Here We Will Do a Numerical Example to Show That the Consumer Equilibrium Makes Sense

Title: A utility maximization example and what happens after the price of one good falls and nominal income is lowered to keep real income constant

Here we will do a numerical example to show that the consumer equilibrium makes sense.

Suppose that the consumer has $8.00 in income (recall that we assume they will spend all of it). They can purchase two goods, A and B. Here are their prices:

PA = $2

PB = $.50 (or 50 cents)

How much will they buy of each good? We need to know how many utils the consumer gets from each good. This is given in the tables below. Notice that as their consumption increases for each good, the marginal utility falls.

Assume that after the 13th unit of good B that you get no more utility.

Notice how MU is calculated for each good. The MU of the first unit of good A is 40 because until you consume one unit, you have no utils. The MU for the second unit is 30.That is because 70 – 40 = 30. The second unit causes utility to rise by 30.

But, we need to know MU divided by price for each good. That is the last column, in green. Each number in the MU column is divided by the price of the good. For the first unit of good A, we have 40 divided by 2, which = 20. For the first unit of good B, we have 15/.5 = 30.

The consumer will by 3 units of A and 4 units of B to maximize utility (or the total number of utils when they spend all of their $8 income). Why? Because at those quantities we have

MUA/PA = MUB/PB = 10

For both good A and good B this is ten (see the last column in green at QA = 3 and QB = 4).

This is consistent with the consumer equilibrium equation. Also, all $8.00 of the consumer’s income is spent since

$2 times 3 units of A = $6 and $.50 times 4 units of B = $2. Adding the $6 and the $2 gives us $8.

There are actually other combinations of QA and QB that will cost exactly $8, but they will not give the consumer as many utils as the combination we have already found. The table below shows how many utils the consumer gets with QA = 3 and QB = 4 and another combination that spends exactly $8 (why does this other combination spend exactly $8?).

You can verify these numbers from the other tables above. Notice that the consumer does not get as many utils from QA = 2 and QB = 8 as from QA = 3 and QB = 4. Notice also that with QA = 2 and QB = 8, the consumer equilibrium is not met. That is

MUA/PA does not equal MUB/PB

(you can verify this by looking at the tables).

So we have verified that the consumer maximizes utility when the quantities of good A and good B that they purchase makes

MUA/PA = MUB/PB true.

Utility theory and the law of demand

What if the price of good A in the previous example falls to $1.00? Then the table for A has to be redone. We get

Now the consumer will buy 5 units of A and 6 units of B. This will cost exactly $8.00 and it will satisfy the consumer equilibrium equation (MUA/PA = MUB/PB). Check the tables to make sure.

But this tells us why the Law of Demand is true.

Here is what happened.

1. Before the price decrease, we had

MUA/PA = MUB/PB

(when QA = 3 and QB = 4)

2. Then price fell.

3. This means that we must now have

MUA/PA > MUB/PB

Since a lower PA means that MUA/PA has risen since its denominator has fallen. And since nothing is happening to the B side, the A side must be greater.

4. But, we already know that when #3 is true, and

MUA/PA > MUB/PB

The consumer will buy more A.

But notice that we only decreased the price of A (all other factors were held constant-nothing else changed like income or the price of B) and it led to the quantity of A increasing. This is exactly what The Law of Demand says. So the real reason why the The Law of Demand is true is because of utility theory.

Maximizing Utility: The rest of the story

Did you realize I pulled a fast one on in the last class? Well, yes, of course, you’re smart. When we lowered the price of A from 2 to 1, all I talked about was the consumer buying more A (to show you the law of demand). You might have noticed that he bought more B, too. But wait, you say, earlier, on the board, I said when

MUA/PA > MUB/PB

the consumer buys more A and less B. So why not in the problem we worked on in the table?

Well, the answer is that when we lower the price of A the consumer’s real income increases (which, you remember, of course, from MACRO, for those of you who took it). To satisfy the law of demand, the real income must be held constant (along with everything else). How do we do that in this case?

We lower the person’s nominal income form $8 to $5 (after the fall in the price of A from 2 to 1). At $5, they can still buy 3 of A and 4 of B (their original purchases). This is because

A) 1*3 = 3 and B) .5*4 = 2 and 3 + 2 = 5.

So we keep the real income constant and the consumer can still buy the original QA and QB. Being able to buy the exact same quantities of goods is what keeping real income constant means.

Now, with the price of A = 1 (and the price of B = .5) and income = 5, how much will the consumer buy of good A and good B? The consumer must now spend exactly $5 and the consumer optimum

MUA/PA = MUB/PB

must still be true. How will we do that?

The consumer could buy 4 of A and 2 of B (which would spend exactly $5-but then MUA/PA < MUB/PB which you can see from the table(10<16)-meaning they should buy less A (less than 4) and more B (more than 2)). 4 of A is too much. A has to be more than 3, so it will increase (since its price fell) and it cannot increase all the way to 4. But how much exactly?

The consumer will buy 3.57143 of A and 2.85714 of B. Exactly $5 is spent. A) 1*3.57143 = 3.57143 and B) .5*2.85714 = 1.42857 and 3.57143 + 1.42857 = 5. But is the consumer optimum true? This part is tricky, but it will be no sweat for smart people like you. You know that that the consumer equilibrium has to be true. If a consumer buys 3.57143 of A, his MUA will be 14.2857. Why? His MUA at QA = 3 is 20 and his MUA at QA = 4 is 10. By buying about 3.57 of A, he moves 57% of the way from 20 to 10. And 57% of 10 is 5.7. And 20 – 5.7 is 14.3 (about 14.2857). We could do something similar with B. The MUB at QB = 2.85714 is 7.14286. We have moved about 86% of the way from 2 units of B to 3. That means we have gone about 86% of the way from 8 (MUB at QB = 2) to 7 (MUB at QB = 3).

So now, the optimum becomes

14.2857/1 = 7.14286/.5 OR

14.2857 = 14.28572 So the consumer optimum is satisfied and all $5 are spent. Also notice that QA has increased (from 3 to 3.57) and QB has fallen (from 4 to 2.86). This is what is supposed to happen. More A, less B.