Calculating the Opportunity Cost of a Good

Calculating the opportunity cost of a good can seem tricky and I remember struggling with this. So, I developed a method (that has never failed me). I hope you find it useful.

Study Suggestions to Help you to Master the Calculation, Remember, and Do Awesome

1. Use a sheet of paper and write each step. If you can, print this page and work it out with me and then by yourself (ensure you answer all questions for review).
   
2. Ask yourself often “do I know why I did this?”. Ensure you know the answer.
   
3. Avoid falling into the trap of reading and rereading the example as this will lead you to misinterpret remembering what comes next for the ability to solve the problem on your own—this is what you need in order to solve different problems.

The Problem

Suppose a country produces only apples and bananas. Moving from one point on its production possibilities frontier (PPF) to another means 5 apples are gained and 4 bananas are forgone. What is the opportunity cost of an apple?

Opportunity Cost of an Apple

An important detail in the question is that, for this country, moving along the PPF, producing 5 apples comes at a cost of 4 bananas, i.e. in order to produce 5 apples, it must forego 4 bananas. In other words, having 5 apples more means having 4 bananas less. Now that we just stated the problem in three different ways, let’s simplify all these words and write it more neatly like this:

+ 5 apples = – 4 bananas   (1)

If you just thought that we’re not supposed to compare apples and oranges, or apples and bananas for that matter, you are right, but the expression above is not meant to be literal. The purpose of (1) is simply to represent the tradeoff the country faces, i.e. 5 apples more, or + 5 apples, means 4 apples less, or – 4 bananas. There’s no such thing as a free lunch, right?

***

Questions for review:

1. Why does (1) represent a tradeoff?

2. Why can we write that 5 apples are equal to -4 bananas (they so clearly are not, duh)?  

***

Before we get to calculating the opportunity cost of an apple, let’s take a little detour that will help clarify the method.

Suppose I am interested in how many bananas the country must forego in order to have, not 5, but half that number of apples, 2.5 apples. We can write this as

+ 2.5 apples = – ??? bananas   (2)

The expression in (2) still represents the tradeoff the country faces, more apples means fewer bananas. But now, on the left-hand side, we have half the apples we had before, 2.5 apples instead of 5 apples. What does this imply for the number of bananas the country must forego in order to have half the apples?

Looking at the ??? in (2) you are probably already thinking that to gain half the apples, the country must now forego half the bananas it had to forego before. You are right:

+ 2.5 apples = – 2 bananas   (3)

What did we do? Well, we wanted to know how many bananas the country would have to forego to have half the apples it had before. So, we divided both sides of (1) by 2 (to get the “half”). Expressions (1) and (3) are equivalent.[1]

This we what we did: 
We divided both sides of (1) by 2: 
+ 5 apples = - 4 bananas
÷ 2           ÷ 2
and got 
+ 2.5 apples = - 2 bananas

Now let’s go back to the initial question… the opportunity cost of 1 apple. That is the number of bananas the country foregoes if it wants to produce 1 more apple, or

+ 1 apples = – ??? bananas   (4)

So, what number should replace the ??? in (4)? How do we go from (1) to (4) keeping the identity true?

Think for yourself.

 

Think for yourself. 

 

Think for yourself.

Well, it makes sense that for the country to have one-fifth of the apples, it must forego one-fifth of the number of bananas it had to forego originally:

1 apple = – 4/5 bananas   (5)

This means that the opportunity cost of an apple is 4/5 bananas or 0.8 bananas.

This we what we did:
We divided both sides of (1) by 5:
+ 5 apples = - 4 bananas
÷ 5           ÷ 5
and got
+ 1 apples = - 4/5 bananas

***

Questions for review:

1. How many bananas should the country forego if it wanted to produce 10 more apples?

2. How many bananas should the country forego if it wanted to produce 200 more apples?

***

Opportunity Cost of a Banana

Let’s look at the same problem but now from a different angle. Instead of calculating the opportunity cost of an apple let’s calculate the opportunity cost of a banana.

