To begin with, let us recall our three-sector product-market equilibrium model given as
C + I + G = C + S + T
To this three-sector model, we now add the foreign trade-the exports (X) and imports (M). with the addition of X and M, the four-sector product-market equilibrium condition is written as
C + I + G + (X - M) = C + S + T
The variables X and M need some explanation and quantification exports (X) of a country depend on a variety of factors governing the foreign demand for its goods and services. The inclusion of foreign demand parameters in the domestic model of a country is neither an easy task nor a necessity for a simplified model. Therefore, X is assumed to be a constant factor, that is,
X = X
As regards imports, imports (m) of a country are a function of a number of factors, however, for the sake of analytical simplicity; imports are treated as the function the country's national income(Y). That is import function takes the following form
M = + mY
Where, M is autonomous import and m is marginal propensity to import, the proportion of marginal national income spent on imports.
With X and M defined, the four- sector product-market equilibrium condition given in can be rewritten as
C+ I + G + X - M - mY = Y = C + S + T
The product-market equilibrium condition can also be expressed as
Y = C + I + G + X - M - mY
Where C = a + by d( where Yd = Y - T = disposable income)
S = - a + (I - B) y (where I - B = mps)
I = I - Hi (where h > 0)
G = G, (where G is constant)
T =T + t y, (where T is constant tax and t is tax rate <1)
By substituting the equilibrium level of income can be expressed as
Y = a + b [Y - (T + t Y)] + I - hi + G +X - M - my
=a + by - b t - bty + I - hi + G + X - M - my
Y = 1 / 1-b+ bt + m (a - b T + I - hi + G + X - M
Y = 1 / 1 - b (1 - t) +m (a - b T + I - hi + G + X - m
Note that the term 1/ (1 - b + bt + m) is tax-trade multiplier which may be redesignated as mu. Also let us designate the sum of the five constants, viz a, i. G, X, and M as A. by substitution these value
Y = mu (a - b T - hi)
(Where mu is tax-trade multiplier and A = a + I + G + X - M)
Equation gives the aggregate demand (AD) function in a four-sector model.