Development of Spatial Modeling

Spatial Price Equilibrium models as opposed to non-spatial equilibrium models, which are generally referred to as net-trade models, explicitly take transport costs from any point of production to the points of consumption into account. Net trade models like the ex-ample formulated below usually consist of a system of equations that must hold simultane-ously.

The price transmission equation (3) usually reflects the trade policies of the country in question vis-à-vis the world market. Under very restrictive assumptions, the price trans-mission equation could account for transportation costs, too. Precisely, this would mean that the destination of exports would be exogenous to the model or that the transportation costs are unique regardless of where exports are shipped to. The reader will note that the same argument applies also for trade policies in net trade model: The model assumes the same instruments to be applied to all imports (and exports in the case of export subsidies or taxes) regardless of their source – a fact which is crucial for the discussion of modelling approaches to be applied for the analysis of the sugar market in the previous chapter.

If bilateral transportation costs (or trade policies) are to be introduced in the model it can no longer be formulated as a system of equations as the net trade model above, but must involve inequalities. This becomes apparent if one tries to add another dimension to the price

transmission equation (3) which would, in the case of constant per unit transport cost, look

tcij per unit transportation costs from i to j

It is obvious that equation (5) can only hold for those pairs of countries where trade actually occurs. In all other cases, the left hand side of the equation would be larger than the right hand side, which corresponds to a real world situation where it is not profitable to ex-port from i to j.⁴⁷ To determine in which direction trade takes place is, however, a task en-dogenous to the model. A model taking into account bilateral transportation costs would thus look like:

As indicated above, the price transmission equation (8) will hold with strict equality in those cases where trade occurs and with inequality in those cases where not.

Lacking proper mathematical algorithms to solve the above problem Enke (1951) showed in his 1951 article how a special equilibrium model with linear export supply (import demand) curves which imply domestic demand and supply curves which are also linear can be solved with an electric circuit. In such a circuit, prices and transportation costs would be analogue to voltages, export supply coefficients would be analogue to electrical resistors and trade flows would be analogue to amperages. If connected, from such a circuit ammeters and

47 Note that the left hand side can never be larger than the right hand side in any equilibrium point, as, in a competitive market, arbitrage would occur in that case and level any differences.

voltmeters can be used to read the local prices and the directions and volumes of trade flows.⁴⁸

One year after the Enke article Samuelson (1952) showed that the spatial equilibrium model can equivalently be cast as a maximum problem. He demonstrates how to construct a

“Net-Social-Payoff”-function including the integrals under the export supply and import de-mand functions minus the transportation cost. With this function maximized, prices and quantities would satisfy the equilibrium conditions of the SPE described above. At the time the article was published, there was, however, no possibility to calculate the maximum of such a function other than experimentally varying the variables of the model.

Under the very restrictive assumption of non-price responsive demand and supply equations such a model could be solved with linear programming techniques, which had been developed recently before. The simplex method developed by Dantzig (1951) allows finding the optimum of a linear function subject to linear constraints as is shown below.⁴⁹

LP ∑

i

i

i X

c * Maximize

j i

i

ij X b

a ≤

∑

subject to

≥0 Xi

with

Xi choice variables

aij, bj, ci coefficients

i = 1,…, n set of variables

j = 1, …, m set of constraints

The simplex method has one particular advantage over the equilibrium formulation in (6) – (10), which is its ability to determine endogenously which of the inequalities will hold with strict equality and which will not. Its disadvantage is that it can only handle linear ob-jective functions. Price responsive supply and/or demand functions would, however, even in their most simple possible form, which is a linear function, lead to a quadratic (i.e. nonlinear) net-social-payoff-function in the Samuelson sense.

An important step towards the solution of a nonlinear SPE model are the Kuhn-Tucker-Conditions (KT-Conditions, (Kuhn and Tucker, 1951)). Together with the so-called

48 In the same paper Enke also demonstrated that for three or less countries the solution is mathematically fairly easy.

constraint qualification, they pose necessary conditions for an optimal point of a Nonlinear Programming (NLP) problem such as (12) below.⁵⁰

NLP

To derive the Kuhn-Tucker Conditions, the Lagrangian of the NLP is formed in (13).

)

The corresponding KT-Conditions of the above problem are:

KT-Conditions

The KT-Conditions show one remarkable difference compared to the first order con-ditions that can be derived from a Lagrangian function describing an optimisation problem

49 Dantzig developed the method in 1947. Isolated from western mathematicians and economists, Leonid V.

Kantorovich found a similar method already in 1939 (Kantorovich, 1939; Montias, 1961).

50 Basically, this means an NLP not fulfilling the constraint qualification may have a maximum or a minimum not fulfilling the KT Conditions. For a detailed explanation of the Constraint Qualification refer to Chiang and Wainwright (Chiang and Wainwright, 2005),Ch. 13.2.

