Let’s consider first a simple model for apartments in a college. Here we are interested to predict the prices of the rooms, and how we can allocate them to students. For simplicity, we will assume that they are all equal except for the location, which could be inner or outer.

Types of variables

Economist will say that parameters for a model, i.e. variables that are fixed for some type of analysis exogenous variables, while the variables of interest of a model endogenous variables. In this setting, it could be the location for a certain room.

Two Principles

Economists often rely on some reasonable assumption before starting to model their economical phenomena, here we present two.

Optimization principle

People try to choose the best patterns of consumption that they can afford.

What is meant here is that people often chose something that they want instead of something that they do not want. This should be an intuitive. Valiant says that when this does not hold, the reason is often not in economical domain.

Equilibrium principle

Prices adjust until the amount that people demand of something is equal to the amount that is supplied.

This might take a while to happen, not always true. It assumes there is a price that is stable for buyers and sellers.

Two important Curves

Demand Curve

Reservation price is the highest amount a person is willing to pay for a certain good.

What is the Demand Curve?

The demand curve is the curve that shows the quantity of a good that people are willing to buy at each price. The demand curve is a function of the price and some relation of the reservation price.

A mathematical description

Say $p$ is the price of a good, and $q$ is the quantity of the good that is bought. The demand curve is a bijective function of $q$ to $p$, that depends on some concentration of people at certain reservation prices in the continuous case. Usually, we observe that as the price goes down, the number of acquired items goes up.

Let’s say $\theta$ is a continuous distribution of people at a certain reservation price, and let’s consider the derivative with respect to the number of goods bought. We can write the demand curve as:

$$ q(p) = \int_{p}^{\infty} \theta(x) dx $$

In practice, every people that has a reservation price higher than $p$ will buy, so with this integral we are just counting how many people we have.

Supply Curve

We can build a similar curve to the demand in exactly the same way. We say there is a selling price for each seller which is the highest price they are willing to sell a good.

Then, you can build a model of the quantity of goods you can sell based on the price of the offering.

A simple mathematical model

$$ q(p) = \int_{-\infty}^{p} \theta(x) dx $$

Which means, if I can sell the price $p$, I will sell it to everyone that has a selling price lower than $p$. This finds the number of people at a certain fixed $p$.

But, often in real scenarios, the variable is not the price, but the number of goods that are available at a certain time. So often the supply curve is just a vertical line, representing the quantity of goods that are available at a certain time.

The equilibrium

The point of contact between the demand curve the the supply straight line is called the equilibrium of the market. There is a easy line of reasoning that motivates why this is an equilibrium. Let’s say $p^{*}$ is the price at the point of contact. If instead of $p^{*}$ we are selling at a price $p > p^{*}$, then the quantity of goods that are bought will be less than the quantity of goods that are willing to be sold, so the price has an incentive to go down if we want to sell everything (e.g. if you have a room, that room is just a liability if it is not rented). Else, if we are selling at a price $p < p^{*}$, then the quantity of goods that are bought will be more than the quantity of goods that are willing to be sold, so there are not enough goods for everybody, and in order to make the spending slower, the price has an incentive to go up (so we have less buyers).

Comparative Statics

Comparative statics attempts to analyze different equilibria in the market and assess which one is better, without worrying about how the market changes from one to the other (this is why it is called statics). Classical arguments are the following:

  1. If I increase supply while having the same demand, the price goes down.
  2. If I increase demand while having the same supply, the price goes up.

And interesting caveat: some policies could move supply and demand at the same time, this is for example when fungible goods are introduced in the market.

Pareto efficiency

We say that an allocation of resources is pareto efficient if there is no other allocation of resources that makes someone better off without making someone worse off.

Monopolist Behaviour

A monopolist is a person that has a monopoly on a certain good. A monopolist can set the price of a good, and the quantity of the good that is sold. A monopolist can also discriminate between different types of people, and sell the good at different prices. In this section, we will examine two types of monopolist behaviour: discriminating and ordinary.

Discriminating Monopolist

Discriminating means the monopolist can set different prices of the same good to different people. Surprisingly, if the monopolist has enough knowledge of the single buyer, then the monopolist has exactly the same allocation as a competitive market. Here, the buyers are the ones that are losing money, as they are paying more than the good is worth to them.

Ordinary Monopolist

Ordinary monopolist means the monopolist cannot discriminate between different people, and has to set a single price for the good. It can be shown that this is highly inefficient, as it corresponds to maximize a rectangular area under the demand curve, without selling all the goods that are available.

This often means the monopolist if willing to charge a price $p > p^{*}$, with the cost of leaving some goods unsold.