1. EE/Econ 458 Introduction to Economics: Producer’s Surplus
J. McCalley
These slides are your main resource for this topic.

2. Supply functions and Producer Surplus
An supply curve characterizes the manner in which the supply of a good will change as its price changes, holding constant all other factors that influence producers’ willingness or ability to pay for the good.
This characteristic is called the inverse supply function. Inverting it gives us the supply function.
Supply increases as price increases (or supply decreases as price decreases).
When supply changes with price, we say the supply is elastic.

3. Supply functions and Producer Surplus
In early market designs, previous to ~2013, wind plants did not participate in the market; they generated as much as the wind dictated, and they were paid the clearing price.
This illustrates an inelastic supply, i.e., a supply that is price insensitive. But most electricity producers behave elastically (today, wind participates in the market behaving more elastically).

4. Supply functions and Producer Surplus
Consider a suppliers w/ following behavior:
below 20 $/MWhr, they shut down their power plant.
at 20 $/MWhr, they produce, supplying up to 4 MWhrs;
if price goes to 30 $/MWhr, they produce up to 7 MWhrs;
if price goes to 50 $/MWhr, they produce up to 50 MWhrs.

5. Supply functions and Producer Surplus
Consider that we have the hourly cost function for a producer C as a function of the amount it produces P.
Two cautions:
Make sure to distinguish price p from production P.
We will assume that C to be in $/MWhr and P to be MW for one hour (MWhr).
We assume C to be convex (we will be maximizing –C equivalent to minimizing C).
C(P) x

6. Supply functions and Producer Surplus
The value C(P) is the amount of money the producer needs to spend in order to produce P MWhrs. And we know its derivative C’, the incremental cost, or, the marginal cost of energy.
The producer faces an energy price of p and needs to find the production level which maximizes profits.
Profits are given by the amount the producer is paid for producing P MWhr less the amount it costs the producer to produce P MWhr. The amount the producer is paid when the price is p is pP. So profits are given by pP-C(P).
In contrast to the consumer’s resource (money), the producer’s resource is energy, P.
Whereas the consumer had a constraint on money, the producer has a constraint on energy. That constraint is 0<P<Pmax.

7. Supply functions and Producer Surplus
So producer needs to solve the following optimization problem:
(1)
In our approach to inequality constrained optimization problems, we first solve the unconstrained problem and then check our solution. Following this approach, we apply first order conditions to the expression inside the curly brackets to obtain:
(2)
Implication?
When the producer sees price p, s/he should produce the amount of energy P which results in
(3)

8. Supply functions and Producer Surplus
When the producer sees price p, s/he should produce the amount of energy P which results in
(3)
What does this tell us?...
Recall that C’(P) expresses the marginal cost for the amount of energy we buy, P.
That is, the marginal cost of energy should be equal to the price of energy in order for the producer to maximize profits.
One can also interpret (3) as the inverse supply function: for each possible quantity P, it gives us the market price of energy p that would induce the producer to sell exactly P.

9. Supply functions and Producer Surplus
Definition: The inverse supply function, p=C’(P), expresses price as a function of supply.
Definition: The demand function, P(p), expresses supply as a function of price.
Example 1: Let
Find the inverse supply function and the supply function.
And so the inverse supply function is:
And the supply function is found by solving for P:
The two functions are plotted on the next slide…

10. Supply functions and Producer Surplus
Inverse supply function
Supply function

11. Supply functions and Producer Surplus
Question: Consider that the producer has a choice of participating in the electric energy market or not. Will it benefit by doing so?
Answer: It depends… on what?
On the difference between
the producer’s total profits when it participates and
its total profits when it does not.
If it does participate, and the energy price is p, then it produces P, it gets paid pP, and it incurs a cost C(P)
If it does not participate, then it produces 0, it gets paid 0, and it incurs a cost C(0) The producer’s surplus is:
(4)
By Fundamental Theorem of Calculus

12. Supply functions and Producer Surplus
(4)
Equation (4) has a nice graphical interpretation illustrated below, where we assume a price p* results in a supply P*.
• The first term of (4) corresponds to the total “box,” shaded by vertical lines, lower side 0:P* and left-hand-side 0:p*.
• The second term of (4) corresponds to the area under the p=C’(x) curve from 0 to P*, which is the area shaded by horizontal lines.

13. Supply functions and Producer Surplus
Question:
Suppose the price of energy is increased from p0 to p1.
What happens to the producer’s surplus?
That is, does price increase cause PS to increase or decrease, and by how much?
How to think about answering this question….?
Get the producer’s surplus at p0, PS(p0), assuming P0
Get the producer’s surplus at p1, PS(p1), assuming P1
Compute PS(p1)-PS(p0)

14. Supply functions and Producer Surplus
1. Get the producers’s surplus at p0, PS(p0), assuming P0
 When price is p0, producer supplies P0 and PS is
2. Get the producer’s surplus at p1, PS(p1), assuming P1
 When price is p1, producer supplies P1 and PS is
3. Compute PS(p1)-PS(p0)

15. Supply functions and Producer Surplus
Group the integrals and subtract the group
because now we can combine the integrals as
Add and subtract p1P0 to the above expression, resulting in

16. Supply functions and Producer Surplus
Add and subtract p1P0 to the above expression, resulting in
Rearrange the terms
Factor P0 from the first two terms and p1 from the second two terms, resulting in
Grouping the second term with the integral term…

17. Supply functions and Producer Surplus
Grouping the second term with the integral term…

18. Supply functions and Producer Surplus
Above says that the change in the producer’s surplus can be decomposed into two parts:
• The new revenues from the increase in price that the producer enjoys from selling the old amount (yellow) and
• The increased profit due to the additional sold amount (red).