Some publications refer to the Kuhn-Tucker-Conditions as the “Karush-Kuhn-Tucker-Conditions” (KKT-Conditions) which reflects the fact that William Karush in his Master’s thesis (Karush, 1939) “obtained identi-cal results twelve years earlier” than they had been published by Kuhn and Tucker in 1951(Tapia and Trosset, 1994), which was, however, only revealed decades later.

with equality constraints, which are the so-called complementary slackness conditions in the third column. In an equality-constrained problem the first order derivatives of the Lagrangian must be zero. The KT-Conditions in contrast require that either the marginal equation be zero or the associated variable (or both). Exploiting complementary slackness makes it pos-sible to determine endogenously which inequality is to hold with strict equality and which not, i.e. to find so-called corner or boundary solutions (Bishop et al., 2001).

With constraint qualification provided for, the KT-Conditions provide necessary con-ditions for a local optimum. For convex optimisation problems, the KT-Concon-ditions even are necessary and sufficient conditions for a global optimum. An optimisation problem is called convex, if the feasible points of a solution form a convex set and if the objective function is convex in case of a minimum problem and concave in case of a maximum problem (Chiang and Wainwright, 2005).⁵¹

It is, however, in most cases not trivial to find the points satisfying these conditions.

In the 1950s, when Samuelson published his article, no efficient algorithm existed to solve a nonlinear program of appropriate size. For instance, attempts to use the gradient method to find a solution of a spatial agricultural sector model of Hokkaido (with price responsive de-mand functions) failed (Takayama, 1992). In that decade there were thus published many studies with non-price responsive demand relying on Linear Programming (LP). The most important agricultural economist in this context was Earl O. Heady. (Heady, 1952; Heady and Candler, 1958).

Applying a modified Simplex-algorithm developed by Wolfe (1959) which was able to deal with Quadratic Programming (QP) problems⁵², Takayama finally succeeded to solve his Hokkaido model and together with G. J. Judge to apply QP to the Samuelson framework (Takayama and Judge, 1964a; Takayama and Judge, 1964b).Ever since, their approach has been applied numerous times to real world problems.

To use the QP approach it is, however, necessary that the equations of the model sat-isfy the so-called “integrability condition”. Otherwise, the net-social-payoff function Samuelson had in mind cannot be derived unambiguously. This is e.g. the case when the matrix of coefficients of the demand and supply functions in the multi product case is not

51 Arrow and Enthoven(1961) showed that, under certain conditions, the sufficiency of the KT-Conditions can also be extended to the case of quasiconvex programming.

symmetric (Takayama and Judge, 1971). In that case, no unique net-social-payoff function 53

can be derived and hence maximized. Other cases in which the model is not integrable are the presence of discriminatory ad valorem tariffs or different interest rates over regions (Arndt et al., 2001; Langyintuo et al., 2005). In those cases, a SPE can be formulated as a Linear Complementarity Problem (LCP). The general form of a LCP is shown in (15) (Cottle et al., 1992):

For a QP it can be shown, that (15) is equivalent to its corresponding KT-Conditions.⁵⁴ QPs form a subset of LCPs for which an efficient solving algorithm called the principal pivoting method had been developed by Cottle and Dantzig (1968) building on the work of Lemke and Howson (1964).

With growing computational capacities and the development of appropriate solution algorithms it became possible to solve mixed complementarity problems (MCP). These are a generalization of LCPs in that they are able to handle non-linear functions instead of merely linear ones on the one hand and a mixture of equality and inequality constraints instead of merely inequality constraints on the other hand (Billups et al., 1997). The general form of a MCP is shown below in (16) (Rutherford, 1995):

MCP

52 Quadratic programming problems are a subset of nonlinear programming problems. The former are restricted to a quadratic objective function and linear constrains, whereas the latter allow for nonlinearities of higher order than quadratic in objective functions and constraints (Bishop et al., 2001).

53 Modellers using supply and demand functions which are merely locally symmetric (as e.g. isoelastic func-tions) encounter a similar problem when calculating welfare effects of policy changes: The results are not un-ambiguous as they depend on the path of integration (See e.g. Grethe (2004) Ch. 5.8).

54 The vector z in this case would correspond to choice and slack variables in (14) and matrix M and vector q correspond to the coefficients of the (linear) first order derivatives of the Lagrangian.

0

Besides linear and nonlinear programs and complementary problems, MCPs are a generalization of many problems in mathematics and optimisation, for instance systems of linear and nonlinear equations. For the practical purpose of SPE modelling, the MCP formu-lation has the advantage that it allows demand and supply functions to be nonlinear.