Optimising solution:

If we consider the problem set up above, then we can write the reduced model as  follows:  

Here, L  and  K  are,  respectively,  the  parametrically 'fixed' total resource endowment of labor and capital. Now, we will make another simplifLing assumption, namely,  f²(L_I,  K_I), and P(L₂, K₂) exhibit constant returns to scale (CRS) i.e.,

We can use  the above optimisation programme to derive major  theorems of international trade as well. We assume that  L₁, L₂, K₁  and K₂ are  fully  used  up at  parametrically given levels of L and K, respectively.

The Lagrangian for this model  is

The  first-order conditions for constrained maximum  are  obtained  by differentiating with respect to the four choice variables L₁, L₂, K_I, K₂  and the two Lagrange multipliers. We get

The second  order conditions consist of restrictions on border-preserving principal minors of the border Hessian determinant formed by differentiating equations  (i) to (vi) again with respect to the L,'s , L, 's  and λ,  's  .

The  border-preserving principal minors  of  K  alternate  in  sign, the  whole determinant having sign +l. Assuming  the sufficient second-order conditions hold,  first order conditions can be solved for  the explicit choice functions.

In  the above, (vii) and  (viii) show  the quantities of each  factor that will  be used by each industry at given output prices and total resource constraints.

The role of  the factor prices is filled by  the Lagrange multipliers λ_L,  and λ_K,