# Binary choices in consumer theory

Back when I was a teaching assistant, we were teaching utility theory and how it relates to price determination. So we were looking at continuous goods (e.g. "food") so as to get marginal utility. But a student asked about buying a bed; the choice is zero or one, a person doesn't need two or more. So is it possible to determine marginal utility in this case, and if so how?

Yes, it is. First, what you describe is not as much binary choice, but situation where you have discrete quantities where any quantity higher than 1 does not bring any benefits (a person can have 2 beds just the second bed is useless). You can calculate marginal utility for such case normally how you would do for other goods that come in discrete quantities (I think you might be abusing the word continuous here since many intro micro courses, assume food is discrete not continuous).

For a discrete quantity, marginal utility, is the utility of the additional of the last unit consumed, hence in discrete case

$$MU = U(n+1) - U(n)$$

Where $$n$$ is the number of units already consumed.

For example, using slices of pizzas, if 0 slices of pizza give you 0 utility, 1 slice of pizza gives you utility of 10 and 2 slices of pizza gives you utility of 15 then we have:

Slices of Pizza  U   MU

0                0   NA
1               10   10
2               15    5


Now if we assume that people only get utility from just having one bed and would never want to buy more, so we assume for $$U(bed)$$ that $$U(0)=0$$, $$U(1)=10$$, $$U(2)=10$$ and $$U(3)=10$$ ... then we have

Beds   U   MU

0      0   NA
1     10   10
2     10    0
3     10    0
...


As you can see above the additional beds just do not add any additional utility so MU is always zero (except from moving from zero beds to one bed), but you can certainly calculate it for discrete case.

• Perhaps it's more appropriate to write $MU=U(x)-U(x-1)$, assuming of course that $1$ is the smallest incremental unit? Or, $MU=U(x)-U(x-\Delta x)$, where $\Delta x$ is such a unit. Nov 2 '21 at 16:51
• @HerrK. You are right my notation was sloppy I adjusted it, thanks for the note
– 1muflon1
Nov 2 '21 at 17:02
• Thanks. So I guess the continuous approach (checking the slope of an indifference curve) wouldn't work here. Maybe we jump from one indifference curve (bed=0) to another indifference curve (bed=1)? Nov 2 '21 at 19:28
• @Daniel you are mixing stuff. Slope of indifference curve is marginal rate of substitution not marginal utility. MU for continuous function is slope of utility MU=U’. Slopes are given by derivatives of functions and for discrete function you can’t compute derivative as derivatives require function to be continuous. However, all these economic concepts have their discrete counterpart. The slope of a utility function U’ is also instantaneous rate of change, the discrete case U(n+1)-U(n) is in a sense a discrete counterpart to that
– 1muflon1
Nov 2 '21 at 19:58
• I'm a bit unclear on why one would want to talk about marginal utilities if choice is binary. Can you say more than $A \succsim B \iff U(A) \geq (B)$? i.e. isn't the problem just trivial? Nov 2 '21 at 20:44