<p>I am trying to model a small system, with limited population and a single firm. Personally I have no economic background but only taken a simple microeconomics course and I am quite interested in playing around with the seems to be simple &quot;supply intersect demand and the equilibrium is found&quot; concept. Suppose the following cases.</p>
<p>Suppose the firm is a farm with only a single land that can produce A amount of food and the marginal production of food with more labor would decrease with some rate related to beta.
Then, using a simple math model, I model the quantity of food Q produced with the amount of labor L. Thus, overall on the firm side we have these equations from profit maximization.</p>
<p><span class="math-container">\begin{align*}
    \text{the quantity produced in terms of labor} = Q &amp;= A  (1-e^{-\beta L})\\
    \frac{\partial Q}{\partial L} &amp;= A\beta e^{-\beta L} \\
    \frac{\partial^2 Q}{\partial L^2} &amp;= -A\beta^2 e^{-\beta L} \\
    L &amp;=\frac{-1}{\beta} log(1-\frac{Q}{A}) \\
    \frac{\partial L }{\partial Q} &amp;= \frac{1}{A\beta(1-\frac{Q}{A})} = \frac{1}{\beta(A-Q)} \\
    A-Q = Ae^{-\beta L}
\end{align*}</span>
profit maximization
<span class="math-container">\begin{align*}
    \text{profit}&amp;=PQ - C \\
    \text{cost} = C &amp;= cL \\
    \text{profit} = PQ - c \frac{-1}{\beta} log(1-\frac{Q}{A}) &amp;= P A  (1-e^{-\beta L}) - cL \\
    \text{when profit maximised} \\ 
    \frac{\partial \text{profit}}{\partial Q} &amp;= 0\\
    P &amp;=  S(Q) =  \frac{c}{\beta(A-Q)}  \\
    \frac{\partial \text{profit}}{\partial L} &amp;= 0\\
    c &amp;= D(L) = P\beta A e^{-\beta L}
\end{align*}</span></p>
<p>For the population, suppose we have N people, B is somewhat the satisfaction or some constant that shows the satisfaction of a meal, f is a constant of the needed amount of food for a single meal.</p>
<p>In a simple case, everyone will work in the farm, or else they cannot buy any food, so the number of labor L is N. We have the following and the equilibrium price P, wage rate c and the quantity produced.</p>
<p>if all people work no matter price
<span class="math-container">\begin{align*}
    L &amp;= N \\
    P &amp;= D(Q) = \frac{BfN}{Q}
\end{align*}</span>
We have
<span class="math-container">\begin{align*}
    P &amp;=  S(Q,c) =  \frac{c}{\beta(A-Q)}  \\
    P &amp;= D(Q) = \frac{BfN}{Q} \\
     c &amp;= D(L,P) = P\beta A e^{-\beta L} \\
     L &amp;= N \\
\end{align*}</span>
<span class="math-container">\begin{align*}
    c &amp;= P\beta A e^{-\beta N} \\
    Q &amp;= A(1-e^{-\beta N}) \\
    P &amp;= \frac{BfN}{A(1-e^{-\beta N})}\\
\end{align*}</span></p>
<p>In a more complex case, some people whether die than work for the farm if the wage is not high enough (if I even consider this &quot;high enough&quot; related to the price of a meal, the equation would be more complicated with beta' in terms of P and Q, so I just treat it simple). Such that we have the following equations to model this, with the max number of labor be N and this behavior is modeled by a constant beta'.
We have the following equation.</p>
<p>if increase with minimum
<span class="math-container">\begin{align*}
    %\text{price of one meal} &amp;= %\frac{Pf}{Q} \\
    P &amp;= D(Q) = \frac{BfN}{Q} \\
     c &amp;= S(L) = N (1-e^{-\beta' L})\\
\end{align*}</span></p>
<p>The tricky part and problem comes from how to solve the equilibrium price of product P and the equilibrium wage c. I have tried quite a bit but cannot figure out how to solve it, due to the P varies with c and c varies with P. Shown below:</p>
<p><span class="math-container">\begin{align*}
    P &amp;=  S(Q,c) =  \frac{c}{\beta(A-Q)}  \\
    P &amp;= D(Q) = \frac{BfN}{Q} \\
     c &amp;= D(L,P) = P\beta A e^{-\beta L} \\
     c &amp;= S(L) = N (1-e^{-\beta' L})\\
\end{align*}</span>
solve
<span class="math-container">\begin{align*}
    \text{1 and 3}\\
    1 &amp;= \frac{\beta A e^{-\beta L}}{\beta(A-Q)} \\
    \text{1 and 4} \\
    P &amp;= \frac{N (1-e^{-\beta' L})}{\beta(A-Q)} \\
\end{align*}</span></p>
<p>In conclusion, I would like to ask if there are anything that I am missing or is this some sort of limitations to my problem? Or are there some assumption in the model that I am playing around with? Any comment and discussion would be great!</p>

