My Challenge in Econ 101 explanation of Marginal Benefit = Marginal Cost

Question

In Econ 101 textbooks, there are lots of examples and emphasis on marginal analysis leading to the greatest equation of all, $MB=MC$.

My challenge is the following and wonder if anyone had a similar issue or a way to get over it:

One of the textbooks examples use consuming pizza or water. First unit brings you the biggest joy. Then it diminishes. So student would understand each additional unit of pizza wouldn't bring the same amount (e.g. initial v. when you are full) utility. But then there comes a time where you are quite full and feel satisfied. At this point, additional unit of consumption would probably make you feel sick or vomit, so it is not "worth the money" to spend on.

The trouble with this example is that it illustrates at margin, benefit is not constant. There comes a time when you would stop consuming. But this doesn't really shed light on why $MB=MC$? How can I do a better job explaining that this "equality" is embedded in all these stories? Is it perhaps the stuffed/eat-until-vomit is a corner solution example?

For pedagogical reason, if you are better at explaining these concepts, please share how you would go about this with an example both interor and corner solution in ECON 101 textbook situation.

My strategy was from example to the graph and to show why the equality should hold leading to connecting the dots on the first order conditions of utility maximization... But this equality is hard to really spell it out on 101 level, that's my challenge.

Thanks!

Answer 1

I suggest this graphic illustration.

You can first plot your utility U as a function of the quantity x of pizza eaten. Utility is expressed in \$. Plot a positive concave function starting form the origins (0 pizza leads to 0 utility), going up to a maximum (at $x=4$ let's say) and then slightly decreasing after $x=4$. The decreasing part means that you are loosing utility by eating too much pizza. Now assume a unit of pizza costs \$1.

You can now plot a second graph illustrating the first order condition. Plot first a decreasing curve corresponding the marginal utility (a straight line if you have chosen a quadratic form to the utility function) starting from $(x,y)=(0,5)$ (or any positive $y$), going down and crossing the X-axis at $x=4$ and being negative after $x=4$. Then, plot the horizontal line $y=1$. The intersection of these two curves gives the optimal quantity of pizza to eat, $x^*$, given the cost.

For $x<x^*$, you should eat more pizza because the marginal unit of pizza eaten worth more than it costs. For $x>x^*$, you have eaten too much pizza comparing to what it costs you. You can insist on two other points. First, imagine you do not pay as you eat pizza but you are invited at a party and food is free. In this case, the actual $MC$ is 0. You will then eat until the point you get sick, meaning $x=4$. In other words, you eat as long as the marginal utility of eating is positive. Second, back to the case in which a pizza costs \$1, you can show that $x^*<4$. It means that you stop eating before the point you get sick. You stop eating before because you have no interest in eating a marginal share of pizza that yields you \$0.5 of utility for instance but costs you \$1.

Edit In the example I give, the marginal cost of eating pizza is monetary, meaning money that you pay. The marginal benefit is the marginal utility received from eating pizza (possibly negative), it encompasses both the "good feeling" of alleviating your hunger but also the "bad feeling" of eating fat and damaging your body. If you want to define $MC$ as the sum of "bad feeling" and monetary cost, and $MB$ as the positive feeling, you need to specify how both feelings increase/decrease when you eat pizza. This is more demanding in terms of hypotheses than using the utility-based approach, and maybe too ad-hoc.

Answer 2

So, you assume a finite stock of money units and also that the only thing any consumer desires is units of happiness. That is, consumers are utility maximizers and they have utilities that are functions of happiness only.

Now, if you impose some arbitrary value - say, one unit of money - for one unit of happiness you could construct any silly example. Say, a frat guy loves beautiful women. He can win the affection of a beautiful woman by taking her on a date that costs 10 units of money. The first day late with a beautiful girl gives him 20 units of happiness. Clearly, the date is worth it! This is beacause he is getting 2 units of happiness per unit of money which has, more or less, doubled the value of his money. Suppose he keeps dating beautiful women and that the 5th date - that costs 10 units - gives him exactly ten units of happiness. Now, suppose he knows date 6 will only give 8 units of happiness. The marginal benefit of the date is too low. He keeps his money and instead trades his units of money 1-1 for units of happiness until he depletes his remaining stock of money.

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Something like that. Any arbitrary example where you can clearly demonstrate that if $MB<MC$, you're "losing" something by continuing to consume that good and $MB>MC$ means you're "gaining" something by continuing to consume that good.

Also - I have no graphing tools (typing from phone) but this example makes use of constant MC and so gives a good visual for what happens when MB is above and/or below MC by presenting two very easily identified regions where the MC curve partitions the graph. Above the curve is "good" and below is "bad".

This usually works well for me.