## Recap on concepts and notation

Remember that we can write marginal cost in three different ways:

$\frac{∂ c(q) }{ ∂q} = c'(q) = MC(q)$ where $c(y)$ is the cost function.

Remember that revenue or "sales" are given by:

$R(q) = p \times q$

## Firm's constraints:

• Technological constraints: what can be produced, how and from what inputs

• The shape of the cost function if determined by production function and prices of inputs.
• Market constraints: how will consumers and other firms react to a given firm’s choice?

• If I charge high price, I cannot sell many units.

## Perfect competition

Key feature of perfect competition: agents take market price as given, outside of any particular firm’s control.

This happen, typically, when many small agents operate in either side of the market.

If the firm is a price-taker at price level $p^*$ (e.g. the firm is one of many many producers) and tries to sell for a higher price, then it will sell zero units.

If the firm charges exactly $p^*$, it can produce/sell pretty much as much as it wants. BUT What is the right quantity??

## Supply function of a competitive firm

We assume the firm wants to maximize profits $\pi(q) = p \times q − c(q)$

If $\pi(q)$ is smooth and has a global maximum, the optimal $q$ is found by setting:

$$\frac{∂ \pi(q) }{ ∂q} = 0$$

As before, we call this the first-order condition (FOC).

## Supply function of a competitive firm

The first-order condition (FOC) implies:

\begin{aligned} \frac{∂ \pi(q) }{ ∂q} &= 0 \\\\ MR - MC(q) &= 0 \\\\ p - MC(q) &= 0 \\\\ p &= MC(q) \\\\ \end{aligned}

The firm's MC curve determines the firm's supply function.

## Supply function of a competitive firm

However, the first order condition is not sufficient: sometimes it identifies a local minimum.

The derivative of profit wrt q is zero also at the bottom of a "valley".

While a local maximum (what we want) is found at the peak of the "hill".

How do we distinguish bottom of the valley from top of the hill?

See graph.

## Supply function of a competitive firm

In the top of the hill slope goes from positive to negative, decreases.

Second-order condition: $\pi''(q) = -c''(q) \leq 0 ~$ or $~ c''(q) \geq 0$

$$\frac{∂ MC(q) }{ ∂q} \geq 0$$

The supply function is not the whole MC(q), but only the segment that is upward-sloping.

Still, not the end of the story...

## When does the firm indeed operate?

Compare profit from operation (q*>0) Vs. shutting down (q=0)

That is: $p q − c_v(q) − F ~$ Vs. $−F$

Operate if: $~ p q − c_v(q) − F \geq (−F)$

That is, if: $p q − c_v(q) \geq 0$

That is, if: $p \geq \frac{c_v(q)}{q} = AVC(q)$

Operate when price covers average variable cost (AVC)

## Supply curve for the firm -- Finally!

Supply curve is the upward-sloping part of MC curve that also lies above the AVC curve.

Notice $q^* = 0$ if $MC < minAVC$.

## Inverse Supply curve

The inverse supply curve is the same equation of the supply curve except we have solved for p:

Mathematically:

$p = c’(q)$

if $c''(q) >0$ and $c'(q)>AVC$

## Supply curve - Example - Steps

Suppose $c(q) = q^2 +1$

1. Calculate MC: MC = 2 q

2. Equate MC = P: $p = 2 q$. This gives the (inverse) supply curve.

3. Make sure $MC \geq AVC$: In this example, for any value of $q>0$ we have that MC > AVC, since: $2 q \geq q$

4. Supply is $q = p/2$ and inverse supply is $p = 2q$

## Short-run industry supply

The supply curve in the "short run of the industry" is the sum of the supplies of all participating firms.

$S(p) = \sum_1^n q^s_i(p)$