Budget Constraints and Utility Maximization

In this section, we will assume that \(\alpha = 0.5\) (i.e. the utility function is: \(u(x_1, x_2) = x_1^{0.5}x_2^{0.5}\)).

Now we introduce the concept of money into our model. Consumers face a budget constraint when choosing to maximize their utility. Given an income \(M\) and prices \(p_1\) for good \(x_1\) and \(p_2\) for good \(x_2\), the consumer can at most spend up to \(M\) for both goods:

\[M \geq p_1x_1 + p_2x_2\]

Since goods will always bring non-negative marginal utility, consumers will try to consume as many goods as they can. Hence, we can rewrite the budget constraint as an equality instead (since if they have more income leftover, they will use it to buy more goods).

\[M = p_1x_1 + p_2x_2\]

This means that any bundle of goods \((x_1,x_2)\) that consumers choose to consume will adhere to the equality above. What does this mean on our graph? Let’s examine the indifference curve plots, assuming that \(M = 32\), and \(p_1 =2\) and \(p_2 = 4\).