# Chapter 3 Representation of market prices

## 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

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 |

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 |

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

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 |

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 |

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:

## 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:

\(M_1\) 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:

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.