^{1}

^{2}

^{3}

The current non-linear programming method does not derive regional economic surpluses and may derive an imprecise maximized value of the total economic surplus. The main reason is that the integrals for supply functions will automatically take regional non-economic producer surpluses into account if any intercepts of supply functions is negative. Consequently, the derived values are always lower than the real regional and total economic surpluses. The unknown regional economic surpluses and the imprecise total economic surplus will limit the suitable application of the model for broader contexts including game theory analysis, international trade policy analysis, and even GDP calculation. This paper recommends two formulae applied for two types of functions, namely original and inverse supply and demand functions, to calculate the regional and total economic surpluses of commodities. The two methods can be converted to each other conveniently, for example by using an inverse matrix of coefficients of original supply and demand functions to solve coefficients of inverse supply and demand functions. A numerical example is used to illustrate the spatial equilibrium model for 2 products and 3 regions with original linear supply and demand functions.

The concept of the spatial equilibrium model was raised in the late 19^{th} century [

Until 2011, there were three common methods applied to derive optimal solutions for spatial equilibrium models. Non-linear programming is the most common method used to solve spatial equilibrium models with a non-linear objective function for the total economic surplus including linear and non-linear supply and demand functions as presented in journal articles and mathematical programming books including Takayama and Judge (1964) [

Linear programming is used to solve spatial equilibrium models with linear supply functions, linear demand functions, and linear objective function of the total transportation cost as presented in Phan, Harrison, and Lamb (2011) [

Spatial equilibrium models have been applied for some commodities in some countries since the 1970s, for example, Hall, Heady, Stoecker, and Sposito (1975) [

In terms of economic-reality aspects, the three methods can derive correctly optimal solutions of regional prices, regional supply quantities, regional demand quantities, regional transport quantities, and the total trans- portation cost. However, the optimal solution derived for the total economic surplus can be imprecise. In addi- tion, the current methods do not calculate regional economic surpluses in aggregate or for each specific commo- dity. Therefore, the following sections examine why the two limitations exist and proposes how to solve preci- sely the economic surpluses of commodities, regional economic surpluses, and the total economic surplus gene- rated by the spatial equilibrium models with linear supply and demand functions.

The spatial equilibrium model is the broadest market equilibrium model [

When the commodity can be traded between two regions, the commodity is transported from a lower price region to a higher price region―from Region 1 to Region 2 in

brium model. And, if the unit transportation cost is higher than the price difference between regions, the two regions are separated and this simple model can be split into two partial equilibrium models [

For a general case―a spatial equilibrium model for multiple products and multiple regions―the theoretical explanation for the commodity movement is that “the quantity traded is the amount such that the total economic surplus of all products of all regions is maximized” or “the quantity traded is the amount such that the total transportation cost of all products between all regions is minimized”. Mathematically, the maximization process leads to the optimal regional prices, regional supply quantities and regional demand quantities, and total economic surplus. The minimization process identifies the optimal quantities traded between regions and the total transportation cost.

In

Mathematically, the behavior of producers is expressed as the supply quantity being a function of supply price,

In almost economics textbooks, the total economic surplus is calculated as the sum of the areas of black and grey triangles presented in

modities minus the total transportation cost or

matical expression for the simple two-region model is presented below.

Maximize the total economic surplus:

Subject to:

Here:

TE: The total economic surplus

In

the total economic surplus of the two regions = the black areas in Regions 1 and 2

+ grey areas in Regions 1 and 2

− the light grey area in Region 1

+ the light grey area in Region 2

− the light grey square in Region 2 (the total transportation cost area).

