6.3 Interpreting the Equilibrium Price Ratio
Interpreting the Equilibrium Price: Scarcity and Preferences
In the previous example, the two agents had Cobb-Douglas utility functions which put equal weight on each good, and there was twice as much good 1 as good 2; and the price ratio worked out so that the price of good 2 was twice that of good 1 (i.e. $p_1/p_2 = 1/2$). This is a feature of competitive equilibrium with Cobb-Douglas preferences, which can be used to illustrate some nice aspects of how the price ratio in competitive equilibrium relates to agents’ relative preferences for the two goods and the goods’ relative scarcity. To show why, we’ll have to do a bit of algebra.
Let’s go back to looking at both $p_1$ and $p_2$. Suppose each agent has normalized Cobb-Douglas preferences of the form \(u(x_1,x_2) = \alpha \ln x_1 + (1-\alpha) \ln x_2\) As we’ve shown many times, this means that each agent will spend fraction $\alpha$ of their income — in this case, the value of their endowment — on good 1, so their gross demand for good 1 will be \(x_1^* = \alpha \left ({p_1e_1 + p_2e_2 \over p_1}\right) = \alpha e_1 + \alpha e_2 \times {p_2 \over p_1}\) Subtracting $e_1$ from both sides, we can see that their net demand for good 1 is \(x_1^* - e_1 = -(1-\alpha)e_1 + \alpha e_2 \times {p_2 \over p_1}\) Now suppose that agent A’s value of $\alpha$ is $a$, while agent B’s value of $\alpha$ is $b$: \(\begin{aligned}u^A(x_1^A,x_2^A) &= a \ln x_1^A + (1-a) \ln x_2^A\\u^B(x_1^B,x_2^B) &= b \ln x_1^B + (1-b) \ln x_2^B\end{aligned}\) If we again assume that A is the supplier and B is the demander, A’s net supply and B’s net demand are given by \(\begin{aligned}s_1^A(p) &= (1-a)e_1^A - ae_2^A \times {p_2 \over p_1}\\d_1^B(p) &= -(1-b)e_1^B + be_2^B \times {p_2 \over p_1}\end{aligned}\) Equating these and solving for the price ratio $p_1/p_2$ gives us \(\begin{aligned} (1-a)e_1^A - ae_2^A \times {p_2 \over p_1} &= -(1-b)e_1^B + be_2^B \times {p_2 \over p_1}\\ (1-a)e_1^A + (1-b)e_1^B &= {p_2 \over p_1}\left(ae_2^A + be_2^B\right)\\ {p_1 \over p_2} &= {ae_2^A + be_2^B \over (1-a)e_1^A + (1-b)e_1^B} \end{aligned}\) OK, if you’ve tuned out because of all the algebra (full disclosure: I would!), just look at that last expression, which is our equilibrium price ratio in terms of four variables:
- Agent A’s preferences put a weight of $a$ on good 1 and $1 -a$ on good 2
- Agent B’s preferences put a weight of $b$ on good 1 and $1 -b$ on good 2
- $e_1^A$ and $e_1^B$ are the endowments of good 1
- $e_2^A$ and $e_2^B$ are the endowments of good 2
So how does the equilibrium price ratio change with each of these?
- An increase in $a$ or $b$ will increase $p_1/p_2$. In other words, the more agents like good 1 relative to good 2, the higher its relative price will be.
- An increase in $e_1^A$ or $e_1^B$ will decrease $p_1/p_2$. In other words, the less scarce good 1 is, the lower its relative price will be.
- An increase in $e_2^A$ or $e_2^B$ will increase $p_1/p_2$. In other words, an increase in the amount of good 2 makes good 1 relatively more scarce, and therefore increases the price of good 1 relative to good 2 (or decreases the price of good 2 relative to good 1)
There are a few interesting special cases:
- Equal preferences: As the example we used to derive equilibrium above, if all agents put an equal weight on each good, so $a = b = 1-a = 1-b = {1 \over 2}$, the equilibrium price is determined entirely by the relative amounts of the two goods (i.e. the dimensions of the Edgeworth Box): \({p_1 \over p_2} = {e_2^A + e_2^B \over e_1^A + e_1^B} = \frac{\text{Total amount of good 2}}{\text{Total amount of good 1}}\)
- Equal endowments: Suppose $e_1^A = e_2^A = e_1^B = e_2^B$, so the Edgeworth Box is a square, and the original allocation is in its center. In this case, the equilibrium price is determined entirely by the relative preferences of the agents: \({p_1 \over p_2} = {a + b \over (1-a) + (1-b)} = \frac{\text{Sum of preference weights on good 1}}{\text{Sum of preference weights on good 2}}\)
More generally, the equilibrium price ratio is jointly determined by the relative preferences for the two goods, and their relative scarcity. This result is also true for utility functions that aren’t Cobb-Douglas, but they’re not necessarily shown as clearly by the formula for the equilibrium price.
Note: I’m going to write a bit more for Thursday’s lecture, but wanted to publish these notes for now. Stay tuned! The quiz won’t cover anything else though… :)