Leontief and Linear Programming |
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Leontief and Linear Programming |
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In the open Leontief model described above, given exogenous final demands, we can solve for quantities uniquely; similarly, given exogenous primary input costs, we can solve for prices uniquely. Thus, prices and quantities seem to be solvable quite independently of each other - a fundamental feature of the Classical system and one quite contrary to the Neoclassical.
However, in the Leontief system final demand c and unit factor returns Bw are given exogenously and may not seem related to one another. If we impose the logical condition that w｢ B｢ x = p｢ c, where w｢ B｢ x are the total factor payments in the economy and p｢ c is the value of the demanded bundle, then we obtain the intuitively plausible idea that owners of factors use their factor returns to make final demands. In this way, then, the quantity side and the price side will not be unrelated to one another, even if one can determine prices independently of quantities and vice-versa.
If we were to rewrite the Leontief system as an appropriate linear programming problem, we actually obtain the result that w｢B｢x = p｢c (Dorfman, Samuelson and Solow, 1958: Ch.9). However, to construct a linear programming problem we must relax the equality restraints imposed earlier and specify an appropriate objective function. Consider the case, then, when we allow an inequality in the quantity restrictions so that:
x ｳ Ax + c
so that we can allow commodities to be produced in excess of input demands and final demand, then it seems we are no longer restricted to a single determinate solution. Supposing our old two-sector case, then we have:
(1-a11)X1 - a12X2 ｳ C1
-a21X1 + (1-a22)X2 ｳ C2
Thus, in Figure 1, the inequality of the quantity equations implies that a solution (X1*, X2*) need no longer be at point X, the intersection of L1 and L2. Instead, we must allow for the possibility of the strict inequality holding for any of the equations in our system. Thus, the inequality in the first equation allows all output combinations in the area below L1 are allowable and the inequality in the second equation implies that all areas above L2 are allowable. The shaded area in Figure 1, thus, represents the output combinations which are feasible given the inequalities.
Figure 1 - Quantity Determination
However, we still want to obtain (X1*, X2*) as the solution, thus we must specify an objective function which yields this. Consider the minimization of total factor payments, w｢B｢x. This is, of course, merely a weighted sum of outputs, i.e. in the two sector case:
w｢B｢x = wb01X1 + wb02X2
We can represent this in Figure 1 by the negatively-sloped locus B with horizontal intercept (w｢B｢x)/wb01 > 0 and slope -b01/b02 < 0. B represents the combinations of outputs X1 and X2 which yield the same factor returns, w｢B｢x. Shifting B rightwards increases total factor payments, whereas shifting it left decreases factor payments. Thus, by minimizing w｢B｢x, then we shift B leftwards to B* where B* intersects the equilibrium point X at (X1*, X2*).
As a consequence, with inequalities, we can view the problem of solving for quantities as a linear programming problem of the following sort:
min w｢ B｢ x
s.t.
x ｳ A｢ x + c
x ｳ 0
Thus, we seek to choose non-negative output levels (x) that minimize factor payments (w｢ B｢ x) subject to the constraint that supply of outputs (x) not exceed demand for outputs (A｢ x + c). Thus, given technology (A, B), the level and composition of final demand (c) and the historically-given real wage (w), we ought to be able to solve for quantities, x* ｳ 0. Notice the important conclusion that, set up this way, output prices do not enter into the determination of quantities.
In linear programming, every primal problem has a dual problem. In this case, we wish to consider the prices-cost equations as inequalities in the following form:
p ｣ Ap + Bw
where price cannot exceed cost of production. In a two-sector system with a single primary input, this implies:
(1-a11)p1 - a21p2 ｣ wa01
-a12p1 + (1-a22)p2 ｣ wa02
which implies, as shown in Figure 2, that we no longer are restricted to solution (p1*, p2*) at the intersection of the M1 and M2 curves (point P). Rather, the first inequality allows us to consider the area above M1 and the second inequality allows combinations in the area below M2. Thus, any price combination in the shaded area in Figure 2 becomes feasible.
Figure 2 - Price Determination
The appropriate objective function now is to maximize the value of final demand, i.e. maximize p｢ c. In a two-sector system, this is written:
p｢ c = p1C1 + p2C2
Thus, we can superimpose a negatively-sloped locus C in figure 2 representing the combinations of p1 and p2 which yield the same value of final demand, p｢ c for a given C1 and C2. Thus, the locus C has horizontal intercept p｢ c/C1 > 0 and slope -C1/C2 < 0. Shifting the C locus to the left decreases the value of final demand, whereas shifting it to the left increases the value of final demand. Thus, maximizing p｢ c, we move to the highest locus C* and thus obtain the equilibrium solution (p1*, p2*) at P, the intersection of the M1 and M2 curves.
