
This algebraic framework is flexible and useful in predicting how economic events and
policy actions will affect real GDP.
Step 5. What is the household saving rate of the economy? What is the government
saving rate of the economy? What is the aggregate saving rate of the economy?
i.
The households’ saving rate (or private saving rate), 𝜎𝑝 , is given by the ratio
between households´ saving (a.k.a. private saving, 𝑆𝑝 ), and disposable income:
𝑆𝑝 = 𝑌𝑑 − 𝐶 = (7.100 − 700) − 5.900 = 500
𝑆𝑝
500
𝜎𝑝 = 𝑌 = 7.100−700 = 0′ 078125 ≅ 7′81%
𝑑
ii.
The Government’ saving rate (or public saving rate), 𝜎𝑃 , is given by the ratio
between government´ saving (a.k.a. public saving, 𝑆𝑃 ), and domestic income:
𝑆𝑃 = 𝑇 − 𝐺 = 700 − 800 = −100
𝜎𝑝 =
iii.
𝑆𝑃
𝑌
−100
= 7.100 = −0′ 01408 ≅ −1′4%
The aggregate saving rate, 𝜎, is the share of total (private and public) saving on
domestic income:
𝜎=
𝑆𝑝 +𝑆𝑃
𝑌
=
500−100
7.100
= 0′056338 ≅ 5′63%
Step 6. Assume that because of changes in the relative prices of domestic and foreign
goods, the marginal propensity to consume domestic goods falls by 10 percentage
points. Calculate the equilibrium output when the marginal propensity to consume
(MPC) is changed from 0.9 to 0.8 and make sense of this change.
𝑌[1 − 0,8] = 140 − 630 + 400 + 800 → 𝑌 ∗ = 3.550
so that ∆𝑌 ∗ = 3.550 − 7.100 = −3.550.
A lower MPC generates less demand for output at every step of the multiplication
process. Note that, in general,
𝑌 = 𝐴𝐶𝑆 + 𝑀𝑃𝐶[𝑌 − 𝑇],
where ACS denotes the autonomous components. Therefore,
𝑑𝑌 = 0 + 𝑑𝑀𝑃𝐶(𝑌 − 𝑇) + 𝑀𝑃𝐶(𝑑𝑌) →
𝑑𝑌[1 − 𝑀𝑃𝐶] = 𝑑𝑀𝑃𝐶(𝑌 − 𝑇) →
(𝑌 − 𝑇)
𝑑𝑌
=
>0
𝑑𝑀𝑃𝐶 (1 − 𝑀𝑃𝐶)
Practice problems on TOPIC 3
Macroeconomics 1
Step 7. Assume instead that because of a surge of business confidence, investment
rises to 500 (ceteris paribus). Calculate the new equilibrium output and make sense of
this change.
𝑌[1 − 0,9] = 140 − 630 + 500 + 800 → 𝑌 ∗ = 8.100
so that ∆𝑌 ∗ = 8.100 − 7.100 = 1.000.
A greater I generate greater initial demand for output, with an expansionary effect
that reverberates through the multiplication process. Note that, in general,
𝑌 = 𝐴𝐶𝑆 + 𝑀𝑃𝐶[𝑌 − 𝑇],
where ACS denotes the autonomous components of spending. Therefore, for 𝑑𝐴𝐶𝑆 = 𝑑𝐼:
𝑑𝑌 = 𝑑𝐴𝐶𝑆 + 𝑀𝑃𝐶(𝑑𝑌 − 0) →
𝑑𝑌[1 − 𝑀𝑃𝐶] = 𝑑𝐼 →
𝑑𝑌
1
=
>0
𝑑𝐼 [1 − 𝑀𝑃𝐶]
Step 8. Ceteris paribus, assume that the policy maker wants to increase public
spending (∆𝐺 > 0) and simultaneously change taxes (∆𝑇𝑥 ) to keep a constant public
budget. Identify how ∆𝑇𝑥 must be related to ∆𝐺 for this goal to be achieved.
The public budget is given by 𝐵 = 𝑇𝑥 − 𝑇𝑟 − 𝐺. For the budget to not change we need
∆𝐵 = ∆𝑇𝑥 − ∆𝐺 = 0 → ∆𝑇𝑥 = ∆𝐺
Step 9. Ceteris paribus, we want to determine which adjustment should be made to
the government spending (∆𝐺) or to taxes (∆𝑇𝑥 ) for real GDP to become its potential
∗
level, estimated to be equal to 𝑌𝑝𝑜𝑡
= 6.000.
Calculate, separately, the value of ∆𝐺 and of ∆𝑇𝑥 that allow the economy to perform
potential GDP and make sense of these changes.
i.
6.000[1 − 0,9] = 140 − 630 + 400 + 𝐺 → 𝐺 = 690
Therefore, ∆𝐺 = 690 − 800 = −110.
ii.
6.000 = 140 + 0′9(6.000 − 𝑇𝑥 + 300) + 400 + 800 → 𝑇𝑥 = 1.122,22
Therefore, that ∆𝑇𝑥 = 1.122,22 − 1.000 = 122,22.
Step 10. Ceteris paribus, you want to determine which adjustment should be
simultaneously made to the government spending (∆𝐺) and to taxes (∆𝑇𝑥 ) for real GDP
∗
to become equal to its potential level, estimated to be 𝑌𝑝𝑜𝑡
= 6.000. Identify how ∆𝑇𝑥
must be related to ∆𝐺 for this goal to be achieved.
