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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.)
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2 Answers 2

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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.

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  • $\begingroup$ Hi @1muflon1 apologies for my delay, this is a superb response! Thank you for going into so much detail, it's exactly what I was looking for. I noticed you answered my other question as well, thank you, I'll respond to that one tomorrow! $\endgroup$
    – CormJack
    Feb 7, 2023 at 3:43
  • $\begingroup$ I've read through a few times now and everything makes sense, one thing I'm slightly uncertain about is you seem to use p as both an exogenous/independent variable, and a parameter. Is this to state that exogenous variables are always parameters. Or is it possible to distinguish between them in some cases? Thank you. $\endgroup$
    – CormJack
    Feb 7, 2023 at 3:45
  • $\begingroup$ @CormJack I was probably not clear enough about this. All parameters are exogenous, but they are not variables because they are held constant. However, a constant is from point of the view of model exogenously given. For example, if an agent has an endowment of 100 dollars that endowment is exogenous, even though its not a variable. But, if you would treat $p$ as an variable, it would be exogenous variable. Sorry if I wasnt clear enough about that $\endgroup$
    – 1muflon1
    Feb 7, 2023 at 14:58
  • $\begingroup$ Fantastic, 1) parameters are always exogenous, 2) A variable can be endogenous or exogenous. 3) Exogenous and Endogenous, are synonyms for independent and dependent. 4) I assume this means we can have multiple dependent variables in a model 5) An endogenous variable can be a choice variable or not? This last one is confusing, the idea of a choice variable being a dependent variable feels contradictory? Thanks again, very close now! $\endgroup$
    – CormJack
    Feb 7, 2023 at 22:31
  • $\begingroup$ @CormJack 1-5 yes, regarding 5 in economics typically endogenous variable will be choice variable of some agent, although there might be more exceptions, but in physics biology etc endogenous variables are often not choice variables. The reason why this might feel contradictory is that typically the word dependent and independent variable are used in statistics. Dependent variable in simple regression goes on LHS and independent variable on RHS, but this is because basic stats models already assume RHS variable is exogenous, you can't (or shouldn't) run regular reg with endogenous regressor $\endgroup$
    – 1muflon1
    Feb 8, 2023 at 10:03
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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.

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  • $\begingroup$ Hi Tony, I appreciate you taking the time to answer. I just have a few follow up questions if you have time: $\endgroup$
    – CormJack
    Feb 5, 2023 at 16:36
  • $\begingroup$ 1) Can we get endogenous variables, that aren’t choice variables? What’s a common example? 2) Are “parameter” and “exogenous” interchangeable. 3) Can you shed some light on the relationship between independent and dependent, as related to the other concepts (endogenous, parameter, etc.) as discussed? Thanks! $\endgroup$
    – CormJack
    Feb 5, 2023 at 16:39
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    $\begingroup$ In general, endogenous and exogenous variables are words used in econometrics and statistics. A choice variable is a common word in micro and macro fields. A choice variable is endogenous, it is endogenously determined within the model. Example: Consider a profit maximization problem, where a firm wishes to maximize profits. What a firm can choose is the amount of inputs to employ and so the amount of output that maxize profits. Inputs are a choice variable, and endogenous, because when you solve this maximization problem, you determine the optimal amounts of inputs $\endgroup$
    – Tony
    Feb 5, 2023 at 21:14
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    $\begingroup$ Same example about profit maximization: if prices are taken as given, because you assume there exists millions of equal firms, and so none of them has market power, then prices are exogenous. Firms cannot control prices to maximze profits. However, I won't call prices parameters, because in macroeconomics, parameters are exogenous (external to the model) but can be calibrated by relying on external estimates. You don't calibrate prices (what we calibrate in macro models, for instance, are dicounting rates). As you see, the jargon changes according to your study field. I hope this can help now $\endgroup$
    – Tony
    Feb 5, 2023 at 21:25
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    $\begingroup$ Calibration means you give values to your parameters based on external studies. For instance, you can find many papers that have estimated the subjective discounting rate. You relies on those studies to give a value (calibrate) to the subjective discounting rate in your model. As I have already said, the jargon changes a bit according to your research field. $\endgroup$
    – Tony
    Feb 7, 2023 at 8:23

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