A Model of Commodity Markets
In this lecture, we’ll build a simple model of a commodity market with a representative consumer and a representative producer (firm).
The consumer takes the market price as given, and decides how many units of the commodity to purchase.
The firm takes the market price as given, and decides how many units of the commodity to produce and sell.
In equilibrium, the quantity demanded by consumers must equal the quantity supplied by firms.
Consumers
Setup
A representative, price-taking consumer decides how many units, \(q_d\), of a commodity to purchase at unit price \(p\). The utility they receive for purchasing \(q_d\) units at price \(p\) is:
\[u(q_d) = v(q_d) - pq_d\]\(v(q_d)\) is the total benefit that the consumer derives when he consumes \(q_d\) units of the commodity. \(pq_d\) is the total cost paid by the consumer.
First order condition
The consumer’s optimization problem is the following:
\[\max_{q_d} ~ v(q_d) - pq_d\]The maximum utility is achieved when the derivative of the consumer’s uitlity function is equal to zero. Taking the derivative and setting it equal to zero yields the first order condition:
\[v^\prime(q_d) - p = 0\]or:
\[v^\prime(q_d) = p ~ ~ ~ ~ (\text{Eq.1})\]Equation (1) is the first order condition of the consumer. It therefore shows the relationship between price and quantity demanded, because this is the equation the consumer uses to determine how many units to consume.
In other words, equation (1) defines the consumer’s demand curve! Technically, it’s the inverse demand curve because it shows \(p\) in terms of \(q_d\). If we re-arrange the equation to write \(q_d\) as a function of \(p\), we’d get the demand curve.
Equation (1) shows that at the consumer’s optimal choice, price equals marginal benefit, because \(v^\prime(q_d)\) is additional benefit from an incremental unit of consumption, when the current consumption is already \(q_d\).
Example: Deriving a demand curve
A representative, price-taking consumer decides how many units, \(q_d\), of a commodity to purchase at unit price \(p\). The utility they receive for purchasing \(q_d\) units at price \(p\) is:
\[u(q_d) = 9q_d - \frac{3}{2} q_d^2 - pq_d\]
- Write down the consumer’s demand curve.
Answer.
Step 1. Write down the consumer’s optimization problem.
\[\max_{q_d} ~ 9q_d - \frac{3}{2}q_d^2 - pq_d\]Step 2. Write down the consumer’s first order condition.
\[9 - 3q_d - p = 0\]Step 3. Rearrange the first order condition so that we write the quantity demanded in terms of \(p\).
\[\begin{aligned} 9 - 3q_d - p &= 0 & \\ 3q_d &= 9 - p ~ ~ ~ ~ & \text{(Add 3q_d to both sides)} \\ q_d &= \frac{9-p}{3} ~ ~ ~ ~ & \text{(Divide both sides by 3)} \end{aligned}\]So \(q_d = \frac{9-p}{3}\) is the consumer’s demand curve.
Firms
Setup
A representative, price-taking firm decides how many units, \(q_s\), of a commodity to produce and sell at unit price \(p\). The firm’s total cost function for producing \(q_s\) units is \(c(q_s)\).
First order condition
The firm’s profit function is revenue minus cost, which can be written as follows:
\[\Pi(q_s) = pq_s - c(q_s)\]The firm’s optimization problem is therefore:
\[\max_{q_s} ~ pq_s - c(q_s)\]Taking the derivative with respect to \(q\) gives u the first order condition:
\[p - c^\prime(q_s) = 0\]or:
\[p = c^\prime(q_s) ~ ~ ~ ~ (\text{Eq.2})\]Equation (2) is the first order condition of the firm. It therefore shows the relationship between price and quantity supplied, because this is the equation the firm uses to determine how many units to produce.
In other words, equation (2) defines the producer’s supply curve! Technically, it’s the inverse supply curve because it shows \(p\) in terms of \(q_s\). But if we re-arrange the equation to write \(q_s\) in terms of \(p\), we’d get the supply curve.
Equation (2) also shows that at the producer’s optimal choice, price equals marginal cost, because \(c^\prime(q_s)\) is the additional cost to produce an incremental unit of output, when the current output is \(q_s\).
Example: Deriving a supply curve
A representative, price-taking firm decides how many units, \(q_s\), of a commodity to produce and sell at unit price \(p\). The firm’s total cost function for producing \(q_s\) units is:
\[c(q_s) = q_s + q_s^2\]
- Write down the firm’s supply curve.
Answer.