The dual of this problem, then, is stated as the following:
max p｢ c
s.t.
p ｣ Ap + Bw
p ｳ 0
thus we seek to choose the non-negative prices (p) that maximize the value of final demand (p｢ c) given that prices do not exceed unit cost of production p ｣ Ap + Bw. Thus, given technology (A, B), the level and composition of final demand (c) and the historically-given real wage (w), we can solve for prices, p* ｳ 0. Notice that here we have the important condition that quantities of output (x) do not enter into the determination of prices.
Having specified the appropriate linear programming problems, let us now turn to the solution. We can rewrite this primal problem of minimizing factor payments as a Lagrangian of the following sort (cf. Takayama, 1974):
L(x, l ) = w｢ B｢ x + l ((I - A｢ )x - c)
where l is a vector of Lagrangian multipliers. The solution to this problem is a pair (x*, l *) such that:
(i) w｢ B｢ x* ｣ w｢ B｢ x for all x ｳ 0, (I - A｢ )x ｳ c
(ii) l *((I - A｢ )x - c) = 0 where l * ｳ 0
The first condition merely states that x* minimizes the objective function; the second condition is the complementary slackness condition: at the solution x*, either a particular constraint is binding (i.e. net outputs equal consumption demand for a particular good j, so that so that (Ij - aj｢)xj = cj where Ij and aj｢ are the jth rows of I and A｢ respectively) or, if not binding ((Ij - aj｢ )xj > cj, thus overproduction of good j), then the associated multiplier will be zero (l j* = 0).
Let us now turn to the dual problem of maximizing the value of final demand. Rewriting this into a Lagrangian form:
L(p, m ) = p｢ c + m [Bw - (I - A)p]
where m is a vector of Lagrangian multipliers associated with this problem. The solution to the dual programming problem can be characterized as a pair (p*, m *):
(i) p｢ *c ｳ p｢ c for all p ｳ 0, Bw ｳ (I - A)p
(ii) m *(Bw - (I - A)p*) = 0 where m * ｳ 0
Now, the first condition states that the solution p* must mazimize the value of net output. The second condition is complementary slackness again: at the solution p*, either a particular constraint is binding (i.e. price equals unit cost of production for a given particular good i, so that biw - (Ii - ai)pi = 0 where bi, ai and Ii are the ith rows of B, A and I respectively) or, if not binding (biw - (Ii - ai)pi > 0), then the associated multiplier will be zero (m i* = 0).
Appealing to the duality theorem of linear programming, we can observe the following:
(i) m j* = xj* for all j = 1, .., n.
(ii) l i* = pi* for all i = 1, .., m.
(iii) p｢ *c = w｢ B｢ x*
Condition (i) states that the multiplier in the dual (m j*) is equivalent to the solution to the primal problem (xj*) and (ii) states that the multiplier in the primal problem (l i*) is equal to the solution the dual problem (wi*). Thus, the first Lagrangian could actually be rewritten as:
L(x, w) = p(I - A｢ )x + w(v - B｢ x)
where we have substituted w for the multiplier l , whereas the second Lagrangian can be rewritten:
L(w, x) = w｢ v + x[Bw - (I - A)p]
where we have substituted x for the multiplier m . Thus, the complementary slackness conditions become:
p*((I - A｢ )x - c) = 0
x*(Bw - (I - A)p*) = 0
where the first implies the familiar "free goods" assumption (if the net output of any good exceeds its consumption demand, then the price of that good is zero) and the second implies the "excess cost" condition (if price falls below unit cost of production for any good, then that good will not be produced). With sufficient assumptions on the Leontief system (e.g. indecomposability of A), the free goods and excess cost conditions will not be applied as all goods will be produced and they will have a positive price.
Condition (iii) is quite interesting since it states that, in equilibrium, factor payments will be equal to final demand - thus confirming our earlier statements. More interesting is that this does not implicate our assumptions about the independence of the determination of quantities and prices. We earlier stated that (A, B, w, c) enter into each of the programming problems. However, as it turns out, B and w turn out to be irrelevant to the determination of equilibrium quantities and c turns out to irrelevant to the determination of equilibrium prices.
We can see this diagramatically in Figures 1 and 2. In Figure 1, we can see that as long as w ｳ 0 and B ｳ 0, then a locus B with any slope will, when minimized, yield the unique solution point X. This is true regardless of the values of the weights on the locus, w and B. Thus, in effect, only A and c really matters to the solution, x*. We are thus back to the regular Leontief equation, where x = A｢ x + c and factor input requirements and factor payments do not affect the solution quantities, x* ｳ 0. Similarly, in Figure 2, we can see that as long as c ｳ 0, then a locus C with any slope will, upon maximization, yield the unique solution point P. Thus, the size and composition of final demands, c, do not affect the determination of equilibrium prices; only w, A and B matter for the determination of equilibrium prices, the values of C1 and C2 are quite irrelevant. Thus, the Leontief assertion, p = Ap + Bw, remains and demands do not affect the equilibrium prices, p* ｳ 0. In short, the Classical conclusions of the independence of the determination of prices and quantities maintain themselves even when expressed in linear programming form.