This is a small break from my normal type of post, but I've become a TA for my school's AP economics (AP = Advanced Placement, for those unfamiliar with the American and Canadian education systems) course, which has left me with a couple of takeaways:

- Calculus should be a prerequisite for economics
- AP Econ/Econ 101 resorts to a lot of inconsistent nonsense in order to explain things to people who don't understand calculus

To remedy my annoyance at introductory economics (which I have confirmed from friends taking the same course at other schools universally explains the demand curve differently from the way it should/the way that is mathematically consistent with the rest of Econ 101), I decided to write down a derivation of the Econ 101 demand and supply curves in consistent way.

Under normal circumstances, I would probably rather criticize the theory for being unrealistic, but being clear about the math going on behind the scenes is all I choose to care about for the moment.

I should preface the math with an explanation of the way Econ 101 usually deals with the demand curve:

There are a lot of people who come to a market that sells one item. Each person is willing to buy the item at any price lower than some arbitrary price, so if the owner of the market comes out and declares a high price, relatively few people will buy the item. Similarly, if the owner declares a low price, many people will buy it.

This explanation results in a weird demand curve with 'steps' at different prices whose width is determined by the number of people with their maximum price at that level. This is entirely different from the smooth curves instructors like to draw to illustrate demand, and inconsistent with the math used when teaching firm behavior (marginal revenue doesn't make sense when the demand curve is a bunch of steps).

Anyway, this is how Econ 101 students (with at least an understanding of derivatives) should be taught supply and demand:

Demand Curve Derivation

Consumers derive a certain amount of utility when they buy units of the good. This utility can be expressed as the function U(Q) where U stands for utility and Q is the quantity of the good that consumers purchase.

Consumers pay the same price for each unit of the good that they buy, so their total cost is PQ where P is the price of the good.

People want to maximize the net benefit they derive from buying units of the good. Mathematically this means maximizing U(Q) - PQ.

We know from calculus that setting the derivative to zero will give us the maximum, so the net benefit maximizing quantity satisfies

U'(Q) - P = 0 or U'(Q) = P

This is the demand curve. The reason it is downward sloping is because of diminishing marginal utility -- the notion that each additional unit of the good is less valuable than the last. This means that U'(Q) is a negative function of Q, necessitating a downward sloping demand curve.

Supply Curve Derivation:

Firms want to maximize profits, which are defined at their total revenue (PQ) minus total costs (C(Q)). They do this given what they know about the demand for their product, so they replace the P in PQ with U'(Q) from the demand curve. Thus, firms maximize

U'(Q)Q - C(Q)

meaning that

d/dQ U'(Q)Q - C'(Q) = 0

which is the same as

U''(Q)Q + U'(Q) - C'(Q) = 0

The supply curve needs to be written as a function of P, so we can just substitute P in for U'(Q) above, yielding

P = C'(Q) - U''(Q)Q

This is the supply curve.

Let's derive the demand and supply curves given example utility and cost functions:

U(Q) = aln(Q)

C(Q) = 1/3(Q-b)^3 - cQ^2 + dQ

In this case, the demand curve should be

P = U'(Q) = a/Q

and the supply curve should be

P = C'(Q) - U''(Q)Q = (Q-b)^2 - 2cQ + d + a/Q

This example gives fancy curves similar to those you might draw as examples, but a simpler example does a better job of showing the types of the linear curves you might see in econ 101/AP Micro

U(Q) = Qa - 0.5bQ^2

C(Q) = Q^2 + cQ

This gives the demand curve

P = a - bQ

and the supply curve

P = 2Q + c - bQ = (2-b)Q + c

Consumers derive a certain amount of utility when they buy units of the good. This utility can be expressed as the function U(Q) where U stands for utility and Q is the quantity of the good that consumers purchase.

Consumers pay the same price for each unit of the good that they buy, so their total cost is PQ where P is the price of the good.

People want to maximize the net benefit they derive from buying units of the good. Mathematically this means maximizing U(Q) - PQ.

We know from calculus that setting the derivative to zero will give us the maximum, so the net benefit maximizing quantity satisfies

U'(Q) - P = 0 or U'(Q) = P

This is the demand curve. The reason it is downward sloping is because of diminishing marginal utility -- the notion that each additional unit of the good is less valuable than the last. This means that U'(Q) is a negative function of Q, necessitating a downward sloping demand curve.

Supply Curve Derivation:

Firms want to maximize profits, which are defined at their total revenue (PQ) minus total costs (C(Q)). They do this given what they know about the demand for their product, so they replace the P in PQ with U'(Q) from the demand curve. Thus, firms maximize

U'(Q)Q - C(Q)

meaning that

d/dQ U'(Q)Q - C'(Q) = 0

which is the same as

U''(Q)Q + U'(Q) - C'(Q) = 0

The supply curve needs to be written as a function of P, so we can just substitute P in for U'(Q) above, yielding

P = C'(Q) - U''(Q)Q

This is the supply curve.

Let's derive the demand and supply curves given example utility and cost functions:

U(Q) = aln(Q)

C(Q) = 1/3(Q-b)^3 - cQ^2 + dQ

In this case, the demand curve should be

P = U'(Q) = a/Q

and the supply curve should be

P = C'(Q) - U''(Q)Q = (Q-b)^2 - 2cQ + d + a/Q

This example gives fancy curves similar to those you might draw as examples, but a simpler example does a better job of showing the types of the linear curves you might see in econ 101/AP Micro

U(Q) = Qa - 0.5bQ^2

C(Q) = Q^2 + cQ

This gives the demand curve

P = a - bQ

and the supply curve

P = 2Q + c - bQ = (2-b)Q + c

John: It's good you are thinking about better ways to teach Intro. A coupla thoughts on your post:

ReplyDelete1. Many Intro texts have a "Diminishing Marginal Utility" explanation of the demand curve. I don't see why it's obviously better to do the same thing in calculus rather than words.

2. Your individual buyer takes P as parametric (sees himself as facing a perfectly elastic supply curve). Your individual seller does not take P as parametric (sees himself facing a downward-sloping demand curve). The standard textbook treatment is symmetric, in that both buyer and seller in a competitive market take P as parametric. And relaxes that assumption later when it considers monopsony and monopoly power.

You're right on the first thing, I just don't like it when people switch between talking about individuals and utility and groups of people and with different reserve prices.

DeleteI made the seller consider the demand curve because I wanted to be able to generalize it to any market structure -- if demand is perfectly elastic P = MC, and as elasticity falls the representative firm becomes more monopolistic. In any case, wouldn't the price taker be the one to take P as parametric? If so, then the way I did it should have been fine for all but perfect competition...

The way you did supply is fine for the general case, and approaches perfect competition in the limit. (Some economists might say it is not strictly a "supply curve", because Qs is a function not just of P, but of elasticity of demand too, but let that pass.)

DeleteBut for symmetry demand should maybe be treated the same way, with a monopsonistic buyer facing an upward-sloping supply curve the general case, approaching competition in the limit. But then bilateral monopoly-monopsony is a problem, which is why we don't do it that way, and start out with both buyer and seller taking P as parametric.

3. If the good is discrete (like cars) then the demand curve really is a step function. (But if you remember that Q is a flow, so you can buy 1 car per 9.87654321 years, it gets smoothed out).

ReplyDelete