Opportunity Costs, Absolute & Comparative Advantage, and Gains from Trade – Another Example

By Anne Alexander

To put some of the most important concepts of this lesson together, here’s an example to give you more to work with.

Suppose that there are two brothers, Click and Clack, who each own an automobile service shop. Both Click and Clack have two skills, repainting cars and rebuilding engines. The following table shows the number of days it takes each brother to repaint one car or rebuild one engine:

Notice that it takes Click less time than his brother BOTH to repaint cars and to rebuild engines. Therefore, Click has an absolute advantage in both repainting cars and in rebuilding engines.

Now, let’s use the table to look at each brother’s opportunity costs of repainting a car and then each brother’s opportunity cost of rebuilding one engine. In this part, we will see that there are a couple of little "tricks" that can be used to figure out opportunity costs of doing one activity in terms of losing time devoted to another activity.

Calculating Click’s Opportunity Costs

First of all, let’s focus on Click’s opportunity cost of repainting one car. In order to repaint that one car, Click has to spend 1½ days of his time. This is 1½ days LESS that he’d have to spend on rebuilding engines. In that 1½ days Click would spend repainting a car, he could instead have rebuilt 1½ cars. How do we know this? One of the days spent by Click repainting could instead be spent by him rebuilding an entire car. In the remaining half day he spends on repainting, another half car could be rebuilt. Thus, the opportunity cost to Click of repainting one car is 1½ cars rebuilt.

Now, what if Click decided instead to rebuild one engine? In the language of opportunity costs, since it takes him one day to rebuild the engine, that’s one less day he has to spend repainting cars. During the one day that he spends rebuilding an engine, he could have repainted 2/3 of a car. How do we know this? We can use one of the little tricks that were referred to above. To calculate Click’s opportunity costs for rebuilding one engine, first put the number of days it takes to rebuild an engine (the activity Click does here) in the numerator (top) of a fraction. Next, put the number of days it takes to repaint a car (the next best use of his time) in the denominator (bottom) of the same fraction. Here, the resulting fraction is (1) / (1½). Then, use the "invert and multiply" rule for dividing fractions (you learned this in junior high!) or your calculator to find the opportunity cost to Clack of rebuilding engines. Here, this means take (1) / (1½)

= (1) / (3/2) = 1 * (2/3) = 2/3. This trick can always be used to find the opportunity cost of one activity in terms of the next best use of a person’s time. The general steps are (1) put the amount of time needed to do the activity in the top of a fraction (2) put the amount of time needed to pursue the other activity in the bottom of the fraction and (3) find the resulting number.

To look intuitively at Click’s opportunity cost of rebuilding an engine, think about it this way. To rebuild one engine, Click needs 1 day. To complete one painted car, on the other hand, Click needs 1½ days. Therefore, if Click were to spend 1 day on repainting, he can only complete 2/3 of one repainting job. Thus, the opportunity cost to Click of rebuilding one engine (which takes him one day) is 2/3 of one car repainted.

Another good thing to notice now is that Click’s opportunity cost for rebuilding an engine is the reciprocal or reverse fraction of his opportunity cost for repainting one car. Remember, the opportunity cost of repainting one car was 1½ = 3/2 of a rebuilt engine. Now on your calculator or using the "invert and multiply" rule for dividing fractions, you can see that (1)/(2/3) = 3/2. The opportunity cost of rebuilding one engine is 2/3 of a repainted car, and 2/3 is the reciprocal of 3/2. Therefore, the opportunity cost of one activity is the reciprocal of the opportunity cost of the other activity. This little trick for calculating one person’s opportunity costs between two activities will always hold true.

Calculating Clack’s Opportunity Costs

Next, let’s look at Clack’s opportunity costs for rebuilding engines. It takes Clack 1½ days to rebuild one engine. During that time, he could repaint ¾ of a car. This is because repainting one car would take Clack 2 full days, while rebuilding the engine takes him only 1½ days. Repainting takes ½ day longer for Clack than rebuilding an engine, so he could paint ¾ of a car in the time he rebuilds one engine. Therefore, the opportunity cost to Clack of rebuilding an engine is ¾ of a repainting job. Now, again use one of the tricks in calculating opportunity costs. To calculate Clack’s opportunity costs for rebuilding one engine, first put the number of days it takes to rebuild and engine (the activity Clack does here) in the numerator (top) of a fraction. Next, put the number of days it takes to repaint a car (the next best use of his time) in the denominator (bottom) of the same fraction. Here, the resulting fraction is (1½) / (2). Then, use the "invert and multiply" rule for dividing fractions to find the opportunity cost to Clack of rebuilding engines. Here, this means take (1½) / (2) = (3/2) / (2) = (3/2) * (1/2) = ¾. Therefore, it costs Clack ¾ of a repainted car to rebuild one engine.