Step 1. Write down the firm’s profit function.
\[\begin{aligned} \Pi(q_s) &= \text{Revenue} - \text{Cost} \\ &= pq_s - c(q_s) \\ &= pq_s - (q_s + q_s^2) \\ &= pq_s - q_s - q_s^2 \end{aligned}\]Step 2. Write down the firm’s optimization problem.
\[\max_{q_s} ~ pq_s - q_s - q_s^2\]Step 3. Write down the firm’s first order condition.
\[p - 1 - 2q_s = 0\]Step 4. Rearrange the first order condition to write \(q_s\) in terms of \(p\).
\[\begin{aligned} p - 1 - 2q_s &= 0 \\ 2q_s &= p - 1 \\ q_s &= \frac{p-1}{2} \end{aligned}\]So \(q_s = \frac{p-1}{2}\) is the producer’s supply curve.
Market equilibrium
In equilibrium, quantity supplied has to equal quantity demanded. That is, \(q_s\) must equal \(q_d\).
This equation, \(q_d = q_s\), is called the “market clearing condition”. It ensures that there is no excess supply or demand in the market which would put downward or upward pressure on prices.
We now have the following three equations:
\[\begin{align} p = v^\prime(q_d) ~ ~ ~ ~ ~ ~ & \text{Consumer's First Order Condition} \\ p = c^\prime(q_s) ~ ~ ~ ~ ~ ~ & \text{Firm's First Order Condition} \\ q_d = q_s ~ ~ ~ ~ ~ ~& \text{Market Clearing Condition} \end{align}\]The first equation is the demand equation, and it comes out of the consumer’s optimizing behavior. The second equation is the supply equation, and it comes out of the firm’s optimizing behavior. The third equation is the market clearing condition.
Together, these equations give us three equations in three unknowns (\(p\), \(q_d\), \(q_s\)). We can solve the system of equations to find the equilibrium price and quantity in the market.
Example: Commodity market equilibrium
A representative, price-taking consumer decides how many units, \(q_d\), of a commodity to purchase at unit price \(p\). The utility they receive for purchasing \(q_d\) units at price \(p\) is:
\[u(q_d) = 9q_d - \frac{3}{2} q_d^2 - pq_d\]A representative, price-taking firm decides how many units, \(q_s\), of a commodity to produce and sell at unit price \(p\). The firm’s total cost function for producing \(q_s\) units is:
\[c(q) = q_s + q_s^2\]
- Write down the consumer’s demand curve.
- Write down the producer’s supply curve.
- Calculate the equilibrium price and quantity.
- Calcualte the consumer utility in equilibrium.
- Calculate the firm profit in equilibrium.
Answer.
Write down the consumer’s demand curve.
We already did this in the previous example.
\[q_d = 3 - \frac{1}{3}p\]Write down the producer’s supply curve.
We did this too.
\[q_s = \frac{1}{2} (p - 1)\]Calculate the equilibrium price and quantity.
To do this, write down the equilibrium condition (\(q_s=q_d\)) and solve for \(p\).
\[\begin{aligned} q_s &= q_d & \\ \frac{p-1}{2} &= \frac{9-p}{3} \\ 3(p-1) &= 2(9-p) \\ 3p - 3 &= 18 - 2p \\ 5p &= 21 \\ p &= 21/5 \\ p &= 4.2 \end{aligned}\]Then, plug \(p=4.2\) into either the supply or demand curve to get \(q\).
\[\begin{aligned} q_d &= \frac{9-p}{3} \\ &= \frac{9-4.2}{3} \\ &= 1.6 \end{aligned}\]So the equilibrium price and quantity are \(p=4.2\), \(q=1.6\).
Calculate the consumer utility.
To do this, just plug \(p\) and \(q\) into the utility function.
\[\begin{aligned} U &= 9q - \tfrac{3}{2}q^2 - pq \\ &= 9(1.6) - \tfrac{3}{2}(1.6)^2 - (4.2)(1.6) \\ &= 3.84 \end{aligned}\]Calculate the firm profit.
To do this, just plug \(p\) and \(q\) into the profit function.
\[\begin{aligned} \Pi &= pq - q - q^2 \\ &= (4.2)(1.6) - 1.6 - (1.6)^2 \\ &= 2.56 \end{aligned}\]Note that if you had drawn the supply and demand curves and calculated the consumer and producer surpluses using triangles, you’d find that the consumer surplus equals the consumer utility, and that the producer surplus equals the firm profit. In most circumstances, they reflect the same concepts.