The outcome of the expression presented in

difficulties. Unlike at aggregate level, at regional level the similar objective function expressed as (

the regional transportation cost) does not represent the economic surplus of a region. For example, in the export

region (Region 1), the calculation of (

gle area (consumer surplus) plus the grey area (only part of producer surplus) minus the light grey area (the export cost). This implies that if researchers would like to know the regional economic surplus of Region 1, they should plus the total export value but not minus the regional transportation cost, or they should apply the formu-

la:

In the import region (Region 2), the calculation of (

not present the economic surplus of a region. This calculation represents the black triangle area (consumer surplus) plus the grey triangle area (producer surplus) plus the light grey square area. The light grey square means the import value includes the regional transportation cost. This implies that if researchers would like to know the regional economic surplus of Region 2, they should minus the total import value but not minus the regional

transportation cost or apply the formula:

According to the above formulae, the regional economic surplus of the export or import region does not take into account the regional transportation costs because the regional transportation cost is already included in the regional import values generated automatically by the application of integrals of inverse supply and demand functions. In addition, the current calculation of the total economic surplus by applying the integrals will not be precise if intercepts of any regional inverse supply functions are negative, as presented in

When the intercepts^{1} of regional supply functions are negative, integrals of regional inverse supply functions will automatically include regional negative non-economic values^{2} in producer costs, and will reduce the producer surpluses, regional economic surpluses and total economic surplus. The reduced surplus is the area of the

black triangle ^{3}, so in order to solve the regional and

total economic surpluses correctly, researchers should take into account the case having negative intercepts of inverse supply functions. The recommended formula to re-calculate regional and total economic surpluses for the simple model having negative intercepts of the supply functions is presented below.

Here:

_{ }

For greater clarity, the recommended formula to re-calculate the regional and total economic surpluses after solving the simple spatial equilibrium model having negative intercepts of the supply functions by the current non-linear programming method is illustrated by six steps in

Although the coefficients between original and inverse functions can be converted rather conveniently^{4}, fewer studies have used spatial equilibrium models with inverse supply and demand functions than that with original supply and demand functions,

Phan and Harrison (2011) [

quantities, regional transport quantities, regional prices, and total transportation cost. However, the solutions differ in terms of how values of objective functions are interpreted. If the objective function of original functions

consumer cost), not the total economic surplus as presented in Sections 1 and 2^{5}.

Similar to the application of integrals for inverse linear supply functions, the application for original linear supply functions also generates an imprecise regional producer surplus if intercepts of these functions are negative. In addition, the objective function of the total economic cost cannot be used to calculate directly the total economic surplus. Therefore, the five steps listed below set out how to calculate the precise regional and total economic surpluses for studies applying spatial equilibrium models with original linear supply and demand functions.

1. The current method, for example the non-linear programming method with the objective function of

Minimize the total economic cost:

subject to:

where:

TEC: Total economic cost

2. The optimal solutions found in Step 1 are used to calculate the intercepts of the regional original linear demand and supply functions. The formulae to calculate the intercepts are presented below.

For supply functions:

For demand functions:

where:

3. The calculated intercepts of regional original demand functions in Step 2 and the given demand own-price coefficients,

where:

4. If the regional supply intercepts are positive, the regional and total economic surpluses can be calculated by the recommended formulae below:

where:

TE: The total economic surplus

5. If the regional supply intercepts are negative, the above regional economic surpluses should be adjusted due to the negative non-economic values of producer surpluses generated automatically by the application of integrals of original linear supply and demand functions. The sum of found regional economic surpluses is the precise total economic surplus. The following are recommended formulae to re-calculate regional and total economic surpluses.

where:

For greater clarity,

A numerical example slightly modified from that of Takayama and Judge (1964) [^{6}.

The unit transportation costs (t_{i},_{rr}_{’}, the cost to transport one unit of product i from region r to region

Suppose that an economist would like to find answers for the following 9 questions:

1. What are the optimal regional prices (supply price,

2. What are optimal regional supply quantities

3. What are optimal regional demand quantities

4. What are optimal regional transport quantities

5. What is the minimized total transportation cost (TC)?

6. What are the optimal regional transportation costs

7. What are the regional economic surpluses and the maximized total economic surplus without taking import and export values and signs of intercepts of supply functions into account (imprecise TE)?