CC BY-SA 4.0
2024-01-26T15:21:29.970Z
Hi! Seems like you know MathJax, why are you pasting in images instead of copying the source it into your question?
Giskard
CC BY-SA 4.0
2024-01-26T15:22:09.880Z
Explaining the notation would be nice! E.g.; in $D(Q) = \frac{B\int N}{Q}$ I have no clue where the $\int$ came from. I am not even sure it is $\int$, because I cannot see the source code (:
Giskard
CC BY-SA 4.0
2024-01-26T15:24:51.617Z
Finally: it is possible that your equation system does not specify a unique equilbrium, it is known that in general equilibrium systems multiple equilibria may arise. Then indeed $c$ would depend on $P$ or vice-versa.
Giskard
CC BY-SA 4.0
2024-01-26T15:43:09.377Z
@Giskard Great thanks, I was in a hurry just now and just copied the image from my overleaf doc, I will edit it later. For the second question ∫ should be the f. the needed amount of food for a single meal, maybe the image is not that clear.
DongAlpha
CC BY-SA 4.0
2024-01-26T15:43:48.977Z
@Giskard I would like to ask why that it is possible to not have a unique eqm? As in reality, there should always be an answer or an outcome of the market.
DongAlpha
CC BY-SA 4.0
2024-01-26T22:22:47.763Z
There is also usually a clear outcome to a dice roll, yet these outcomes are not unique.
Giskard
CC BY-SA 4.0
2024-01-26T22:25:20.783Z
However this is not reality, this is a mathematical model (not to be confused!) and equations frequently have multiple solutions.
Giskard
<supply-and-demand><labor-economics><production-function>
Toy model equilibrium calculation: How a small system's firm determines wage and price?
DongAlpha
# Toy model equilibrium calculation: How a small system's firm determines wage and price?

(Asked by: username_0 on 2024-01-26)

I am trying to model a small system, with limited population and a single firm. Personally I have no economic background but only taken a simple microeconomics course and I am quite interested in playing around with the seems to be simple "supply intersect demand and the equilibrium is found" concept. Suppose the following cases.

Suppose the firm is a farm with only a single land that can produce A amount of food and the marginal production of food with more labor would decrease with some rate related to beta.
Then, using a simple math model, I model the quantity of food Q produced with the amount of labor L. Thus, overall on the firm side we have these equations from profit maximization.

\begin{align\*}
\text{the quantity produced in terms of labor} = Q &= A (1-e^{-\beta L})\\
\frac{\partial Q}{\partial L} &= A\beta e^{-\beta L} \\
\frac{\partial^2 Q}{\partial L^2} &= -A\beta^2 e^{-\beta L} \\
L &=\frac{-1}{\beta} log(1-\frac{Q}{A}) \\
\frac{\partial L }{\partial Q} &= \frac{1}{A\beta(1-\frac{Q}{A})} = \frac{1}{\beta(A-Q)} \\
A-Q = Ae^{-\beta L}
\end{align\*}
profit maximization
\begin{align\*}
\text{profit}&=PQ - C \\
\text{cost} = C &= cL \\
\text{profit} = PQ - c \frac{-1}{\beta} log(1-\frac{Q}{A}) &= P A (1-e^{-\beta L}) - cL \\
\text{when profit maximised} \\
\frac{\partial \text{profit}}{\partial Q} &= 0\\
P &= S(Q) = \frac{c}{\beta(A-Q)} \\
\frac{\partial \text{profit}}{\partial L} &= 0\\
c &= D(L) = P\beta A e^{-\beta L}
\end{align\*}

For the population, suppose we have N people, B is somewhat the satisfaction or some constant that shows the satisfaction of a meal, f is a constant of the needed amount of food for a single meal.

