<p>One of the most fundamental distinctions in economics is that between stocks, measured at a point in time, and flows, measured over a period of time or as instantaneous rates. The construction of dynamic economic models naturally leads to differential equations in which the rate of change of a stock variable depends on one or more flow variables.</p>
<p>An example of an economic model centred on a system of two such differential equations is the Dasgupta-Heal-Solow-Stiglitz model.  The stock variables in this model are man-made capital <span class="math-container">$K$</span> and non-renewable resource <span class="math-container">$S$</span> (the simplifying assumption is made that there is just one type of each).</p>
<p>In outline, the model takes output <span class="math-container">$Y$</span> to be a function of capital <span class="math-container">$K$</span> and use of the resource <span class="math-container">$R$</span> (a further simplifying assumption is to ignore labour input), and the rate of change of the stock of capital to equal output less consumption <span class="math-container">$C$</span>:</p>
<p><span class="math-container">$$\dot{K}=Y-C=f(K,R)-C$$</span></p>
<p>The rate of change of the stock of non-renewable resource is simply the rate of use of the resource:</p>
<p><span class="math-container">$$\dot{S}=-R$$</span></p>
<p>The objective is to maximise the discounted present value of the utility <span class="math-container">$U$</span> derived from consumption:</p>
<p><span class="math-container">$$\max_C \int_{t=0}^T e^{-\rho t}U(C(t))dt$$</span></p>
<p>For particular output and utility functions, solutions may be obtainable using the method of dynamic optimization (see for example <a href="https://www.cireqmontreal.com/wp-content/uploads/cahiers/04-2010-cah.pdf" rel="nofollow noreferrer">Benchekroun &amp; Withagen 2010</a>).</p>

CC BY-SA 4.0
2021-10-05T21:43:08.980Z
2021-10-10T10:40:48.023Z
2021-10-10T10:40:48.023Z
Adam Bailey
<p>More than a question, I'm looking for suggestions of models of system of differential equations applied in economics or maybe in another area of interes.</p>
<p>I have a research homework and I have to make an analysis of the model and present it in front of a class and now I'm in the process of choosing the model what I'm going to talk about so any suggestion will help me a lot :)</p>

CC BY-SA 4.0
2021-10-04T10:50:40.047Z
Are you interested in models with one differential equation, or (as "system" suggests) are you looking for models with more than one differential equation?
Adam Bailey
CC BY-SA 4.0
2021-10-05T18:33:45.060Z
@AdamBailey models with more than one differential equation
Pech
<mathematical-economics><nonexpert><difference-equations>
System of differential equations in economics and areas of interest
Pech
# System of differential equations in economics and areas of interest

(Asked by: username_0 on 2021-10-03)

More than a question, I'm looking for suggestions of models of system of differential equations applied in economics or maybe in another area of interes.

I have a research homework and I have to make an analysis of the model and present it in front of a class and now I'm in the process of choosing the model what I'm going to talk about so any suggestion will help me a lot :)

**Tags:** mathematical-economics, nonexpert, difference-equations
> **Comments:**
> **username_1**: Are you interested in models with one differential equation, or (as "system" suggests) are you looking for models with more than one differential equation?  
> **username_0**: @AdamBailey models with more than one differential equation  

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## 1 answers

### Answer by username_1 on 2021-10-05. Score: 4 | Accepted answer

One of the most fundamental distinctions in economics is that between stocks, measured at a point in time, and flows, measured over a period of time or as instantaneous rates. The construction of dynamic economic models naturally leads to differential equations in which the rate of change of a stock variable depends on one or more flow variables.

An example of an economic model centred on a system of two such differential equations is the Dasgupta-Heal-Solow-Stiglitz model. The stock variables in this model are man-made capital $K$ and non-renewable resource $S$ (the simplifying assumption is made that there is just one type of each).

In outline, the model takes output $Y$ to be a function of capital $K$ and use of the resource $R$ (a further simplifying assumption is to ignore labour input), and the rate of change of the stock of capital to equal output less consumption $C$:

$$\dot{K}=Y-C=f(K,R)-C$$

The rate of change of the stock of non-renewable resource is simply the rate of use of the resource:

$$\dot{S}=-R$$

The objective is to maximise the discounted present value of the utility $U$ derived from consumption:

$$\max\_C \int\_{t=0}^T e^{-\rho t}U(C(t))dt$$

For particular output and utility functions, solutions may be obtainable using the method of dynamic optimization (see for example [Benchekroun & Withagen 2010](https://www.cireqmontreal.com/wp-content/uploads/cahiers/04-2010-cah.pdf)).


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