What is Clack’s opportunity cost of repainting one car? It takes him two days to do one repaint job. One and a half of those days could have been used to rebuild one engine. The other half-day could have been used to rebuild 1/3 of an engine. Thus, repainting one car costs Clack 1 1/3 rebuilt engines. You can use the trick demonstrated above to see this again.

Notice again that Clack’s opportunity cost of repainting one car is the reciprocal of his opportunity cost of rebuilding one engine. Remember that the opportunity cost to Clack of rebuilt engine is ¾ of a repainted car. On the other hand it costs Clack

1 1/3 = 4/3 rebuilt engine to repaint one car. Notice that the reciprocal of ¾ is

(1) / (3/4) = 4/3. The other trick we learned for calculating opportunity costs holds again.

Comparative Advantage in Rebuilding Engines and Repainting Cars

Now that we know each brother’s opportunity cost for producing rebuilt engines and repainted cars, we can see who has a comparative advantage in each activity. Remember that Click had an absolute advantage in both activities. But by using each brother’s opportunity costs, we can find out if there is any reason why the two may trade activities between themselves. First, let’s recap each brother’s opportunity cost for each activity.

Rebuilt engines: (2) It costs Click 2/3 of a repainted car to rebuild one engine; (2) It costs Clack ¾ of a repainted car to rebuild one engine.

Repainted cars: (1) It costs Click 1½ engines rebuilt to repaint one car; (2) It costs Clack 1 1/3 engines rebuilt to repaint one car.

Now it’s easily seen why there may be trade between the brothers even though Click has an absolute advantage in both rebuilding engines and repainting cars. It costs Click less to rebuild one engine than it costs Clack. Why? Click’s opportunity cost for rebuilding one engine (2/3 of a paint job) are lower than Clack’s (3/4 of a paint job). On the other hand, it costs Clack less to repaint one car than Click. Why? Clack’s opportunity cost of repainting a car (4/3 of a rebuilt engine) are lower than Click’s (3/2 of a rebuilt engine). Therefore, in terms of efficiency, Click should spend more time rebuilding engines, and Clack should spend more time repainting cars.

Production Possibilities Tables and Graphs for Click and Clack

Now let’s look at each of the brother’s production possibilities when they both work a 30-day month. Once we’ve done this, we can combine all of our results and see why Click and Clack could make themselves better off by trading their abilities with each other rather than having both of them doing all of their own repainting and engine rebuilding. The following tables each summarize three points on the brothers’ production possibility frontiers for a 30-day month. One possibility in each table shows a brother’s production in one month when they devote all of their time to engine rebuilding and none to repainting. Another point in the table shows their production in one month when all of their time is devoted to repainting and none to engine rebuilding. The last point in the table shows their output of repainting and rebuilding when their time is split between activities.

Click’s Production Possibility Frontier:

Clack’s Production Possibility Frontier:

With this information in hand, we can graph each of the brother’s production possibility frontiers. This will aid us in visualizing their possibilities for trade.

Production Possibilities Relationships for Click and Clack

Now, let’s combine the PPF’s tables and graphs with what we know about simple lines to find the exact relationship given in each brother’s PPF. We’re going to do this in order to make talking about trade between the brothers even easier.

First, think about the vertical axis measurement in the graphs above (Rebuilt Engines) as what Mankiw calls in your text the "y-coordinate." Then think of the horizontal axis measurement (Repainted Cars) as the "x-coordinate." Finally, remember that the equation for a simple line is:

y = mx + b

Where "b" is the intercept of the line and "m" is the slope of the line. Remember that the slope of line is defined as , or simply "the change in y given a change in x."

