# Principles of Economics/Specialization and Gains From Trade

The idea of scarcity is all-encompassing in economics. For review purposes, scarcity is the idea that our unlimited desires are never satisfied because the amount of resources we have is limited. It is not surprising to hear that scarcity is involved with gains from trade – the concept that people will gain more from trading or working with other rational individuals as opposed to working by him or herself on a task he or she does not have the pleasure of understanding or doing. Let's put this into more concrete terms with an example.

## Gains from Trade: An Example

Individual Talent
James Portley Inventing and designing products

In the table above, there are two people. Say those two people are co-founders of MetalPlastic, a company that makes and designs phones. The two situations presented below are more of a thought experiment since this is not a real-life example.

### The Situation with No Trade

James Portley would like to advertise his new product that may very well change the face of the phone industry. However, since he cannot convincingly give any reason as to why you, the seller, should buy this product, he will fail to change the face of the phone industry – his desire was not met. Quincy Adams would like to advertise a new phone. Since there are no new phones in the market, Adams will have to find a new phone somewhere else; otherwise, he may not find a new phone to advertise. Since Quincy Adams cannot make a new phone and advertise the phone, he will never fulfill his desires.

Imagine that James Portley and Quincy Adams met one day. The pair noticed their individual talents and thought to work together. The co-founder group settled for the company name MetalPlastic. The pair started working together. After Portley made his new product, Adams decided to risk the money they have to present their new product. It was a success story. Because Quincy was able to convince financial investors to invest in the phone, they were able to make a profit selling what they wanted. Each person's individual desires were met – to meet their goals.

### Conclusion to Take Away

By specializing their talents, the co-founder group were able to fulfill their individual desires. Quincy made a trade with James, either intentionally or not: if James made a product, Quincy will help advertise the good. Keep in mind, however, that James could have declined. If he did decline, there would be no reason to specialize. Same goes if James decided to trade with Quincy.

It is important to realize this lesson: rational individuals will gain from the trade if and only if each individual agrees that trading is better than by working by themselves. This consequence of rationality is built into its definition. A person will weigh the costs to the benefits. If the benefits weigh more, the person will cooperate with the trade; if the costs weigh more, the person will not cooperate. Either way, each individual checked to see if the opportunity cost of working with another person granted them a better "deal" than by working by themselves. We now understand gains from trade.

### The Catch of this Simple Example

"While it may be true that working with other individuals is beneficial when you do not know how to do a task, it does not necessarily mean that working with other individuals is always best if the task you do is either similar or identical. Are there gains from trade in every situation imaginable?"

The answer to that question is "no, not always, but most of the time." The next section will explain why in great detail.

Imagine you have two people who work in the same factory, Boxing Glass. Their names are Harry and Steven. Harry can make two times more glasses and 3 times more boxes compared to Steven. We would say that Harry has an absolute advantage to Steven – Harry can make way more of both resources compared to Steven. Steven decides to alleviate both their workloads by working together. Should Harry make the trade?

### Production Possibility Frontier

The two lines of productions for each person, Harry and Steven, are shown. The lines are colored differently. For any individual who is color blind, Harry is the top diagonal line, and Steven is the bottom diagonal line. The dot represents the level of production they wish to produce.

A production possibility frontier (PPF, for short) is a graphical curve or line that represents the production of two goods for any entity. For this instance, we will make our math easier by using a line instead of a curve.

The figure above shows two lines: Harry's and Steven's. Since Harry's line is above and to the right of Steven's, Harry has an absolute advantage to Steven. For our purposes, we want to know Harry's opportunity cost of doing work. We have two methods to figure that out.

Method How to use it
Find the slope ${\displaystyle y=mx+b}$  or ${\displaystyle m={y_{2}-y_{1} \over x_{2}-x_{1}}}$ .
Unitary opportunity cost Divide the two productions accordingly.

Each method will be shown below:

#### Calculating Opportunity Cost: Method 1

Before we start calculating, let's first review what the slope ${\displaystyle m}$  means for us. The slope ${\displaystyle m}$  is the rate of change in ${\displaystyle y}$  over the change in ${\displaystyle x}$ . The rate of change is simply a division of two points, ${\displaystyle x}$  and ${\displaystyle y}$ , that are subtracted. The two different values of ${\displaystyle y}$  are subtracted in the numerator (top number of a fraction) and the two different values of ${\displaystyle x}$  are subtracted in the denominator (bottom number of a fraction). The different values of ${\displaystyle x}$  will be deliniated by different subscripts (little numbers next to and below the variable), ${\displaystyle x_{1}}$  and ${\displaystyle x_{2}}$ . The same goes for the different values of ${\displaystyle y}$ . You can only find the slope once you know two different ordered pairs, ${\displaystyle (x_{1},y_{1})}$  and ${\displaystyle (x_{2},y_{2})}$ , of the line. This entire concept is summarized as

${\displaystyle m={y_{2}-y_{1} \over x_{2}-x_{1}}}$

The line of a function needs the slope of a function, but it also one other constant value before a linear function can be formed. Those values are called intercepts. The most common intercept (and the one we will focus on in this economics WikiBooks) is the ${\displaystyle y}$ -intercept, defined as ${\displaystyle b}$  in our formula. Simply put, the ${\displaystyle y}$ -intercept is defined as the ordered pair ${\displaystyle (0,b)}$ . It is the value of y that will intersect with the vertical ${\displaystyle y}$  column of our ${\displaystyle xy}$ -graph.

