17.5 Step 3: Solve for the Equilibrium Price Ratio

We’ve now found that firms will choose to produce the total quantities \(\begin{aligned} Y_1^\star(p_1,p_2) &= {100 \over \sqrt{1 + \left({p_2 \over p_1}\right)^2}}\\ Y_2^\star(p_1,p_2) &= {100 \over \sqrt{1 + \left({p_1 \over p_2}\right)^2}} \end{aligned}\) and consumers, with budgets determined by the GDP $M = p_1Y_1^\star + p_2Y_2^\star$, will choose bundles \(\begin{aligned} X_1^\star(p_1,p_2) &= 100\alpha \sqrt{1 + \left({p_2 \over p_1}\right)^2}\\ X_2^\star(p_1,p_2) &= 100 (1-\alpha) \sqrt{1 + \left({p_1 \over p_2}\right)^2} \end{aligned}\) Our notion of equilibrium will set prices such that $X_1^\star(p_1,p_2) = Y_1^\star(p_1,p_2)$ and $X_2^\star(p_1,p_2) = Y_2^\star(p_1,p_2)$.

If we set $X_1^\star(p_1,p_2) = Y_1^\star(p_1,p_2)$ and solve, we get \(\begin{aligned} 100\alpha \sqrt{1 + \left({p_2 \over p_1}\right)^2} &= {100 \over \sqrt{1 + \left({p_2 \over p_1}\right)^2}}\\ 1 + \left({p_2 \over p_1}\right)^2 &= {1 \over \alpha}\\ {p_1 \over p_2} &= \sqrt{\alpha \over 1 - \alpha} \end{aligned}\) For example, if $\alpha = 0.64$, then \({p_1 \over p_2} = \sqrt{0.64 \over 0.36} = {4 \over 3}\) For example, if $p_1 = 4$ and $p_2 = 3$, then \(\begin{aligned} Y_1^\star(p_1,p_2) &= {100 \over \sqrt{1 + \left({3 \over 4}\right)^2}} = 80\\ Y_2^\star(p_1,p_2) &= {100 \over \sqrt{1 + \left({4 \over 3}\right)^2}} = 60\\ X_1^\star(p_1,p_2) &= 64 \sqrt{1 + \left({3 \over 4}\right)^2} = 80\\ X_2^\star(p_1,p_2) &= 36 \sqrt{1 + \left({4 \over 3}\right)^2} = 60 \end{aligned}\)

Note that while we only solved for equilibrium in one market, because there were only two goods, the equilibrium in the other market followed automatically. This result is known as Walras’ Law: if all markets but one are in equilibrium, then the remaining market must also be in equilibrium. Visually, this is because the equilibrium quantities are a point along the PPF; so if one coordinate is determined, the other one is as well.

We can see this in the graph below. You can also see that the same outcome occurs if $p_1 = 2$ and $p_2 = 1.5$, or any prices in a 4:3 ratio:

We can further see how the equilibrium price ratio will respond to consumer preferences, denoted by $\alpha$. Try changing $\alpha$ using the slider below; the price ratio will automatically adjust to keep the economy in equilibrium. The higher $\alpha$ is, the more consumers like good 1, and the more they’ll demand it, driving the relative price of good 1 higher and causing the economy to shift its production point to the right along the PPF.