Now, for Click’s production possibilities, notice that the y-coordinate intercept (or in the equation of a line, the term "b") is 30 rebuilt engines – that is, when Click produces no repainted cars, he can produce at most 30 rebuilt engines. Thus, the intercept of Click’s PPF is 30. Now, let’s turn to the slope of Click’s PPF. Remember that every repainted car "costs" Click 1½ = 3/2 engines rebuilt. Therefore, for every car Click repaints, he must rebuild 3/2 fewer cars. For Click’s PPF, we can see that the slope corresponds to the opportunity cost of repainting cars – the y-coordinate (rebuilt engines) decreases by 3/2 (Click must give up 3/2 of an engine built) when the x-coordinate (repainted cars) changes (or if Click paints one more car).

Putting all this together, Click’s PPF can be written as:

(Rebuilt Engines) = 30 – (3/2) ´ (Repainted Cars)

Now let’s do this same thing for Clack. Clack’s y-coordinate intercept (the term b in the equation of a line) is 20 rebuilt engines – therefore, when Clack only rebuilds engines, he can produce 20 of them. Here, Clack’s PPF intercept is 20 rebuilt engines. Next, the slope of Clack’s PPF is once again the opportunity cost of his repainting one car. Remember that when Clack repainted one car, he had to give up 1 1/3 = 4/3 rebuilt engines. Therefore, the slope of Clack’s PPF is 4/3 – the y-coordinate (rebuilt engines) decreases by 4/3 (Clack must give up 4/3 engines built) when the x-coordinate (repainted cars) changes (or if Clack paints one more car)

Putting all this together, Clack’s PPF can be written:

(Rebuilt Engines) = 20 – (4/3) ´ (Repainted Cars)

Can Trade Make Click and Clack Better Off?

To find out the answer to this question, let’s start off with a situation where Click and Clack don’t trade at all. Suppose that at first, Click does all of his own engine rebuilding and Clack does all of his own repainting. Neither brother asks the other brother to do any of their work. If this is where the brothers start out, then Click is initially producing 30 rebuilt engines and no repainted cars per month (point A in his PPF). Clack, on the other hand, is currently producing no rebuilt engines and 15 repainted cars per month (point B on his PPF). Now, remember that we said above that Click has a comparative advantage in engine rebuilding, while Clack has comparative advantage in repainting cars. Let’s see how each can use his comparative advantage to make himself better off.

Suppose that the two brothers come up with the following deal. Click offers to rebuild 7 engines for Clack. In return Clack agrees to repaint 5 cars for Click. How does this trade make each brother better off? First, notice that if Click offers to build 7 engines for Clack, then he is still building a total of 30 engines – 23 for himself, 7 for his brother. Also, notice that if Clack paints 5 cars for Click, he’s still painting a total of 15 cars – 5 for Click, 10 for himself.

Now let’s look at why the brothers would choose to make this deal. With the trade, Click moves from having only 30 engines rebuilt and no repainted cars to having 23 engines for his own AND 5 repainted cars. Now if Click were to try to produce these 5 repainted cars by himself, then due to his limitations he would only be able to rebuild 22½ engines. This can be see by looking at Click’s PPF:

(Rebuilt Engines) = 30 – (3/2) ´ (Repainted Cars)

Rebuilt Engines = 30 – (3/2) ´ (5)

Rebuilt Engines = 22 ½ when Click repaints the 5 cars by himself instead of having Clack produce them.

Compare that 22½ rebuilt engines if he paints by himself to the 23 rebuilt engines he could have with trade. This shows that trade makes Click better off. You can also see visually on Click’s PPF why he’s better off with trade – point A’ shows where he ends up after the trade.

We can use the same rationale as to why the trade makes Clack better off. Clack, at the end of the trade with his brother, will have 7 rebuilt engines and 10 repainted cars. If Clack were to try and rebuild those 7 engines by himself, he would only be able to produce 9¾ repainted cars due to his limitations. Using Clack’s PPF, we can see this:

(Rebuilt Engines) = 20 – (4/3) ´ (Repainted Cars)

7 = 20 – (4/3) ´ (Repainted Cars)

13 = (4/3) ´ (Repainted Cars)

Repainted Cars = 9 ¾ when Clack rebuilds 7 engines by himself rather than having Click produce them.

Compare that 9 ¾ repainted cars with the 10 he gets when he trades with his brother. Therefore, Clack is also better off with trade. Again, you can see this visually on Clack’s PPF – point B’ shows where Clack ends up after the trade.

Now we’ve gone through many of the most important concepts in your readings to illustrate opportunity costs, production possibilities, comparative advantage, graphs and lines, and gains from trade. With this example and your readings in the book, you are now an expert in some of the most fundamental economic concepts!