Function | Region | ||
---|---|---|---|

Region 1 | Region 2 | Region 3 | |

Product 1 | |||

Product 2 | |||

Note: D is demand quantity, S is supply quantity, p^{d} is demand price, p^{s} is supply price, r is the subscript for region (r = 1, 2, 3) and i is the subscript for product (i = 1, 2).

Source region | Destination region | ||
---|---|---|---|

Region 1 | Region 2 | Region 3 | |

Product 1 | |||

Region 1 | 0 | 2 | 2 |

Region 2 | 2 | 0 | 1 |

Region 3 | 2 | 1 | 0 |

Product 2 | |||

Region 1 | 0 | 3 | 3 |

Region 2 | 3 | 0 | 2 |

Region 3 | 3 | 2 | 0 |

8. What is the maximized total economic surplus when taking signs of intercepts of supply functions into account (precise TE)?

9. What are optimal regional economic surpluses (precise

A non-linear programming model written in a GAMS file with non-linear CONOPT Solver facilities is used to solve for regional prices, regional supply quantities, regional demand quantities, quantities traded between regions, and regional import and export values, as presented in ^{7}.

The optimal solutions in

Region | Regional price | Regional demand quantity | Regional supply quantity | |||
---|---|---|---|---|---|---|

Commodity 1 | Commodity 2 | Commodity 1 | Commodity 2 | Commodity 1 | Commodity 2 | |

Region 1 | 11.00 | 11.64 | 101.00 | 195.27 | 65.50 | 120.36 |

Region 2 | 9.00 | 8.64 | 64.00 | 35.91 | 134.50 | 160.23 |

Region 3 | 10.00 | 10.64 | 90.00 | 154.27 | 55.00 | 104.86 |

From | To | Quantity transported | Import value | Export value | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Region r | Region r' | Com. 1 | Com. 2 | Com. 1 | Com. 2 | Total | Com. 1 | Com. 2 | Total | |

Region 2 | Region 1 | 35.5 | 74.91 | 390.5 | 871.7 | 1262.2 | 319.5 | 646.95 | 966.5 | |

Region 2 | Region 3 | 35 | 49.41 | 350.0 | 525.5 | 875.5 | 315.0 | 426.7 | 741.7 | |

Total | 70.5 | 124.32 | 740.5 | 1397.2 | 2137.7 | 634.5 | 1073.7 | 1708.2 | ||

Note: Com. 1 means commodity 1; Com. 2 means commodity 2.

If an economist does not take into account signs of the intercepts of original linear supply functions, and values of regional imports and exports, the imprecise total economic surplus generated by the GAMS programming is 6224.6. The number of 6224.6 is commonly solved by the non-linear programming method. The imprecise regional economic surpluses for the two cases are presented in

^{8}.

If an economist takes into account signs of the intercepts of original linear supply functions, the intercepts of the supply functions at equilibrium points found in Questions 1 to 4 will be obtained. GAMS programming produces regional intercepts of the original linear supply functions and intercepts as in

Region | Commodity 1 | Commodity 2 | Total | |
---|---|---|---|---|

Region 1 | 71.0 | 224.7 | 295.7 | |

Region 2 | 0.0 | 0.0 | 0.0 | |

Region 3 | 35.0 | 98.8 | 133.8 | |

Total | 106.0 | 323.5 | 429.5 |

Region | Signs of intercepts of supply functions are not taken into account | Signs of intercepts of supply functions and regional import and export values are not taken into account | ||||
---|---|---|---|---|---|---|

