These notes cover chapters 4 and 5 from Williamson.

In these chapters, we build a simple economic model called a competitive equilibrium. Note that "competitive equilibrium" doesn't refer to a single specific model, but rather refers to a class of economic models, which all share some commonalities. If we write one of these models out, it will look something like this:

Given {some set of exogenous parameters}, our {set of endogenous parameters} will be in competitive equilibrium if the following conditions are satisfied:

- Each agent's choice variables solve that agent's optimization problem.
- Markets Clear
- Any extra conditions, like government budget balance.

Let's go into a bit more detail about each of these pieces. The key features of a competitive equilibrium model are as follows:

- The model will have some set of
, which are fixed in place from outside the model.*exo*genous parameters - There will also be
, which are determined within the model.*endo*genous parameters - In particular, we will have some parameters for
**prices**. - There are one or more
**price-taking agents**, each solving their own optimization problem. - For example, we typically have a "Representative Consumer", and a "Representative Firm". The consumer is trying to maximize their utility subject to some budgetary constraints, and the firm is trying to maximize profits by using some sort of technology that allows them to transform inputs into outputs.
- Each agent will have "choice variables", which are some subset of the equilibrium's endogenous parameters, and that agent will treat all of the other parameters in the equilibrium model as if they were exogenous.
- For example, the firm's profits are endogenous to the equilibrium, and determined by the firm's decisions. But the consumer will treat the firm's profits as if they're given.
- By price-taking, we mean that the agents will always treat the prices as exogenous. This is what puts the
*Competitive*in a Competitive Equilibrium. So although the prices will be endogenous to the equilibrium as a whole, each individual agent will behave as if their actions have no effect on the price-level. - There will be
**Market Clearing Conditions**, which ensure that the choices of individual agents are consistent with one another. - This is the
*Equilibrium*part of a Competitive Equilibrium. - And finally, sometimes there may be extra conditions, representing some other relationship between the parameters.
- For example, we could require that the government budget is balanced.

To build a model like this, we have to make assumptions about which agents are in the economy, what markets they are interacting with, what kinds of choices each agent can make, etc. The details of these assumptions will change the structure of our model. Then if the model behaves in a way which is consistent with the real world, it hints that our assumptions may be reasonable.

But more importantly, if the model makes bad predictions, and doesn't match up with what we see, this suggests that one or more of our assumptions are innacurate. And this can be just as informative, if not more so.

The point of representing the economy algebraically in models like these isn't that math-notation makes an argument true. But rather that math-notation forces us to be rigourous and specific about our assumptions. And this allows us to more meaningfully evaluate and learn from those assumptions.

In chapters 4 and 5, we build a very simple competitive equilibrium. And we make many assumptions to simplify the math. The point of these chapters isn't to have you memorize the specific assumptions we are using in this model and treat them as gospel truth about the world. Rather, the aim is to practice the process of taking a description of the economy and expressing it as a model like these ones.

In later chapters, we will change our assumptions about the structure of the economy, and build different competitive equilibrium models. For example, we might open the economy to trade, or add multiple time periods.

In the model for chapters 4 and 5, we make plenty of assumptions about the economy that definitely aren't true. For example:

- We assume that the only two goods are an aggregate consumption good, \(C\), and units of labor, \(N\).
- We assume that money is neutral, prices don't affect agents' decisions, and so we can put things in real terms, setting the price of \(C\) to \(1\), and using \(w\) to represent the real wage.
- We assume that there is only a single period of time. No agents can make any decisions which affect the future.
- Likewise, even though we assume capital, \(K\), is needed for production, we just make \(K\) exogenous. No time means nobody can make decisions about investment.

Both of these first two assumptions are fairly reasonable for looking at aggregate economic activity. We will add multiple time periods into the model in the second half of the course. And if we have time near the end of the semester, we may look at what happens when we change our assumptions about the role of prices.

The model for chapters 4 and 5, in it's full form, looks like this:

Given exogenous \(\left\{ h,z,K,g \right\}\), a competitive equilibrium is a set of allocations \(\left\{ C,l,N_s,N_d,T \right\}\) and a real wage \(\left\{ w \right\}\) which satisfy the following:

**Consumer Optimization**: Taking the real wage as given, the representative consumer chooses an allocation \(\left\{ C,l,N_s \right\}\) such that \(N_s = h-l\) and such that their allocation solves: \[\max_{C,l}\; U(C,l)\] subject to the constraints: \begin{gather} c\geq 0, \quad \quad h \geq l \geq 0 \tag{NonNeg}\\ c \leq w\cdot(h-l) + \pi - T \tag{Budget}\\ \end{gather}**Firm Optimization**: Taking the real wage as given, the representative firm chooses their labor demand, \(N_d\) to maximize profits: \[\max_{N_d}\; \pi \] where profits are equal to revenue minus costs: \[ \pi = Y - w N_d \] output is determined by some production function: \[Y = zF(K,N_d)\] and subject to the constraint that \(N_d \geq 0\).**Markets Clear:**\begin{gather} N_d = N_s \tag{labor}\\ C + g = Y = zF(K,N) \tag{goods}\\ \end{gather}- The government's budget is balanced: \(T=g\)

In the above model, we are using the following variables to represent different things in our economy:

- \(h\) is the consumer's endowment of time which can be split between leisure and labor.
- \(z\) is the total-factor-productivity.
- \(K\) is the stock of capital.
- \(g\) is the level of government spending, which needs to be funded.
- \(C\) is the amount of the consumption good the consumer has.
- \(l\) is the amount of time the consumer allocates to leisure.
- \(N_s\) is the labor supplied, the amount of time the consumer allocates to labor.
- \(N_d\) is the labor demanded.
- \(T\) is a lump-sum tax placed on the consumer.
- \(\pi\) is the firm's profits, which are paid to the consumer as dividends.
- \(Y\) is just shorthand for the firm's output.
- And \(w\) is the real wage, the number of units of consumption goods the consumer earns for each unit of labor supplied.