Economics proceeds by developing models of social phenomena, which are a simplified representation of reality.

For example, let use consider the market for apartments in a medium-size mid-western college town. We may suppose that the apartments closer to the university are more desired since people have to walk less to go to the university, we will only consider this distance in out model, up to a maximum distance.

Our model does not cover the apartments outside of this distance or ring, and we say that their price is an exogenous variable, while the price of the closer apartments is an endogenous variable.

Whenever we try to explain the behavior of human being we need to have a framework on which our analysis can be based. We will follow two simple principles:

• the optimization principle: people try to choose the best patterns of consumption that they can afford
• the equilibrium principle: prices adjust until the amount that people demand of something is equal to the amount that is supplied.

The first principle is almost a tautology, indeed it seems unreasonable that people would not try to get things that they don't want. The second is a bit more problematic, it may happen that supply and demand are not compatible, but for most cases they are. For example, the price of apartments is fairly stable over a short time-frame. We will ignore long time frames in this initial analysis.

We call the reservation price a person's maximum willingness to pay for something. It is the highest price that a given person will accept and still purchase the good.

In our example, the number of apartments that will be rented at a given price \(p^*\) will just be the number of people who have a reservation price greater than or equal to \(p^*\). For if the market price is \(p^*\), then everyone who is willing to pay at lest \(p^*\) for an apartment will want an apartment in the inner ring, and everyone who is not willing to pay \(p^*\) will choose to live in the outer ring.

The demand curve is a plot showing the price and the number of people who are willing to pay that price or more. Since data is discrete, that would be the best curve that fits the data.

We refer to a competitive market the situation where there are many independent landlords who are each out to rent their apartments for the highest price the market will bear.

Let us now consider the case where there are many landlords who all operate independently. If all landlords are trying to do the best they can and if the renters are fully informed about the prices the landlords charge, then the equilibrium price of all apartments in the inner ring must be the same. The argument is not difficult. Suppose instead that there is some high price \(p_h\) and some low price \(p_l\) being charged for apartments. The people who are renting their apartments for a high price could go to a landlord renting for a low price and offer to pay a rent somewhere between \(p_h\) and \(p_l\). A transaction at such a price would make both the renter and the landlord better off. To the extent that all parties are seeking to further their own interests and are aware of the alternative prices being charged, a situation with different prices being charged for the same good cannot persist in equilibrium.

In out case, considering a short timeframe that does not account for new houses being built, the supply curve would be a vertical line. Whatever price is being charged, the same number of apartments will be rented.

We refer to the equilibrium price \(p^*\) as the price where the quantity of apartments demanded equals the quantity supplied. At this price, each consumer who is willing to pay at least \(p^*\) is able to find an apartment to rent, and each landlord will be able to rent apartments at the going market price. Neither the consumers nor the landlords have any readon to change their behaviour, hence the term equilibrium.

Comparative statics is the study of the behaviour of the equilibrium price when various aspects of the market change. Here we study only the change from one equilibra to another, hence the term static, we do not consider how that movement takes place for now.

As an exercise, suppose that a developer decides to turn several of the apartments into condominiums, what will happen to the price of the remaining apartments? If we suppose that the people who where renting the apartments will buy them, then the demand for apartments will have been reduces as much as the supply, so the equilibrium price remains unchanged.

Let's consider another example, where the city council decides that there should be a tax on apartments of 50 dollars a year, what will this do to the price of apartments? We observe that the supply curve doesn't change as there are just as many apartments after the tax as before the tax, and the demand curve doesn't change either, therefore the price does not change as a result of the tax. In essence, the the demand does not change, the landlords cannot raise the price to compensate for the tax, or not enough people would afford to occuply the apartments and the landlords will not be able to get the rented price. Note that we are assuming that the supply of apartments remains fixed.

So far we have assumed a competitive market. This is not the only type of markets, here are other ways:

• the discriminating monopolist: here we consider a situation where there is only one dominant landlord who own all of the apartments (or a number of individual landlords cooperating) also known as a monopoly. Suppose the landlord could decide to auction them off one by one to the highest bidders, in this situation exactly the same people will get the apartments as in the case of the market solution, and the last person to rent an apartment pays the price \(p^*\). The discriminating monopolist attempts to maximize his own profits leading to the same allocation of apartments as a competitive market but with different amounts paid by each person.
• the ordinary monopolist: what if the monopolist wwere forced to rent all apartments at the same price?If he chooses a low price he will rent more apartments but he may end up making less money than if he sets a higher price. What he will do instead is not to rent all apartments, but to rent only a fraction of them so that it maximizes the provit from the highest bidders.
• rent control: let us consider the case where the city imposes a maximum rent that can be charged for apartments \(p_{max}\). Suppose \(p_{max} < p^*\), then we would have a situation of eccess demand where there are more people who are willing to rent apartments at \(p_{max}\). Who gets the apartment is outside the scope of the model.

To compare different economics institutions, we introduce the Pareto efficiency which is defined in the following way: if we can find a way to make some people better off without making anybody else worse off, we have a Pareto improvement. If an allocation allows a Pareto improvement, it is called Pareto inefficient; if an allocation is such that no Pareto improvements are possible, it is called Pareto efficient. The idea is that, if there is a way to make someone better off without hurting anyone else, why not do it?

We define a consumption budle \((x_1, x_2)\) as a pair representing how much the consumer is choosing to consume of good \(1\), \(x_1\), and how much the consumer is choosing to consume of good \(2\), \(x_2\). We suppose that we can observe the prices of the two goods \((p_1, p_2)\) and the amount of money the consumer has to spend \(m\)

Then, a budget constraint is:

\[ p_1 x_1 + p_2 x_2 \le m \]

where \(p_1 x_1\) is the amount of money the consumer is spending on good \(1\), and \(p_2x_2\) for good \(2\).

The consumption bundle may be generalized as such. If we are interested in studying a consumer's demand for milk, we might let \(x_1\) measure his or her consumption of milk and let \(x_2\) stand for everything else the consumer might want to consume in dollars, so \(p_2 = 1\). The budget constraint would then look like:

\[p_1x_1 + x_2 \le m\]

which is a special case of the original budget constraint, so everything that applies in the general case also applies here.

The budget line is the set of bundles that cost exactly \(m\):

\[p_1x_1 + p_2x_2 = m\]

These are the bundles of goods that just exhaust the consumer's income, which is a line. We say that the derivative of the line is the opportunity cost of consuming good \(1\). In order to consume more of good \(1\) you have to give up some consumption of good \(2\) which is measured by the slope of the budget line.

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