Let’s write (1) again…

+ 5 apples = – 4 bananas   (1)

Although that minus sign in (1) is still helping us to think about the problem in terms of tradeoffs, now we actually want to know how much it will cost, in terms of apples, to have 1 more banana. Let’s write that idea instead:

– 5 apples = + 4 bananas   (6)

***

Questions for review:

1. How does (6) differ from (1)?

2. How are (6) and (1) similar?

***

To give us more practice, let’s repeat the exercise we did before. Ask yourself. How many apples must the country forego to produce only half that number of bananas, so 2 more bananas instead of 4 more bananas?

We’d write that as

– ??? apples = + 2 bananas   (7)

The expression in (7) still represents the tradeoff the country faces, more bananas means fewer apples. But now, on the right-hand side, we have half the bananas we had before or 2 bananas instead of 4 bananas. So… what does this imply for the number of apples the country must forego in order to have half the bananas?

Looking at the ??? in (7) you are probably already thinking that to gain half the bananas, the country must now forego half the apples it had to forego before. You are right:

– 2.5 apples = + 2 bananas  (8)

Here, we wanted to know how many apples the country would have to forego to have half the bananas it had before. So, we divided both sides of (6) by 2 (to get the “half”). Expressions (6) and (8) are equivalent (just like expressions (1) and (3) were; actually, they all are).

This we what we did:
We divided both sides of (6) by 2:
- 5 apples = + 4 bananas
÷ 2           ÷ 2
and got
- 2.5 apples = + 2 bananas

Yep, that makes sense. If, to produce 4 more bananas the country would have to forego producing 5 apples, then to produce only 2, half of 4, the country would have to forego half the apples as well, 2.5 apples. This is the exact result we had before, in (3).

Now let’s think about how many apples the country must forego to produce just 1 more banana; this is the opportunity cost of a banana:

– ??? apples = + 1 bananas   (9)

Now what is the number we need to divide (6) to get to just one banana?

Think for yourself.

 

Think for yourself.

  

Think for yourself.

Well, it makes sense that for the country to have one-fourth of the bananas, it should forego one-fourth of the number of apples it had to forego originally:

– 5/4 apples = + 1 banana   (10)

This we what we did:
We divided both sides of (6) by 4:
- 5 apples = + 4 bananas
÷ 4           ÷ 4
and got
- 5/4 apples = + 1 bananas

This means that the opportunity cost of 1 banana is 5/4 of an apple, or 1.25 apples, an apple and a quarter.

So, although we cannot really compare apples and oranges, when we are told how we must give up one good for the other, we can effectively use that as a basis for our calculations.

Something Interesting

The opportunity cost of 1 apple is 0.8 bananas. And the opportunity cost of 1 banana is 1.25 apples. These numbers are related. Can you see how?

They are the inverse of each other. If you must forego 0.8 bananas to have 1 apple, it must be the case that you must forego 1.25 apples (1/0.8) to have 1 banana.

[1] For the sake of completeness, note that we assume constant opportunity costs throughout, meaning that the tradeoff between apples and bananas does not depend on the scale, i.e. it does not matter if I am talking about a few items of fruit or a few thousand, the terms of the exchange remain constant. Graphically, the PPF would look like a straight line (instead of a curve).

Practice

1. In one afternoon, John can either wash 3 cars or mow two lawns. What is the opportunity cost of washing one car? What is the opportunity cost of mowing four lawns; how long will this take?
2. It takes Sophia two hours to clean her bedroom. It takes Sophia four hours to clean her kitchen. What is the opportunity cost of cleaning her bedroom?

Note how question 2 differs from question 1. Specifically, I do not tell you how Sophie trades off cleaning one room for another, e.g. I do not say in x number of hours, Sophia can clean y bedrooms or clean z kitchens. Instead, I tell you how many hours it will take her to clean each room. To set up the equality we’ve been using we must ensure each side of the equality represents the same units of time, that is the only way to compare apples and oranges, when they represent the same amount of resources.

Did you get all the way here? Congratulations! You deserve a Mars bar!

© 2018 Joana Girante. All rights reserved.