Decoding Endogenous vs Exogenous - Parameter vs Decision Variable - and Independent vs Dependent

Question

this is a topic that i feel is very implicit in a lot of economics, but is some times brushed over in interest of getting strait to the model or the maths. But often i realise i don't actually know exactly what i'm doing/modelling.

So I would love people's thoughts, truly understanding these terms and differentiating between them in context:

Example:

Let's take the following constrained optimisation problem (I'm assuming i can think of this as a "model" e.g. of consumption choice, under resource constraint?

• $U(x_1, x_2) = (x_1x_2)^{1/2} s.t. G(x_1, x_2, p_1, p_2, m) = p_1x_1 + p_2x_2 - m = 0$

Where normally i understand we may not explicitly write the constraint as a function of m, but i wan't to in this case to help develop the fullest understanding:

Paramater vs Decision Variable:

• $p$ & $m$ are our parameters of the model, and specifically they are parameters of the function $G$ and, $x_i$ are our decision variables in the model and of both functions?

Endogenous Vs Exogenous:

I'm using the following as a definition of endogenous variable, is this correct and appropriate?

An endogenous variable is a variable that depends on other variables in a statistical and/or economic model. If the value changes for an endogenous variable, it is because there are changes to its relationships with other variables in the same model. Therefore, it is similar to a dependent variable because both are influenced by one or more independent variables.

1. In this model i would assume that $p$ and $m$ are Exogenous? As they do not depend on other variables in the model, they are set externally?

2. I assume $x_i$ is Endogenous, because the amount we set for $x_1$ depends on the value of $p$ and $m$ e.g. this is made explicitly in the compensated demand function:

$x_i^* = \frac{m}{2p_i}$

Is this correct? Because we are choosing our optimal level of $x_i$ based on our model assumptions (e.g. preference assumptions), and the specific values of the parameters, $p$ and $m$.

Independent vs Dependent:

In layman's terms, I'm seeing Independent variable's as the one's we change e.g. the x axis, and dependent variable is the variable we measure e.g. the y axis.

1. So in this case would our independent variable be the $x_i's$ and the dependent variable actually be the functions $U$ and $G$.
2. Or would our dependent variable the optimal value of $x$ that we determine, our $x_i^* = \frac{m}{2p_i}$

Further Complications: I also want to discuss when the state of these variables may change, i.e. when looking at the maximum value function, exogenous parameters can become endogenous i believe. But i will save this for a separate question, thanks!

Questions:

1. Am i correct?
2. Is there a simple way to think of this going forward
3. What are the important implications of getting this right and wrong (for example understanding dependencies when taking derivatives etc.)

Answer 1

Am i correct?

Most of the things mentioned are correct. Some things that are incorrect include:

• Making a difference between exogenous and independent and endogenous and dependent variable. Those are synonyms. Exogenous variable is also independent variable and endogenous variable is also dependent variable.

In fact, in your model $x_i$ would be dependent variable. If you would want to estimate it using statistical model you would have to use something like 2SLS which uses system of equations.

• Also functions themselves are not variables, so they should not be labeled dependent or independent variables. However, the variable that consist of utility let's call it $U$ can be function of other variables ($U(x_i$), but you should not say function is dependent variable, rather utility is dependent variable.

Is there a simple way to think of this going forward

Simple way of thinking about this is as follows:

• If a variable is considered fixed within a model it would be parameter. This does not mean it has some precise value, as the value could be lets say $p$ for price. But if $p$ does not change it would be considered parameter.

• If the variable can manipulated by the agent in the model then it is a choice variable. For example, if the $m$ would depend on labor supply of the agent it would be a choice variable.

• Exogenous/independent variable is variable that is not caused by any other variables in the model. For example, in your model $p$ is not caused/affected by $x_i$ so $p$ is exogenous. However, $x_i$ is affected by other variables so it is endogenous/dependent.

What are the important implications of getting this right and wrong (for example understanding dependencies when taking derivatives etc.)

Clearly if you confuse what is a choice variable and what is parameter and take derivate wrt $p$ instead of $x$ you will get wrong result.

When it comes to exogenous and endogenous variables, in a theoretical model if you confuse them you will have wrong understanding of the mechanism, and when you try to statistically fit the model to the data you will have wrong estimates.

For example, if you would just run regression of $x$ on $U$ you would get wrong estimates of effect of consumption on utility if you treat $x$ as exogenous, instead of estimating it using something like 2SLS.

Answer 2

I will put things much more easy.

What you presented is a a very simple utility constrained maximization problem. What is endogenous is what is determined within the model. The solutions for this problem are $x_1^{\ast}$ and $x_2^{\ast}$, that are the consumption levels that maximize your utility function. Actually, more than endogenous, $x_1$ and $x_2$ are called choice variables, because you choose the level of $x_1$ and $x_2$ maximizing utility.

Prices and the level of wealth ($m$) are simply taken as given. You cannot control them to solve your maximization problem. That's why they can be referred to be exogenous.