17.4 Step 2: Consumer Optimization

Once firms produce the bundle that maximizes their profits, their entire proceeds accrue to individuals: the firm owners who get the profits, and the workers who earn wages. Therefore the total income earned by individuals in this society from all sources must be the same GDP we just calculated: \(M(p_1,p_2) = 100\sqrt{p_1^2 + p_2^2}\) Let’s assume that consumers’ preferences over the total amount of goods 1 and 2 consumed, $X_1$ and $X_2$, may be represented by the utility function \(u(X_1,X_2) = \alpha \ln X_1 + (1-\alpha) \ln X_2\) This is a little different than saying that every single consumer has this utility function, or even that a “representative consumer” has this utility function; rather, it’s saying that the market demand functions for goods 1 and 2 would be consistent with a single consumer who had all the money in society and these preferences. By the Cobb-Douglas rule, we know that such a consumer would spend fraction $\alpha$ of their income on good 1, and fraction $1-\alpha$ on good 2: \(\begin{aligned} X_1^\star(p_1,p_2) &= {\alpha M(p_1,p_2) \over p_1} = 100\alpha \sqrt{1 + \left({p_2 \over p_1}\right)^2}\\ X_2^\star(p_1,p_2) &= {(1 - \alpha) M(p_1,p_2) \over p_2} = 100 (1-\alpha) \sqrt{1 + \left({p_1 \over p_2}\right)^2} \end{aligned}\) Again, note that this optimal bundle depends only on the price ratio $p_1/p_2$, just as with the firms’ optimal choice.

We can see this bundle in the diagram below. For reference, the PPF is shown, along with the point that firms will choose to produce (shown as a green dot that you’ll see if you change the prices):

Try changing the parameters $p_1$, $p_2$, and $\alpha$ in the diagram above. Changing the prices affects the budget line; changing $\alpha$ affects the optimal point along the budget line.

Note that, in general, the ideal consumption bundle lies outside the PPF. This doesn’t mean that consumers actually consume a bundle outside of the PPF. It means that if prices aren’t in equilibrium, they will want to buy a bundle outside the PPF; but there will be a shortage of one good, causing upward pressure on the price of that good. To see why, let’s look at how we calculate the equilibrium price, and then see what happens when prices are not in equilibrium.