You could use the CobbDouglas production functions as a simple example:
(1) C = ALCaTC1a
and
(1’) F = BLFbTF1b,
where C(F) is the quantity of clothing (food) being produced, LC(LF) is the quantity of labour employed in cloth (food) production, and TC(TF) is the quantity of land employed in cloth (food) production. The coefficient A(B) is a positive constant reflecting technology.
First, define the unit isoquants, showing the amount of land used given the employment of labour in producing each unit of cloth (food). These are:
(2) TC = [A1LCa]1/(1a)
and
(2’) TF = [B1LFb]1/(1b) .(1)
The firstorder conditions, coupled with linear homogeneity, allow the derivation of lines that show factor price ratios that are consistent with various input combinations. Following Krugman and Obstfeld, these are denoted as CC and FF respectively. The equation for the line CC (the cloth industry's expansion path) is:
(3) r = (1/a1)(LC/TC)w ,
where w is the wage rate (payment per unit of labour) and r is the rental rate (per unit of land). The corresponding FF line, similarly defined, is:
(3’) r = (1/b1)(LF/TF)w.
So far, the model transforms the expansion paths in the two industries into equations involving factor prices in each of the two industries. To move closer to closing the model requires that product prices be related to the information above. This is accomplished by imposing equilibrium conditions for competitive markets, that price equals marginal cost (MC) in each industry. Since MC = w/MPL in each industry, this imposition implies the following:
(4) PC = w(Aa)1(1/a1)a1(w/r)a1
and
(4’) PF = w(Bb)1(1/b1)b1(w/r)b1.
These two equations provide this critical relationship between the ratio of input prices and that of product prices:
(5) (w/r) = [Aa(1/a1)1a(Bb1)1(1/b1)b1(PC/PF)]1/(ab).
Determining how much labour and land are used in each of the two industries can now close the model. The expansion paths give the key to this determination.
(6) TC = (w/r)(1/(a1))LC
and
(6’) TF = (w/r)(1/(b1))LF.
But, LF = LLC and TF = TTC.
Appropriate substitution allows TC and TF to be stated as a linear function of LC. The resulting equations are:
(7) TC = kLC
and
(7') TF = mLmLC
where k=(w/r)(1a)/a and m = (w/r)(1b)/b.
