1 Models

Applied Macroeconomics
Chapter 1. Macroeconomic Models
Prof. Kevin Lawler

Dept. of Economics
Kuwait University
BASIC MACROECONOMIC MODELS
• Reference: W.J.M. Heijman: Applied Macroeconomics

Study Objectives
• To solve basic macroeconomic models
• To learn the difference between ex ante and ex post investment
• To solve a simple dynamic macroeconomic model
• To study the Haavelmo effect
• To study the various effects of taxation
• To learn to compute the ’wedge’
• To show the interdependency between economies
A closed economy with fixed prices
• Within a Keynesian framework this case can be modelled as

follows (Exogenous variables are underscored), no government:
EQUILIBRIUM OUTPUT
• The equilibrium condition implies Y = ED. Solving this model we

get:
• with 1/(1 − γ) for the so-called multiplier. Figure 2.1 gives a

geometrical presentation of the model y*=eq. income.
Stocks ↓
Producers ↑ production
Stocks ↑
Producers ↓ production
STOCKS ADJUSTMENT
• The essential feature of a stable equilibrium is that when it gets

distorted, after a certain amount of time, it will be restored by
some mechanism. In this case the mechanism works as follows
(with V for stocks on hand):
• ED > Y(I > S) ⇒ V ↓⇒ Y ↑,
• ED < Y(I < S) ⇒ V ↑⇒ Y ↓,
• ED = Y(I = S) ⇒ V constant ⇒Y constant.
• With an unstable equilibrium the equilibrium is not restored
after distortion
SIMULTANEOUS EQUATION MODEL
• The above model is the most simple case of a simultaneous

equations model. That is, the model consists of a system of
equations. In such models variables are classified as endogenous
and exogenous. Endogenous variables are determined by the
model itself, while exogenous variables are determined from
outside the model.
• The reduced form of the model above becomes:

EX ANTE –EX POST CASES
• Considering the model above and contrasting savings with

investment, it is important to make a distinction between ex ante
and ex post investments. In the model discussed above, people can
spend income on consumption goods or they can save it (Y = C +
S).
• Household saving is assumed to be matched by the ex post
investment of firms (S = I) [note that the relationships S=I and I=S
are interpreted differently].
• Whether there is equilibrium or not depends on what portion of
firms want to retain. So it depends on their ex ante (desired)
investment. If ex post (realized) investment equals ex ante
investment then there are no unplanned changes in the inventories
of firms.
Dynamics
• Dynamics is of most importance as we see from the above.

Therefore, we shall now give an illustration of how to solve a
simple model in a continuous time framework. In this case the
model can be written as:
CHANGES IN INCOME OVER TIME
• Compared to the first model we have added the first-order

differential equation to it dY/dt. This equation describes the
changes in income over time. The solution of this equation by
means of substitution we get:
• dY/dt = α(γY + C + I − Y) = −α(1 − γ)Y +α(C + I).
• The general solution for this differential equation is:

INCOME PATHS TO STEADY STATE
figure 2 the dynamics of national income

Closed economy, fixed prices, government budget
• The main function of the government in the economy is to levy

taxes and to spend it for the benefit of the society. Taking that into
account the model has to be adapted as follows
GOVERNMENT SECTORS –TAX AND SPENDING
MARGINAL PRODUCTIVITY-a
• In the model the demand for labor Nd equals the coefficient α

times production Y. The coefficient alpha is called the labor
coefficient: the amount of labor needed to produce one unit of
production. Rewriting the model gives:
• Now, assume that the government wants to stimulate the economy

while leaving the budget surplus B unchanged. If the surplus is not
to be changed, then:
• ΔB = ΔT −ΔG =0 ⇒ΔT = ΔG.
• Assume that ΔC = ΔI = 0, then
HAAVELMO AND BALANCED BUDGETS -BB
• This implies that income Y increases by the same amount as the

increase in taxation and government spending. This effect is
called the Haavelmo effect.
• In the last model taxation was completely exogenous. If the
taxation is partly endogenous the taxation equation becomes:
• T = τY + T. τ=tax rate
• Rewriting the model and assuming that ΔI = ΔC = 0 gives:
TARGETS AND INSTRUMENTS
• So, also in this case the Haavelmo effect occurs.

• Now, assume that the government has two targets: full employment
and a balanced government budget. Indicating the targets with a
hat: ΔB = ΔBˆ, ΔU = ΔUˆ . In this case we also have two
instruments (indicated with a tilde): public spending ΔG = ΔG˜
and taxation ΔT = ΔT˜ .
• The Tinbergen condition (no of targets=no of instruments) for
solving this problem is fulfilled. The solution procedure is as
follows. We know the reduced form equation for the case of a
partly endogenous tax revenue (2.10). We also know that:
ΔBˆ = ΔT −ΔG˜ = τΔY +ΔT˜ −ΔG˜ , and
ΔUˆ = ΔNs −αΔY.
RHS =TAGET/EXOGENOUS VARIABLES- ENDOGENOUS
ON LHS
• Writing the target variables and exogenous variables on the right

hand side and the endogenous and instrument variables on the left
hand side of the equal sign, the problem can now be formulated as
follows (with all variables written as deviations like ΔY, ΔG˜ etc.):
MATRIX SOLUTION
CROWDING OUT EFFECTS
• A possible effect of the government stimulating the economy is

that the interest rate goes up, because the government borrows
money to finance its extra expenses. This means that private
investments are discouraged. This is called the crowding out
effect.
• An adaptation of the tax function is based on the idea that when

the taxation rate τ increases, tax payers tend to pay less tax. The
function which then arises is:
T = τY − τaY.
α = the "tax avoidance coefficient".