Example: Another commodity market equilibrium
A representative, price-taking consumer decides how many units, \(q_d\), of a commodity to purchase at unit price \(p\). The utility they receive for purchasing \(q_d\) units at price \(p\) is:
\[u(q_d) = 9 \ln q_d - pq_d\]A representative, price-taking firm decides how many units, \(q_s\), of a commodity to produce and sell at unit price \(p\). The firm’s total cost function for producing \(q_s\) units is:
\[c(q) = \frac{1}{2} q_s^2\]
- Write down the consumer’s inverse demand curve. (The inverse demand curve simply writes \(p\) in terms of \(q_d\).)
- Write down the producer’s inverse supply curve.
- Calculate the equilibrium price and quantity.
- Calcualte the consumer utility in equilibrium.
- Calculate the firm profit in equilibrium.
Answer.
Write down the consumer’s inverse demand curve.
Step 1. Write down the consumer’s optimization problem.
\[\max_{q_d} ~ 9 \ln q_d - pq_d\]Step 2. Write down the consumer’s first order condition.
\[\frac{9}{q_d} - p = 0\]Step 3. Rearrange the first order condition to write \(p\) as a function of quantity demanded.
\[\begin{aligned} \frac{9}{q_d} - p &= 0 \\ p &= \frac{9}{q_d} \end{aligned}\]Write down the producer’s inverse supply curve.
Step 1. Write down the firm’s profit function.
\[\begin{aligned} \Pi(q_s) &= pq_s - c(q_s) \\ &= pq_s - \frac{1}{2}q_s^2 \end{aligned}\]Step 2. Write down the firm’s optimization problem.
\[\max_{q_s} ~ pq_s - \frac{1}{2}q_s^2\]Step 3. Write down the firm’s first order condition.
\[p - q_s = 0\]Step 4. Rearrange the first order condition to write \(p\) as a function of quantity supplied.
\[p = q_s\]Calculate the equilibrium price and quantity.
In equilibrium, the inverse supply curve and the inverse demand curve should equal each other at the same quantity. So set the inverse supply and demand curves equal to each other and solve for \(q\).
\[\begin{aligned} q &= \frac{9}{q} \\ q^2 &= 9 \\ q &= \sqrt{9} \\ q &= 3 \end{aligned}\]To get the equilibrium price, plug \(q=3\) into either the inverse supply curve or the inverse demand curve.
\[\begin{aligned} p &= q_s ~ ~ ~ ~ & \text{(inverse supply curve)} \\ &= 3 \end{aligned}\]So the equilibrium price and quantity are \(p=3\), \(q=3\).
Calculate the consumer utility and firm profit.
To get consumer utility, plug \(p\) and \(q\) into the consumer’s utility function. To get firm profit, plug \(p\) and \(q\) into the profit function.
\[\begin{aligned} U &= 9\ln q - pq \\ &= 9\ln(3) - 3\times3 \\ &= 0.8875 \end{aligned}\] \[\begin{aligned} \Pi &= pq - \tfrac{1}{2}q^2 \\ &= 3\times3 - \tfrac{1}{2}(3)^2 \\ &= 4.5 \end{aligned}\]
Marginal benefit equals marginal cost
In equilibrium, \(q_d = q_s\). Let’s write both quantities as simply \(q\). The consumer’s first order condition tells us that
\[p = v^\prime(q)\]and the firm’s first order condition tells us that
\[p = c^\prime(q)\]Combing them, we get:
\[v^\prime(q) = p = c^\prime(q)\]This shows us that in the equilibrium of the commodity market, the consumer’s marginal benefit equals the firm’s marginal cost. That is a standard result in introductory econ classes and is at the heart of why the competitive market equilibrium maximizes total surplus.
Big Picture Review
- We model a commodity market with two main groups of actors: consumers and producers (or firms).
- Consumers take the market price as given, and choose how many units of the commodity to purchase to maximize their utility.
- Their utility is modeled by something called the “utility function.” It represents the benefits, psychological or otherwise, that the consumer derives from purchasing the good.
- Producers take the market price as given, and choose how many units of the commodity to produce to maximize their profit.
- A firm’s profit is equal to its revenue minus its cost.
- The cost function determines how much it costs the firm to produce any given amount of output.
- The key drivers of a consumer’s decision making is their utility function and the price.
- The key drivers of a firm’s decision making is their cost function and the price.
- The demand curve arises out of the first order condition of the consumer’s optimization problem.
- The supply curve arises out of the first order condition of the producer’s optimization problem.
- In equilibrium, the price adjusts such that the quantity produced matches the quantity consumed.