In Example 4.3 we used a specific indirect utility function to illustrate the lump sum principle that an income tax reduces utility to a lesser extent than a sales tax that garners the same revenue. Here you are asked to: a. Show this result graphically for a two-good case by showing the budget constraints that must prevail under each tax. (Hint: First draw the sales tax case. Then show that the budget constraint for an income tax that collects the same revenue must pass through the point chosen under the sales tax but will offer options preferable to the individual.) b. Show that if an individual consumes the two goods in fixed proportions, the lump sum principle does not hold because both taxes reduce utility by the same amount. c. Discuss whether the lump sum principle holds for the many-good case too.

First, we need to establish the budget constraint for the sales tax case. The sales tax will increase the prices of both goods, so the budget constraint will have a steeper slope. Let's represent the goods on the x and y axes and draw the budget constraint for this case. Next, we need to establish the budget constraint for the income tax case. The income tax decreases the individual's income while keeping the prices of goods constant. Thus, the budget constraint will have the same slope as before, but it will be shifted inward due to the reduced income. Importantly, this new budget constraint must pass through the point chosen under the sales tax case, as they generate the same revenue for the government. Now we can demonstrate that options exist on the budget constraint for the income tax case which lie above the budget constraint for the sales tax case. Since individuals choose the bundle of goods that maximizes their utility given their budget constraint, these preferable options indicate that the income tax reduces utility to a lesser extent than the sales tax.

When an individual consumes goods in fixed proportions, their utility depends only on the quantity of the consumed goods, not on the prices. In this case, if their income is reduced due to an income tax, it will cause them to consume a smaller quantity of the goods. Conversely, if the prices of the goods increase due to a sales tax, they will also consume a smaller quantity. Since the utility is determined solely by the quantity of consumed goods in fixed proportions, both the income tax and sales tax will reduce the utility by the same amount in this context, as they lead the consumer to purchase a smaller quantity of the goods. The lump sum principle does not hold in this case.

In the many-good case, it becomes more complicated to analyze the lump sum principle, as the effects of taxes will depend on the individual's preferences and consumption patterns. However, the general concept remains that income tax reduces utility to a lesser extent than a sales tax that collects the same revenue. If the good is a normal good, and as income decreases due to the income tax, the consumption of that good will likely decrease. However, an individual can reallocate their budget among the many goods to limit the decrease in total utility. In contrast, the sales tax increases the prices of all goods, limiting the ability to find better alternatives to maintain utility. In conclusion, the lump sum principle is likely to hold in the many-good case, but the exact outcome depends on various assumptions and individual preferences.