1. Salient models of general equilibrium
1.5. A Koopmans–Kantorovich variant: a linear model based on fixed coefficients
The early models of general equilibrium using fixed input coefficients represented production in an ex post manner, treating them simply as given average input coefficients, which could be observed once equilibrium is reached. Walras noted that they were in fact variables, depending on prices, but considered this fact a negligible technical detail, which can be ignored in an abstract model. In a later edition of his book he derived them from the marginal conditions of cost minimization. As for Leontief the use of fixed input coefficients was a pragmatic necessity dictated by the availability of statistical data as well as computational techniques and facilities.
It was, however, not so much the use of fixed coefficients that raised theoretical concerns, but rather the neglect of technological choice and joint production, the proper representation of the technology. Smooth classical production functions, allowing for substitutability between pairs of inputs and outputs in a wide range, offered an alternative and they became standard tools in neoclassical microeconomic theory. Von Neumann (1937), on the other hand, in his model of equilibrium growth demonstrated that the technology allowing for technological choice and joint production can be represented using fixed input-output coefficients.
It was Koopmans (1951) who laid down the axiomatic foundations of production theory, in general and the linear activity model, in particular. For Koopmans the choice between smooth, differentiable production functions or fixed coefficients was not a theoretical, but a practical problem, which should be governed by the purpose of the model (pure or applied), mathematical and computational algorithms and techniques, the availability of statistical data and so on.
In applied models of optimal resource allocation and choice of techniques, based on detailed representation of technology, the linear activity model combined with the method of linear programming proved to offer a more suitable approach than the models based on smooth, differentiable production functions. The former can be based on the knowledge of discrete technological variants, whereas the estimation of production functions is severely constrained in practice. The linear input-output programming approach dominated for many years the applied macroeconomic modelling for policy analysis.
The linear activity model rests on the assumption that the technology can be represented as the nonnegative combinations of finite number of elementary activities. Let us denote by vector ãj ∈ Rn the unit net input-output coefficients of the jth elementary activity, where ãij is positive if the output of good i exceeds its input, negative if its input is larger than its output, zero otherwise. Let us denote by x = (xj) the vector the (nonnegative) levels of the various elementary activities. The technology defined by linear activity model is the following:
T = {t: t =
∑
= m
j 1
xj⋅ãj = Ãx, x ≥ 0},
where à is the (unit) input-output coefficient matrix, where the unit levels of the elementary activities, and thus that of the unit coefficients, can be chosen arbitrarily.
The production set generated by the set of the following input-output coefficient matrix (two inputs and one output, four elementary activities) is illustrated on Figure 1.1 and 1.2.
à =
1,0 1,0 1,0 1,0
0,1 -0,3 -0,2 -0,4
-0,5 -0,4 -0,3 -0,2
-Figure 1.1
The technological set defined by a matrix Ã
The assumptions of proportionality and additivity imply constant returns to scale and lack of production externalities, and that the technology is a convex polyhedral cone (see Figure 1.2). The production set generated by the linear activity model is a piecewise linear variant of the one defined as follows:
T = {t: t =
∑
= m
j 1
t(j), Fj(t(j)) = 0},
where Fj are the production functions used in the Paretian–Hicksian system of general equilibrium, and they are all homogenous of degree zero.
Figure 1.2
The input requirement set and the unit isoquant
the input requirement set at unit level of output
AKOOPMANS–KANTOROVICH MODEL OF GENERAL EQUILIBRIUM
We will present now a completely linear variant of the Paretian–Hicksian system of general equilibrium based on the linear activity description of technology. As we will show it in the next chapter, we can easily solve this model by means of a linear programming problem.
The necessary conditions of equilibrium are as follows:
(E0) feasible activity levels and prices:
x, y, p ≥ 0,
(E1) producers maximize profit:
a) pÃ≤ 0, but b) pÃx = 0 (E2) consumers’ choice and Walras’s law:
a) y = y(p, e ) = e⋅c
/
psy, where b) e = pa,
(E3) all commodity markets are clearing:
a) a + Ãx ≥ y, but b) pa + pÃx = py.
The commodity composition of consumption (sy) is considered to be fixed in order to maintain the linearity of the equilibrium conditions, as much as possible. The notable exceptions are the so-called complementarity conditions E1/b and E3/b, which state that the profit is maximal (zero) in the case of activities used in equilibrium and that the price of the commodities in excess supply is zero in equilibrium.
Consumption is modelled here as if consumers’ choice would be the outcome of an optimal decision in the case of a Leontief-type utility function:
u(y1, y2, … , yn) = min
{
(y1/ s1y, y2/ s2y, … , yn/ sny}.Condition E2/a is so far nonlinear, but we can linearize it by choosing the unit basket of consumption as numeraire, by setting its value to one, that is psy = 1, when we get
yi(p, e) = e⋅siy
.
It can be easily seen that the equilibrium conditions are equivalent to the optimality conditions of the following Kantorovich-type linear programming problem:
Primal problem Dual problem
x ≥ 0, y ≥ 0 p ≥ 0
(p) a + Ãx ≥ y⋅sy pà ≤ 0 (x)
psy ≥ 1 (y)
y → max! pa → min!
where y = e
/
psy is the level of consumption.Thus, we have again demonstrated the close connection that exists between the models of nation-wide optimal resource allocation and the macro-models of general equilibrium. As we
will show it in the next chapter that close conceptual similarity led modellers for policy analysis to switch to computable general equilibrium from linear programming models in the second half of the 1990s.
1.6. A step towards computable models: Johansen’s model of general equilibrium