# 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.