Commodity 1 | Commodity 2 | Total | Commodity 1 | Commodity 2 | Total | |

Region 1 | 622.1 | 2295.3 | 2917.4 | 1012.6 | 3167.0 | 4179.6 |

Region 2 | 811.7 | 482.1 | 1293.8 | 177.2 | −591.5 | −414.3 |

Region 3 | 553.1 | 1460.3 | 2013.3 | 903.1 | 1985.8 | 2888.8 |

Total | 1986.9 | 4237.7 | 6224.6 | 2092.9 | 4561.2 | 6654.1 |

Region | Commodity 1 | Commodity 2 |
---|---|---|

Region 1 | −44.5 | −54.2 |

Region 2 | −45.5 | −55.7 |

Region 3 | −45.0 | −54.7 |

Region | Commodity 1 | Commodity 2 | Total |
---|---|---|---|

Region 1 | 99.01 | 97.9 | 196.91 |

Region 2 | 51.76 | 62 | 113.76 |

Region 3 | 101.25 | 99.7 | 200.95 |

Total | 252.02 | 259.6 | 511.62 |

If the economist takes into account the losses presented in

The values in

The optimal solution of the total economic surplus obtained by the current non-linear programming method can be imprecise. If the intercepts of supply functions are positive, the optimal solution is correct. If the intercepts of

Consumer and producer surpluses by region | Commodity 1 | Commodity 2 | Total |
---|---|---|---|

Region 1 | 721.1 | 2393.2 | 3114.3 |

CS_{1} | 510.1 | 1905.9 | 2415.9 |

PS_{1} | 211.1 | 487.3 | 698.4 |

Region 2 | 863.5 | 544.1 | 1407.6 |

CS_{2} | 409.6 | 32.2 | 441.8 |

PS_{2} | 453.9 | 511.9 | 965.8 |

Region 3 | 654.3 | 1559.9 | 2214.2 |

CS_{3} | 506.3 | 1189.4 | 1695.7 |

PS_{3} | 148.1 | 370.5 | 518.5 |

Total | 2238.9 | 4497.2 | 6736.1 |

CS | 1425.9 | 3127.5 | 4553.4 |

PS | 813.0 | 1369.7 | 2182.7 |

supply functions are negative, the estimated optimal solutions will be smaller than the correct ones because the application of integrals of supply functions automatically takes negative non-economic values into account. The three current methods do not calculate regional economic surpluses. If applying the similar objective functions of the current non-linear programming methods at the aggregate level for regional levels, regional economic surpluses cannot be correct.

For the spatial equilibrium model with inverse linear supply and demand functions, to calculate the regional economic surpluses, the financial values of regional imports and exports and signs of regional intercepts of the supply functions should be taken into account. The sum of regional economic surpluses is then the precise total economic surplus. For the spatial equilibrium model with original linear supply and demand functions, to calculate the regional economic surpluses, the values of the whole regional areas under the demand curves, signs of regional intercepts of the supply functions should be taken into account. The sum of regional economic surpluses will then be the precise total economic surplus.

These two proposed methods to calculate the precise regional and total economic surpluses can substitute for each other flexibly because the coefficients of inverse supply and demand functions can be converted to those of original supply and demand functions and vice versa simply by using the matrix inverse command in Excel. These two methods to calculate the regional and total economic surpluses can be expanded to other types of functions, for example logarithm functions, and to add time variable, for example a year variable to form a space-time equilibrium model. The more precise calculation of regional economic surpluses and total economic surplus will facilitate the application of spatial equilibrium models to more complicated contexts, for example how changes of a country’s policies or exogenous factors will affect other countries’ economic surpluses, and the total economic surplus. This analysis can support for some basic ideas of GAME theory, for example how a person’s choice will impact on benefits to others and total benefits, and international trade theory, for example the more producer surplus of a commodity of a country has, the higher comparative advantage the commodity is. Equally, the more precise calculation can assist to estimate value added (VA), Gross Domestic Product (GDP), and Gross National Product (GNP).

Phan SyHieu,SteveHarrison,11,DominicSmith, (2015) Recommended Methods to Re-Calculate Regional and Total Economic Surpluses after Solving Spatial Equilibrium Models by the Non-Linear Programming Method. Modern Economy,06,520-534. doi: 10.4236/me.2015.65051