|
Article Excerpt
1. BACKGROUND
Samuelson (1952) introduced a number of insights regarding prices and quantities in spatial commodity networks. Based on welfare maximization, Samuelson (1952) showed that under carefully specified conditions, a linear programming problem characterizing the spatial aspects of the multiregional economic system is embedded. Samuelson (1952) neither claims nor implies that "Competitive markets exhibit netback price relationships among regional prices." We disprove netback relationships by example herein. Samuelson (1952) does not imply that "Everyone benefits when transportation congestion is eradicated." We disprove that by example. Samuelson (1952) does not imply that "Supply region prices must be lower than demand region prices by at least the magnitude of intervening transportation cost from supply to demand." That too is disproved by example. Samuelson (1952) does not imply that "Transportation cost drives a wedge between supply and demand prices, and it doesn't matter how high it gets." That too is demonstrably incorrect. Samuelson (1952) never asserted that "Linear programming is an adequate model of market equilibrium." This paper disproves by example misconceptions about prices and flows in such systems by generalizing the Samuelson model to a Walrasian form and applying comparative statics.
2. WALRASIAN EQUILIBRIUM IN A SYSTEM WITH TWO SUPPLY REGIONS AND TWO DEMAND REGIONS INTERCONNECTED BY A TRANSPORTATION GRID
Consider the spatial structure in Figure 1 in which there are two supply regions at the bottom of the diagram (Supply Regions 1 and 2), each containing a linear indirect supply function:
[p.sub.1] = 1 + 1/2 [q.sub.1] [p.sub.2] = 1 + 1/2 [2q.sub.2] (1)
which is easily invertible to create direct supply functions. (1) There are two demand regions at the top of the diagram (Demand Regions A and B), each containing a linear indirect demand function: (2)
[p.sub.A] = 5 - [q.sub.A] [p.sub.B] = 5 - 1/2 [q.sub.B] (2)
which is easily invertible to create direct demand functions. (3) There is a transportation system interconnecting each supply region with each demand region. (4) We postulate that the throughput capacity of each transportation link is unbounded (no transportation "congestion") with no barriers to entry, constant returns to scale, no transportation losses, and fixed volumetric costs specific to each route or link in the grid in Figure 1 (depicted inside the triangles). (5) Figure 1 distinguishes four interconnected markets (one each at locations 1, 2, A, and B) each with its own distinct price. (6)
[FIGURE 1 OMITTED]
We develop the system of equations/inequalities that characterizes Walrasian equilibrium for Figure 1. The supply and demand relationships are obvious. Using transportation link 1-to-A as an example, seen in Figure 1 to cost $1 per unit with unlimited capacity available, two conditions must be satisfied:
1. A positive quantity of commodity flows from Region 1-to-A only if the price differential across the 1-to-A transportation link is equal to or larger than the transportation cost of $1, i.e., [q.sub.1A] > [??] [p.sub.A] - [p.sub.1] [greater than or equal to] $1
2. If the price differential across the 1-to-A link is strictly less than the cost of transportation of $1, zero commodity will be transported across that link because the cost of such transportation is higher than the market is willing to pay, i.e., [p.sub.A]-[p.sub.1] < $1 [??] [q.sub.1A] = 0.
We write the two foregoing equations for the 1-to-A link (and analogously for the other three links):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
Quantity balances for markets A, B, 1, and 2, which must hold in all four markets under Walrasian equilibrium, are the following:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
Twelve equations/inequalities characterize Walrasian equilibrium in Figure 1, equations (1)-(4). The unknowns are the four prices [p.sub.A], [p.sub.B], [p.sub.1], [p.sub.2]; the four quantities [q.sub.A], [q.sub.B], [q.sub.1], [q.sub.2]; and the four transportation link flows [q.sub.1A], [q.sub.1B], [q.sub.2A], [q.sub.2B].
2.1 Reference Case
The reference case equilibrium is displayed in Figure 2, which can be verified by substituting the numbers directly into equations (1)-(4). The figure depicts the prices and flows through the various transportation links. We array the results from Figure 2 in Table 1 to facilitate comparisons of the reference case to subsequent cases. Figure 2 disproves any notion that "netback relationships" hold across transportation links in spatial markets. A netback calculation begins with the observed price in a demand market (e.g., the price $4.136 in demand Region B). Thereafter, the netback calculation subtracts the transportation cost $2 entering from supply Region 1 to demand Region B from the price $4.136 in demand Region B and argues that the result should be the price in supply Region 1. According to netback logic, $4.136--$2 = $2.136 should be the price in supply Region 1. Netback reasoning is problematic for market prices, as we shall see from the four netback calculations in Figure 2:
* B back to 1: $4.136--$2 = $2.136, which is the market price in region 1.
* B back to 2: $4.136--$0.50 = $3.636, which is the market price in region 2.
* A back to 1: $3.136--$1 = $2.136, which is the market price in region 1.
* A back to 2: $3.136--$2 = $1.136, which is not the market price in Region 2.
The netback calculation $1.136 is well below the true market price $3.636 in region 2.
Alas, netback pricing does not work in this simple two supply-two demand region setting and thus cannot be argued to work in larger, more complex, or more general settings (7). Netbacks cannot be argued to work in any setting. Spatially distributed market prices do not obey netback relationships. This has been misunderstood since Samuelson (1952), who was very careful to state that the complementary slackness conditions in his embedded linear programming model governed prices, not netbacks.
[FIGURE 2 OMITTED]
Notice in Figure 2 there is zero commodity flow from 2 to A, a link across which the netback calculation does not work. The netback calculation does not work along this path from 2 to A because the path is not competitive; nobody wants to use it. Netback calculations hold along transportation links that experience positive flows at equilibrium, but not zero flows.
That...
|