In a simple case, everyone will work in the farm, or else they cannot buy any food, so the number of labor L is N. We have the following and the equilibrium price P, wage rate c and the quantity produced.

if all people work no matter price
\begin{align\*}
L &= N \\
P &= D(Q) = \frac{BfN}{Q}
\end{align\*}
We have
\begin{align\*}
P &= S(Q,c) = \frac{c}{\beta(A-Q)} \\
P &= D(Q) = \frac{BfN}{Q} \\
c &= D(L,P) = P\beta A e^{-\beta L} \\
L &= N \\
\end{align\*}
\begin{align\*}
c &= P\beta A e^{-\beta N} \\
Q &= A(1-e^{-\beta N}) \\
P &= \frac{BfN}{A(1-e^{-\beta N})}\\
\end{align\*}

In a more complex case, some people whether die than work for the farm if the wage is not high enough (if I even consider this "high enough" related to the price of a meal, the equation would be more complicated with beta' in terms of P and Q, so I just treat it simple). Such that we have the following equations to model this, with the max number of labor be N and this behavior is modeled by a constant beta'.
We have the following equation.

if increase with minimum
\begin{align\*}
%\text{price of one meal} &= %\frac{Pf}{Q} \\
P &= D(Q) = \frac{BfN}{Q} \\
c &= S(L) = N (1-e^{-\beta' L})\\
\end{align\*}

The tricky part and problem comes from how to solve the equilibrium price of product P and the equilibrium wage c. I have tried quite a bit but cannot figure out how to solve it, due to the P varies with c and c varies with P. Shown below:

\begin{align\*}
P &= S(Q,c) = \frac{c}{\beta(A-Q)} \\
P &= D(Q) = \frac{BfN}{Q} \\
c &= D(L,P) = P\beta A e^{-\beta L} \\
c &= S(L) = N (1-e^{-\beta' L})\\
\end{align\*}
solve
\begin{align\*}
\text{1 and 3}\\
1 &= \frac{\beta A e^{-\beta L}}{\beta(A-Q)} \\
\text{1 and 4} \\
P &= \frac{N (1-e^{-\beta' L})}{\beta(A-Q)} \\
\end{align\*}

In conclusion, I would like to ask if there are anything that I am missing or is this some sort of limitations to my problem? Or are there some assumption in the model that I am playing around with? Any comment and discussion would be great!

**Tags:** supply-and-demand, labor-economics, production-function

> **Comments:**
> **username_1**: Hi! Seems like you know MathJax, why are you pasting in images instead of copying the source it into your question?  
> **username_1**: Explaining the notation would be nice! E.g.; in $D(Q) = \frac{B\int N}{Q}$ I have no clue where the $\int$ came from. I am not even sure it is $\int$, because I cannot see the source code (:  
> **username_1**: Finally: it is possible that your equation system does not specify a unique equilbrium, it is known that in general equilibrium systems multiple equilibria may arise. Then indeed $c$ would depend on $P$ or vice-versa.  
> **username_0**: @username_1 Great thanks, I was in a hurry just now and just copied the image from my overleaf doc, I will edit it later. For the second question ∫ should be the f. the needed amount of food for a single meal, maybe the image is not that clear.  
> **username_0**: @username_1 I would like to ask why that it is possible to not have a unique eqm? As in reality, there should always be an answer or an outcome of the market.  
> **username_1**: There is also usually a clear outcome to a dice roll, yet these outcomes are not unique.  
> **username_1**: However this is not reality, this is a mathematical model (not to be confused!) and equations frequently have multiple solutions.  

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## 0 answers

