<p>I am new to economics, but come from a statistics and math background, so the control theory and dynamic optimization ideas are familiar to me. What is a little confusing is that in most control theory problems, there is a system state <span class="math-container">$x_t$</span> and then a set of controls <span class="math-container">$u_t$</span>. So the optimization is done based upon the interaction of the system state and the corresponding controls given that state.</p>
<p>I was reading some texts on dynamic macroeconomics, including the Ljungqvist and Sargent book, as well as the book <em>Dynamic Macroeconomics: Quantitative Methods and Applications</em>, which also seems quite nice. The starts with the usual dynamic intertemporal utility optimization problem:</p>
<p><span class="math-container">$$
max_{c} \quad \mathbb{E}(\sum^{\infty}_{t=0} \beta^tu(c_t))
$$</span></p>
<p>In this case, <span class="math-container">$c_t$</span> represents the household's consumption choices over time. The optimization is formulated as either a finite time horizon or infinite time horizon problem.</p>
<p>So what is confusing me, and what the basis of my question is, is the that there is no definition of an exogenous state in this problem. For example, if I have a 1 good economy that only produces ice cream, then given the equation above, I would optimize my intertemporal consumption of ice cream to maximize my utility. HOWEVER, there is no conception of seasons or weather, or hot/code days in this model. There are very clear dynamics to the seasons which would essentially be modelled with a <span class="math-container">$sin$</span> wave or something, to show variations in the temperature over time. And it is pretty clear that a household's demand for ice cream would be higher on hot days and in hot seasons rather in cold seasons. So a more appropriate problem should be something like:</p>
<p><span class="math-container">$$
max_{c} \quad \mathbb{E}(\sum^{\infty}_{t=0} \beta^tu(x_t, c_t))
$$</span></p>
<p>where <span class="math-container">$x_t$</span> represents some temperature dynamics that varies by season.</p>
<p>I imagine that Economists are aware of this question, but I was not sure why this setup of a state vector and control vector is not the standard setup introduce in dynamic economics? Another way of asking this question, is whether there are any technical reasons why the problem is not formulated with a state and control vector. By technical, I mean that the original problems formulation relies up the existence of an optimal vector <span class="math-container">$c_t$</span> that satisfies the problem, and the existence and uniqueness of that solution depends upon the Contraction Mapping theorem. So does inclusion of this addtional state vector <span class="math-container">$x_t$</span> violate the conditions of the Contraction Mapping theorem, or does it perhaps create multiple equilibria that creates problems for the original model? I am wondering if this technical issue is what causes the common model setup to not include terms for some exogenous &quot;state&quot; variable that could affect the utility function.</p>

CC BY-SA 4.0
2022-07-25T20:22:44.897Z
No, there is no problem with adding state variables. Your assumption that common models do not include state variables is not correct. The Ljungqvist and Sargent book gives many examples of models with exogenous (e.g. government policy) or endogenous state variables (e.g. capital).
Fića
CC BY-SA 4.0
2022-07-25T21:55:05.890Z
@Fića thanks for the response. Good to know that add state variables is common. In so many control theory books, the state and control variables are introduced right from the start, so the system is $\dot{x} = Ax + Bu + \epsilon$, where $x$ is the state, $u$ is the control, and $\epsilon$ is a disturbance in the state. Hence, I was confused when I only saw the control variable $c_t$ in the initial recursive macroecon formulations. But I will look in the Ljungvist and Sargent book to at the government policy models to see if I can find these examples.
krishnab
CC BY-SA 4.0
2023-08-23T03:43:38.140Z
Also, Weber's text, "Optimal Control Theory with Applications in Economics", should have examples with controls or atleast reference some examples.
mark leeds
<macroeconomics><utility><dynamic-optimization><recursive-macroeconomics>
dynamic macroeconomics: Does the system "state" affect the dynamic optimization problem?
krishnab
# dynamic macroeconomics: Does the system "state" affect the dynamic optimization problem?

(Asked by: username_0 on 2022-07-25)

I am new to economics, but come from a statistics and math background, so the control theory and dynamic optimization ideas are familiar to me. What is a little confusing is that in most control theory problems, there is a system state $x\_t$ and then a set of controls $u\_t$. So the optimization is done based upon the interaction of the system state and the corresponding controls given that state.

I was reading some texts on dynamic macroeconomics, including the Ljungqvist and Sargent book, as well as the book *Dynamic Macroeconomics: Quantitative Methods and Applications*, which also seems quite nice. The starts with the usual dynamic intertemporal utility optimization problem:

$$
max\_{c} \quad \mathbb{E}(\sum^{\infty}\_{t=0} \beta^tu(c\_t))
$$

In this case, $c\_t$ represents the household's consumption choices over time. The optimization is formulated as either a finite time horizon or infinite time horizon problem.

So what is confusing me, and what the basis of my question is, is the that there is no definition of an exogenous state in this problem. For example, if I have a 1 good economy that only produces ice cream, then given the equation above, I would optimize my intertemporal consumption of ice cream to maximize my utility. HOWEVER, there is no conception of seasons or weather, or hot/code days in this model. There are very clear dynamics to the seasons which would essentially be modelled with a $sin$ wave or something, to show variations in the temperature over time. And it is pretty clear that a household's demand for ice cream would be higher on hot days and in hot seasons rather in cold seasons. So a more appropriate problem should be something like:

$$
max\_{c} \quad \mathbb{E}(\sum^{\infty}\_{t=0} \beta^tu(x\_t, c\_t))
$$

where $x\_t$ represents some temperature dynamics that varies by season.

I imagine that Economists are aware of this question, but I was not sure why this setup of a state vector and control vector is not the standard setup introduce in dynamic economics? Another way of asking this question, is whether there are any technical reasons why the problem is not formulated with a state and control vector. By technical, I mean that the original problems formulation relies up the existence of an optimal vector $c\_t$ that satisfies the problem, and the existence and uniqueness of that solution depends upon the Contraction Mapping theorem. So does inclusion of this addtional state vector $x\_t$ violate the conditions of the Contraction Mapping theorem, or does it perhaps create multiple equilibria that creates problems for the original model? I am wondering if this technical issue is what causes the common model setup to not include terms for some exogenous "state" variable that could affect the utility function.

**Tags:** macroeconomics, utility, dynamic-optimization, recursive-macroeconomics

> **Comments:**
> **username_1**: No, there is no problem with adding state variables. Your assumption that common models do not include state variables is not correct. The Ljungqvist and Sargent book gives many examples of models with exogenous (e.g. government policy) or endogenous state variables (e.g. capital).  
> **username_0**: @username_1 thanks for the response. Good to know that add state variables is common. In so many control theory books, the state and control variables are introduced right from the start, so the system is $\dot{x} = Ax + Bu + \epsilon$, where $x$ is the state, $u$ is the control, and $\epsilon$ is a disturbance in the state. Hence, I was confused when I only saw the control variable $c_t$ in the initial recursive macroecon formulations. But I will look in the Ljungvist and Sargent book to at the government policy models to see if I can find these examples.  
> **username_2**: Also, Weber's text, "Optimal Control Theory with Applications in Economics", should have examples with controls or atleast reference some examples.  

---

## 0 answers

