# Perfectly Competitive Markets in Math

June 26, 2020 · 5:53 am · EconMath

Perfect Competition, or as I like to abbreviate it, Perfekt, is your standard paired Supply vs Demand market graph along with a firm graph where Marginal Revenue = Demand = Average Revenue = Price.

You can interact with it below: (See this page to interact with only this applet)

Mode

$p_{mkt} =60$

$p_{eq} =60.85119791515859$

$p_{min} =55.14677965740265$

Profit per unit: $p_{eq} - p_{mkt} = 0.8511979151585933$

Economic state? Approximately Normal Profit

Should the firm shutdown in the short run? No.

## Definitions

$C_S = S(0)$ is the intersection between $S$ and the y-axis.

$C_D = D(0)$ is the intersection between $D$ and the y-axis.

$C_{MC}$ is an arbitrary constant that shifts $MC$ up or down.

$p_{mkt}$ is the market price.

$p_{eq}$ is the profit-maximizing price.

$p_{min}$ is the minimum of $AVC$.

## The default $MC$ function is awfully complicated!!

Yup, I was having trouble finding a decent function such that all of the cool things like the intersection between $P$ and $ATC$ and $P_{mkt}$ and $AVC$, so I asked Taiga to engineer one using Desmos:

The final function is very cursed:

$MC(q) = \frac{4.4(0.134q + 4.7)^2 - 12(0.134q + 4.7) + 50}{3.2(0.134q + 4.7) - 13.14} 8.4 - 125$

### Thought Process

The equation was written so that it could be manipulated as easily as possible. We already knew the general shape was going to have a sharp descent from the positive y-axis, and then slowly “climb up”. The “climb up” part could be approximated by a function that has a slant asymptote. Therefore, we began with a rational function with a quotient whose degree was 1 for a nice linear asymptote. This is why the numerator looks like a quadratic function (think of $0.134q + 4.7$ as $X$) and the denominator looks linear. Then, to perform linear transformations, we replaced $X$ with $mq + b$ for horizontal stretching/squishing and shifting and took the entire function and replaced it with $MR + B$ for vertical stretching/squishing and shifting.

## Why?

As I was watching Matt Pedlow’s AP Microeconomics series for this year’s exam, I noticed that we can actually define many of the relationships between quantity and price using discrete calculus. So I thought it would be interesting to talk about how that would span out.

Also, it’s always nice to be able to have an interactive visualization of things, and I haven’t seen anything for microeconomics anywhere else.

## Mathematical Formalization

Let me show you how some of these functions are calculated from $MC$.

### Defining $TC$ in terms of $MC$

The definition of Marginal Cost ($MC$) is the difference in the Total Cost ($TC$) of producing the currenth $n$th good minus the $TC$ of producing the last good, $(n-1)$th. Using notation from my post on discrete derivatives, we can define $MC$ as so:

\begin{aligned} MC(q) &\coloneqq TC(q) - TC(q - 1) \\ &= \Delta TC(q-1) \\ MC(q + 1) &= \Delta TC(q) \\ MC(q+1) \Delta I(q) &= \Delta TC(q) \\ \sum_{q = 0}^q MC(q + 1) \Delta I(q) &= \sum_{q = 0}^q \Delta TC(q) \\ &= TC(q) - TC(0) \\ TC(q) &= TC(0) + \sum_{q = 0}^q MC(q + 1) \Delta I(q) \\ \end{aligned}

By the way, both $MC$ and $TC$ map $Q \to P$, where $Q$ is the set of nonnegative integers,1 and $P$ is the set of reals with a unit of economic value, such as dollars.2 (In the app above, $Q$ has a different meaning: the domain that will be plotted in the graph.)

As a side note, we define a special case for $MC(0) \coloneqq TC(0)$ since the general definition results in a term $TC(-1)$, even though $-1 \notin Q$.

### Defining $ATC$ in terms of $MC$

We can define Average Total Cost ($ATC$) similarly, but beware that since it is calculated with respect to the quantity of goods, its domain is actually $Q - \{0\}$. That is, it doesn’t make sense to talk about the average cost per good when no goods have been produced. So, $ATC \colon Q-\{0\} \to P$, and

$ATC(q) \coloneqq \frac{TC(q)}{q}$

We can also define $MC$ in terms of $ATC$:

\begin{aligned} ATC(q) &= \frac{TC(q)}{q} \\ q \cdot ATC(q) &= TC(q) \\ \Delta [q \cdot ATC](q) &= \Delta TC(q) \\ q \Delta ATC(q) &= MC(q + 1) \\ MC(q) &= (q + 1) \Delta ATC(q + 1) \\ \end{aligned}

## Market Price

Since algebraically finding the market price is hard, this approximates by finding the closest quantity $q_{mkt}$ such that

$p_{mkt} \coloneqq S(q_{mkt}) = D(q_{mkt}).$

And if

$p_{eq} \coloneqq MC(q_{eq}) = ATC(q_{eq})$

is close enough to $p_{mkt}$, about $1$ unit, then we’ll say the firm is making approximately normal profit.

Normally, the firm tries to produce at the price $p_c = MR(q_c) = MC(q_c)$. For Perfekt firms, $MR(q) = p_{mkt}$; that is, they are price takers. So,

$p_c = p_{mkt} = MC(q_c).$

We can compare $p_{eq}$ and $p_c$ to find the sign of the economic profit the firm is making:

ComparisonEconomic State
$p_c < p_{eq}$Economic Loss
$p_c = p_{eq}$Normal Profit
$p_c > p_{eq}$Economic Profit

## Shutdown Point

If the firm is making an economic loss, we can also find

$p_{min} \coloneqq \min_{q \in Q} (AVC(q))$

and compare $p_{c}$ and $p_{min}$ like so:

ComparisonShould the firm shutdown?
$p_{min} < p_{c}$No
$p_{min} = p_{c}$Doesn’t Matter3
$p_{min} > p_{c}$Yes

1. The visualization connects points with consecutive quantities with a line segment, which is why the curves look continuous.
2. Note that this definition is not formal.
3. With respect to loss minimization.

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