There are two main ideas exploited on this problem set: 1) consumers allocate their expenditures across all possible goods such that for any two goods A and B, MU_A/P_A = MU_B/P_B, where MU_A and MU_B denote the utility the consumer gets from the marginal unit of good A and good B respectively and P_A and P_B denote the prices per unit of good A and good B respectively; 2) comparing budget sets and sometimes exploiting information on consumption choices, it is often possible to compare the well being of consumers at different points in time or under different programs.

The first idea is exploited in problems 1, 2, 4, and 6. The marginal utility condition is developed in pages 43-50 in the text. Review these pages. A critical feature of the formula MU_A/P_A = MU_B/P_B is that it refers only to the marginal unit of each good, where the marginal unit is meant to denote the last unit consumed and the very next unit that could be consumed (as long as units are defined as very small, you can think of these two units equivalently). It indicates that in order for consumers to be using their money to get the most possible utility, they should be consuming amounts of goods A and B such that the utility they get from the last unit consumed of good A (this is what is meant by MU_A) divided by the price of good A, P_A, is equal to the utility they get from the last unit consumed of good B (this is what is meant by MU_B) divided by the price of good B, P_B. You should review the logic behind this condition. Invent a numerical example in which it did not hold and analyze how you could reallocate your expenditures (this means consume more of good A and less of good B or less of good A and more of good B) to increase your utility. A convenient example is developed in problem 1. Notice that in problem 1 the setup discusses the satisfaction the consumer gets from the last beer and the last cheeseburger. Satisfaction is a synonym for utility. How would you represent the information provided in problem 1 in terms of the marginal utility-price formula? You can also apply the same formula to problem 2. Although this problem is not framed in terms of utility but in terms of average test scores, you should be able to see the parallel to utility maximization (maximization of the student’s average test score is equivalent to maximization of the student’s total test scores in all three subjects, and the total test score can be thought of analogously to total utility). The same marginal condition, expressed in terms of test scores, needs to be satisfied in order for the student to be allocating his/her limited hours to maximize his/her total (or equivalently average) test scores.

It is critical to understand that the formula MU_A/P_A = MU_B/P_B is a condition that will be satisfied only for the marginal unit consumed for each good. It does not apply to any other unit other than the marginal unit. In general, we cannot say anything about the utility the consumer gets from individual units consumed of each good other than the marginal unit, which means we cannot make any statements about the total utility or average utility per unit the consumer gets from all the units consumed of one good versus another. The only thing we can say is that for a consumer that maximizes his/her utility, the marginal utility divided by price for each good must be equal. The formula MU_A/P_A = MU_B/P_B can also be manipulated algebraically to express it as MU_A/MU_B = P_A/P_B. This illustrates that for a consumer that maximizes his/her utility, the utility from the marginal unit of good A relative to the utility from the marginal unit of good B must be equal to the ratio of the prices of good A to good B. If good A is more expensive than good B, what does this imply about the utility the consumer gets from the marginal unit of good A relative to the utility the consumer gets from the marginal utility of good B? The answer to this question underlies the reasoning you need to apply to problem 6. Suppose you are given information about the amount of two goods A and B a consumer consumes but you are not told the prices the consumer pays for the two goods. Can you infer from the formula MU_A/MU_B = P_A/P_B whether the consumer gets more utility from the marginal unit of good A than good B? If you knew the prices for goods A and B as well as the quantities of good A and B the consumer consumed, could you use the formula MU_A/MU_B = P_A/P_B to infer anything about the total utility the consumer got from all units of good A consumed versus all units of good B consumed? The answers to these questions underlie the reasoning you need to apply to problem 4.

Problems 3 and 5 involve comparing the well being of a consumer at two different points in time. This involves using the diagrammatic model of consumer choice developed on pages 60-70 of the textbook. The diagrammatic model involves budget lines and indifference curves. The budget line reflects the possible combinations of available goods the consumer could purchase given his/her income and the prices of the goods. In the case of two goods, denoted as goods A and B, if you knew the consumer’s income and the prices of the two goods, you could plot the budget line on a graph. This is discussed on pages 65-66 of the textbook. In many problems, you are not given specific information about income and prices and so cannot plot the exact locations of the relevant budget lines, but you are given information about income and prices at two different points in time that enables you to locate the relative positions of the consumer’s budget line at the two points in time. For example, you might be told that between time 1 and time 2, the consumer’s income doubled and the prices of the two goods he/she consumes did not change. If you represented the two budget lines in the same graph, this would enable you to determine that the consumer’s budget line at time 2 was parallel to the consumer’s budget line at time 1 and further from the origin than the consumer’s budget line at time 1. Alternatively, you might have been told that between time 1 and time 2 the consumer’s income doubled, the price of one good did not change, and the price of the other good tripled. In this case, you would be able to demonstrate graphically that the budget lines at times 1 and 2 crossed. You should experiment with some numerical examples of your own to corroborate these claims. Assume a specific income and specific prices of two goods A and B. In the first case, double income but leave the prices of the two goods unchanged and plot the original and new budget lines on the same graph following the discussion on pages 65-66. In the second case double income, leave one price the same and triple the other price and plot the two budget lines on the same graph. You should begin to see general patterns. You will need to exploit this kind of reasoning on both problems 3 and 5 and also on Quiz 2.

Sometimes plotting budget lines for two different times on the same graph is revealing about whether a consumer was better or worse off over time. This can be inferred when one budget line is entirely inside the other. This is called an unequivocal shift in real income. You should be able to see why you can order the consumer’s well being in this case using just the position of the consumer’s budget line in two different periods. If the budget lines for two different periods cross, you need more information to be able to tell whether the consumer was better off in one period or the other. Sometimes this can be resolved with information about the particular consumption bundle the consumer chose in one period. This is developed further on problem 5. The principle is simple. Suppose the consumer’s budget line at time 1 crosses the consumer’s budget line at time 2 when they are represented on the same graph. Suppose you knew the combination of goods the consumer consumed at time 1. This corresponds to a particular point on the time 1 budget line. If you located this point graphically and it was on or inside the budget line for time 2, it would indicate that the point chosen at time 1 could be purchased at time 2. In other words, the consumer had sufficient income at time 2, given the prices prevailing at time 2, to be able to purchase the consumption bundle chosen at time 1. If you could purchase what you used to purchase, could you be any worse off? As long as your preferences had not changed, you could not be worse off if you could still purchase what you used to purchase, as you would still have the option of duplicating your old choice. It is possible, however, that you might be better off given new consumption possibilities. These themes are developed on problem 5. They illustrate a general point: even if budget lines for two different time periods cross, often knowing something about actual consumption choices in one or both periods can be revealing about whether the consumer was better or worse off in the two alternative periods.