∗
For GDP to become 𝑌𝑝𝑜𝑡
= 6.000, we need to produce ∆𝑌 = 6.000 − 7.100 = 1.100.
Therefore ∆𝑌 = 0′ 9(− ∆𝑇𝑥 ) + ∆𝐺 → 1.100 = 0′ 9(− ∆𝑇𝑥 ) + ∆𝐺 → ∆𝐺 = 1.100 + 0′ 9 ∆𝑇𝑥
Practice problems on TOPIC 3
Macroeconomics 1
For example, ∆𝑇𝑥 = 100 must be accompanied by ∆𝐺 = 1.190.
Step 11. What are the endogenous variables, the exogenous variables, and the
parameters of the Keynesian Cross model?
Endogenous = 𝑌, 𝑌𝑑 , 𝐶, 𝑆𝑃 , 𝑆𝑝 , 𝑆, 𝜎𝑃 , 𝜎𝑝 , 𝜎
Exogenous = No exogenous variables
Parameters = 𝑐0 , 𝑐1 , 𝑇𝑥 , 𝑇𝑟 , 𝑇, 𝐼, 𝐺
PROBLEM 1
Repeat the first 7 steps of the step-by-step exercise by assuming proportional taxes to
income: 𝑇 = 𝜏𝑌, with 𝜏 = 0′3. Repeat also step 11.
PROBLEM 2
Let an economy with the following parameters and variables
• Y = Real GDP - national income
• T = Net Taxes = Tx - Tr
• Tx = Taxes = 0.2Y
• Tr = 100
• Yd = Disposable income = (Y – T)
• C = Consumption = 200 + 0’7(Y – Td)
• I = Investment = 600
• G = Government spending = 1.000
• EX = Exports = 300
• IM = Imports = 100
a. Calculate the equilibrium GDP for this economy (remember Y = AE).
b. What change in the economy´s tax rate would be required to produce an economic
expansion equal to 55 (change in GDP)?
c. Ceteris paribus, assume that the potential level of GDP is 5.000.
i.
What level of investment expenditure (I) would be necessary to reach
that level?
Practice problems on TOPIC 3
ii.
Macroeconomics 1
What level of government spending would be necessary to reach that
level? How is your answer different from part i)?
d. Suppose now that government expenditure is increased by 100 and the economy´s
tax rate is increased by 5 percentage points (that is, from 20% to 25%). Ceteris
paribus, what is the impact of this fiscal policy mix onto the equilibrium GDP? As a
consequence of that, how would you expect the Government budget to be affected
by this change?
PROBLEM 3
Consider the following information for the country of Incomia.
Category
Amount
Autonomous Consumption
250
MPC
0.9
Tax Rate
0.25
Transfers
150
Investment
800
Government Expenditure
100
Exports
20
Imports
10
a. What is the equilibrium level of GDP in Incomia according to Keynesian Cross model?
b. Suppose that there is a decrease in Exports by 10. What is the new equilibrium level of
GDP in the income-expenditure model? What is the difference between the original and
new GDP because of a decrease in Exports?
c. More in general, how does GDP change due to a change in Export by ∆𝐸𝑋 < 0?
d. Suppose that MPC becomes equal to 0.8. What is the new spending multiplier?
e. More in general, how does the GDP multiplier change due to a change in MPC by
∆𝑀𝑃𝐶 < 0.?
PROBLEM 4
Practice problems on TOPIC 3
Macroeconomics 1
The following diagram displays the time path for the US private saving rate (red line)
and the US aggregate saving rate (blue line) during the period 1998-2008. How is it
possible that the aggregate saving rate has been negative during a few months in
2005?
PROBLEM 5
Suppose a standard closed Keynesian-Cross economy with the following parameterization:
•
𝑐0 = 300,
•
𝑐1 = 0.7
•
𝐼 = 175,
•
𝐺 = 100
•
𝑇𝑥 = 50
•
𝑇𝑟 = 0
a) What are aggregate output and consumption in equilibrium?
b) Determine private saving, government saving and aggregate saving in equilibrium.
(c) What is the overall marginal propensity to save (saving rate) in this economy and
how is it related to the propensity to consume?
c) Determine aggregate output, consumption and saving, if the propensity to consume
cm decreases to 𝑐0 = 0.5. Compare your findings with the results for 𝑐1 = 0.7
d) How much has the government to raise its expenditures after the drop in the
propensity to consume, if it wants to stabilize output and keep taxes constant?
Practice problems on TOPIC 3
Macroeconomics 1
BONUS & CHALLENGING PROBLEM
Suppose taxes depend on the level of income in the following way: Tt = Ta +tYt . In
addition, change the parameterization to cm = 0.6, I = 150, G = 250, Ta = 20 and t =
0.2. (a) Determine aggregate output, consumption and taxes in equilibrium for Ca =
100 and Ca = 80. (b) Assume that the taxes are fixed at the level determined in part
(a) for the case of Ca = 100. Calculate again the level of income and consumption in
equilibrium for Ca = 100 and Ca = 80. (c) Why is a policy where taxes depend on the
level of income called an “automatic stabilizer“? (Hint: Compare the impact of a
decline in autonomous consumption of 20 units when taxes are endogenous with the
case of autonomous taxes.) (d) Compare the impact of a change in autonomous
consumption on the size of the budget deficit when taxes are endogenous with the
case of autonomous taxes