Laffer curve
• In this case, taxation T is a function of the taxation rate, with Y

standing for the maximum (official) income that a society can
obtain α and for the "tax avoidance coefficient". Figure 2.3
presents a picture of this function. The curve shown is the so-
called Laffer curve. The implication of this is that it is unwise for
the government to set the taxation rate too high. The tax rate τ
that generates the maximum tax rate revenue is called the
optimum tax rate
• In general one can decompose the tax rate into three different
(sub) tax rates:
1. premiums and contributions (for social insurances) on wages paid
by employers (τ1);
2. taxes and premiums on wages paid by employees (τ2);
3. taxes on products paid by consumers (e.g. a value added tax) (τ3).
OPTIMAL TAXES AND INCENTIVES TO WORK
Figure 2.3: Laffer curve with Y = 100 and different values for a
WAGE
• Employers are confronted with gross wages W plus employer

premiums τ1W deflated by production price P. So the real wage
cost equals W +τ1W/P = W(1 +τ1)/P .The supply of labor depends on
the consumption wages. This is defined as net wages deflated by
the price of consumption
• Given this the wedge (the difference between real wage costs and
net wage) can be derived. Relative to the real wage costs the wedge
ω equals:
International trade, fixed prices, government budget
• In this model economic interaction with foreign countries is taken

into account. We assume that exports are exogenous, while imports
depend on national income. Balance of trade can now be defined as
the difference between exports and imports. The full model can
now be described as follows:
• Y = C + I + G + X − F,
• C = γ(Y − T) + C,
FULL MODEL –TRADE –GOVT AND MULTIPLIERS
TYPES OF MULTIPLIER – AND LEAKAGES
• Further we can deduce from the above model that:

A two-country model of international trade
• To give some notion about macroeconomic interdependency in a

multi-country world, we now examine a simple model of two
countries (1 and 2). We assume that country 1 imports the quantity
exported by country 2 and that country 2 imports the quantity that
country 1 exports. Furthermore, the propensities to spend γ and the
imports coefficients are equal μ for both countries. The model can
be expressed mathematically by the following equations:
NOTATION
MODELING
2*2*2
• From this result we can draw the following conclusion: if A1 ≠ A2

⇒ Y1 ≠ Y2 ⇒ μY2 ≠ μY1 ⇒ X2 ≠ X1 ⇒ B1,B2 ≠ 0.Generally the
conclusions to be drawn from (2.18) are that in order to achieve the
highest possible income while maintaining equilibrium on the trade
balance simultaneously, the two countries should agree on
equalizing and maximizing their autonomous spending A. Now
we have reviewed some basic issues regarding macroeconomic
models, the next step is to provide a more extended framework to
analyze macroeconomic policy.

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Applied Macroeconomics

Chapter 1. Macroeconomic Models

Prof. Kevin Lawler

• Reference: W.J.M. Heijman: Applied Macroeconomics

• Within a Keynesian framework this case can be modelled as

• The equilibrium condition implies Y = ED. Solving this model we

• with 1/(1 − γ) for the so-called multiplier. Figure 2.1 gives a

• The essential feature of a stable equilibrium is that when it gets

• The above model is the most simple case of a simultaneous

• The reduced form of the model above becomes:

• Considering the model above and contrasting savings with

• Dynamics is of most importance as we see from the above.

• Compared to the first model we have added the first-order

• dY/dt = α(γY + C + I − Y) = −α(1 − γ)Y +α(C + I).

• The general solution for this differential equation is:

figure 2 the dynamics of national income

• The main function of the government in the economy is to levy

• In the model the demand for labor Nd equals the coefficient α

• Now, assume that the government wants to stimulate the economy

• This implies that income Y increases by the same amount as the

• So, also in this case the Haavelmo effect occurs.

• Writing the target variables and exogenous variables on the right

• A possible effect of the government stimulating the economy is

• An adaptation of the tax function is based on the idea that when

α = the "tax avoidance coefficient".

• In this case, taxation T is a function of the taxation rate, with Y

• Employers are confronted with gross wages W plus employer

• In this model economic interaction with foreign countries is taken

• Further we can deduce from the above model that:

• To give some notion about macroeconomic interdependency in a

• From this result we can draw the following conclusion: if A1 ≠ A2

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