requires a strictly positive level of capital and labor. Inputs are chosen to maximize profits. Each firm can be viewed as maximizing profits in two steps. First, it chooses the input bundle ( j , k j ) that minimizes the cost of producing y j units of output. The corresponding cost function is given by C j ( w, r ) y j ≡ min j ,k j w j + rk j : y j ≤ f j ( j , k j ) , j = 1 , 2 and satisfies conditions C 1 – C 6. In the second step, given the cost function C j ( · ) y j , the firm solves the optimization problem Π j ( p j , w, r ) ≡ max y j p j y j − C j ( · ) y j The optimal choice of y j must satisfy the following complemen- tary slackness condition y j ≥ 0; p j − C j ( · ) ≤ 0; and p j − C j ( · ) y j = 0 Hence, in a competitive equilibrium only zero profits are possible.
2. The Preliminaries 21 2.2.3 Characterization of equilibrium Restricting our analysis to the case where both sectors are open, i.e. Y 1 , Y 2 > 0 , equilibrium is defined by a set of factor prices and output levels ( w, r, Y 1 , Y 2 ) ∈ R 4 ++ satisfying the following four conditions: Firms earn zero profits in each output market, C 1 ( w, r ) − p 1 = 0 (2.8) C 2 ( w, r ) − p 2 = 0 (2.9) Labor and capital markets clear, 2 j =1 C j w ( w, r ) Y j = L (2.10) 2 j =1 C j r ( w, r ) Y j = K (2.11) Expressions (2.8) and (2.9) require that the marginal cost of production in sector j be equal to the per-unit output price for the sector. Expression (2.10) ensures the aggregate demand for labor from the two sectors is equal to the endowment of labor L. Likewise, expression (2.11) ensures the capital market clears. In principle, since (2.8) and (2.9) consists of two equations in the unknowns w and r, the solution may be written as w = W ( p 1 , p 2 ) (2.12) r = R ( p 1 , p 2 ) (2.13) Notice that endowments do not appear as arguments in these equations. This result obtains because the number of traded goods equals the number of endowed factors of production. Substituting (2.12) and (2.13) into the factor market clear- ing conditions (2.10) and (2.11) yields two linear equations with input-output coeﬃcients C j i ( W ( p 1 , p 2 ) , R ( p 1 , p 2 )), i, j = 1 , 2 , and unknowns Y 1 and Y 2 . The system is linear because the prices p 1 , p 2 are exogenous in which case C j i ( · ) is a scalar value. Assum- ing both sectors produce at positive levels, denote the solution
22 2. The Preliminaries to the resulting system as Y j = Y j ( p 1 , p 2 , L, K ) , j = 1 , 2 (2.14) When both sectors produce at positive levels, and the elas- ticity of factor substitution between L j and K j is the same for both technologies, then the solution w ∗ , r ∗ satisfying the zero profit conditions is unique. Furthermore, it follows from (2.5) that W ( · ) and R ( · ) are homogeneous of degree one in p 1 and p 2 , 3 while the supply functions (2.14) are homogeneous of degree zero in prices, and of degree one in endowments L and K. The factor rental rate equations (2.12), (2.13) and the sup- ply functions (2.14) can each be used to determine the gross domestic product function G ( p 1 , p 2 , L, K ) = W ( p 1 , p 2 ) L + R ( p 1 , p 2 ) K (2.15) = p 1 Y 1 ( p 1 , p 2 , L, K ) + p 2 Y 2 ( p 1 , p 2 , L, K ) Equation (2.15) indicates that in this model, GDP measured via the cost of production or via the value of output, yields the same result. As noted in the previous section, we can also derive