Chapter 3 Representation of market prices | Mathematical methods for Economic analysis

3.1 The market offers

When having access to a market, everyone can ask the buyers and vendors about their prices, and write them down on a piece of paper. This could look like this:

3.1.1 From the buyers

Table 3.1: Price of Wheat expressed in Cabbages
Buyer	Quantity of Wheat bought	Price in Cabbages
Buyer 1	1 Wheat	1.4 Cabbage
Buyer 2	1 Wheat	1.1 Cabbage
Buyer 3	1 Wheat	0.9 Cabbage

Table 3.2: Price of Salad expressed in Money
Buyer	Quantity of Salads bought	Price in Money
Buyer 1	1 Salad	1.4 Money
Buyer 2	1 Salad	1.3 Money
Buyer 3	1 Salad	1 Money

Table 3.3: Price of Cabbage expressed in Rice
Buyer	Quantity of Cabbages bought	Price in Rice
Buyer 1	1 Cabbage	1.3 Rice
Buyer 2	1 Cabbage	1 Rice
Buyer 3	1 Cabbage	0.7 Rice

3.1.2 From the vendors

Table 3.4: Price of Money expressed in Tomatoes
Vendor	Quantity of Money sold	Price in Tomatoes
Vendor 1	1 Money	1 Tomato
Vendor 2	0.75 Money	1 Tomato
Vendor 3	0.5 Money	1 Tomato

Table 3.5: Price of Salad expressed in Cabbages
Vendor	Quantity of Salads sold	Price in Cabbages
Vendor 1	2 Salad	1 Cabbage
Vendor 2	1.5 Salad	1 Cabbage
Vendor 3	1 Salad	1 Cabbage

Table 3.6: Price of Rice expressed in Tomatoes
Vendor	Quantity of Rice sold	Price in Tomatoes
Vendor 1	1.5 Rice	1 Tomato
Vendor 2	1.25 Rice	1 Tomato
Vendor 3	0.8 Rice	1 Tomato

3.2 Graph representation

Such tables are tedious to read and analyse, and so we can represent all this in a better way.

First, we need to enumerate the trade possibilities: each time there is someone willing to buy a product \(P_1\) for a product \(P_2\), we represent it with an arrow going from \(P_1\) to \(P_2\): \(P_1 \rightarrow P_2\).

From table 3.1: \(Wheat \,\to\, Cabbage\)
From table 3.2: \(Salad \,\to\, Money\)
From table 3.3: \(Cabbage \,\to\, Rice\)

From table 3.4: \(Tomato \,\to\, Money\)
From table 3.5: \(Cabbage \,\to\, Salad\)
From table 3.6: \(Tomato \,\to\, Rice\)

Second, as long as we are not bying too much on the market, we only need to remember the most advantageous prices. So for each arrow \(P_1 \,\to\, P_2\) we drew, we assume that we have one unit of \(P_1\) and we are looking for the maximum quantity of \(P_2\) we can get: this will be the coefficient assigned to the arrow.

From table 3.1: \(Wheat \xrightarrow{1.4} Cabbage\)
From table 3.2: \(Salad \xrightarrow{1.4} Money\)
From table 3.3: \(Cabbage \xrightarrow{1.3} Rice\)

From table 3.4: \(Tomato \xrightarrow{1} Money\)
From table 3.5: \(Cabbage \xrightarrow{2} Salad\)
From table 3.6: \(Tomato \xrightarrow{1.5} Rice\)

Finally, we have to join these arrows together in a general weighted and directed graph, summarizing the offers available on the market:

Figure 3.1: Graph representation of the market

3.3 Arbitrage opportunities

Reading the graph 3.1, we can see that if we have \(1\) unit of wheat, we can exchange it for \(1.4\) unit of cabbages, which we can in turn exchange for \(1.4 \times 2 = 2.8\) units of salads. With these salads we can get \(2.8 \times 1.4 = 3.92\) units of money.

We can then rely on graph theory to find arbitrage opportunities: arbitrage opportunities are the cycles that we find in the graph, such that \(1\) unit of one product can be exchanged (in one or several steps) for more units of the same product.

3.4 Exchange rates and real prices

The graph representation of the market offers makes it easier to study some economic problems. For example, some people could ask themselves if fixed exchange rates between two countries would structurally affect the real prices, and if so in what way.

Let us represent the markets of these two countries:

Figure 3.2: Graph representation of the market (floating rate)

\(M_1\) and \(M_2\) represent the monies of the two countries. One unit of \(M_1\) is worth \(\mu_2\) units of \(M_2\), and one unit of \(M_2\) is worth \(\mu_1\) units of \(M_1\). For the sake of simplicity, let us say that \(\mu_1 = \mu_2^{-1}\).

Is it possible that one unit of \(M_1\) is exchanged for one unit of \(M_2\) (fixed exanche rate) while the real prices stay the same? Let us look at this graph:

Figure 3.3: Graph representation of the market (fixed rate)

The real prices are the same than on the previous graph! From this we can conclude that there is no direct influence of the exchange rate on the prices, except through price rigidity and lack of information.