Finally, once all is done, we add in a changing variable, ${\displaystyle x}$ -values, to then define the equation ${\displaystyle y=mx+b}$ . The equation will determine ${\displaystyle mx+b}$  to find all possible ${\displaystyle y}$ -values. A graphical representation will appear for all ordered pairs ${\displaystyle (x,y)}$ , where each point will be connected linearly.

Now that we know what a slope is, we can now find it in our graph.

 Example 1: Find the opportunity cost of Harry's box production to glass production by finding the slope of the PPF. The point at which Harry chooses to produce is at ${\displaystyle (3,2)}$  or 3 boxes and 2 glasses. There is a multitude of ways to find the slope in this instance, yet only one such calculation will be shown. Let ${\displaystyle (x_{2},y_{2})=(3,2)}$ , and let ${\displaystyle (x_{1},y_{1})=(0,4)}$ . We can now find the slope. ${\displaystyle m={(2)-(4) \over (3)-(0)}}$  ${\displaystyle m={-2 \over 3}}$  ${\displaystyle m=-{2 \over 3}}$  The calculation above represents the opportunity cost of ${\displaystyle 2\,Q_{1}}$  (glasses) for every ${\displaystyle 3\,Q_{2}}$  (boxes). While this is fine, we are looking for the opportunity cost of boxes to glasses. Since the glasses are in the numerator, and the boxes are in the denominator, we will just have to switch the places of the numbers. This is called taking the reciprocal, which we will represent as ${\displaystyle m_{r,h}={Q_{2} \over Q_{1}}}$ , for Harry's reciprocal slope. This means our final answer is ${\displaystyle m_{r,h}=-{3 \over 2}}$

Note that the way the fraction is written represents the opportunity cost of the situation. The final answer above tells us that 3 boxes is the opportunity cost of 2 glasses. This simply means that 3 boxes are lost to make 2 glasses. This represents the loss of ${\displaystyle 3\,Q_{2}}$  for every ${\displaystyle 2\,Q_{1}}$ .

Final point to understand before we move on, the slope of the answer above can be found using any point along the PPF of Steven's production line. For example, for the ordered pair ${\displaystyle (x,y)}$ , defined as ${\displaystyle (Q_{2},Q_{1})}$  for this graph, if ${\displaystyle (x_{2},y_{2})=(6,0)}$  and ${\displaystyle (x_{1},y_{1})=(0,4)}$ , Harry's reciprocal slope ${\displaystyle m_{r,h}={x_{2}-x_{1} \over y_{2}-y_{1}}}$  would be defined as

${\displaystyle {(6)-(0) \over (0-4)}={6 \over -4}=-{3 \over 2}=-1.5}$

The same answer as given in the example would be derived and evaluated. This is true because the PPF is linear, meaning the same "rate of change" is used for all points ${\displaystyle (Q_{2},Q_{1})}$ , where ${\displaystyle Q_{2}}$  and ${\displaystyle Q_{1}}$  is any positive rational number defined by the equation of the graph ${\displaystyle y=mx+b}$ . Let's look at an example to find those points of the graph.

 Example 2: Harry would like to know at which points he could produce the fraction quantity of four-thousand boxes and six-thousand glasses. Let ${\displaystyle y=Q_{1}}$  and ${\displaystyle x=Q_{2}}$ . Find the equation of the line ${\displaystyle Q_{1}=mQ_{2}+b}$  by deriving ${\displaystyle m}$  and ${\displaystyle b}$ . Before we get confused, let's make sure we understand what each variable stands for. Remember that ${\displaystyle Q_{1}=y}$  and ${\displaystyle Q_{2}=x}$ . Plugging (more formally known as "substituting") those values in, we find that the equation we are looking for is ${\displaystyle y=mx+b}$ . This is simply just an equation for a line. In which case, let's go ahead and find the values for ${\displaystyle m}$  and ${\displaystyle b}$ . Because we already know the slope of the line ${\displaystyle m=-{2 \over 3}}$ , let's try to find ${\displaystyle b}$ . The point at which the line intersects ${\displaystyle y}$  seems to be at ${\displaystyle (0,4)}$ . Therefore, let's substitute those values into the equation to yield the final answer, keeping in mind that ${\displaystyle y=Q_{1}}$  and ${\displaystyle x=Q_{2}}$ : ${\displaystyle Q_{1}=-\left({2 \over 3}\right)Q_{2}+(4)}$  ${\displaystyle Q_{1}=-{2 \over 3}Q_{2}+4}$

The answer above is called a function. The function uses only one input, usually ${\displaystyle x}$  to find an output, usually ${\displaystyle f(x)}$ . For example, ${\displaystyle y=mx+b}$  finds the output ${\displaystyle y}$  by evaluating every input ${\displaystyle x}$ , given that ${\displaystyle m}$  and ${\displaystyle b}$  are constant. Usually, a question will define the output. Here's another example question that could have been asked, which is identical to the one above, only defined differently:

 Example 3: Harry would like to know at which points he could produce the fraction quantity of four-thousand boxes and six-thousand glasses. Let ${\displaystyle x=Q_{2}}$ . Find the function ${\displaystyle H(Q_{2})=mQ_{2}+b}$  by deriving ${\displaystyle m}$  and ${\displaystyle b}$ . Here, the "defined output" would have been ${\displaystyle H(Q_{2})}$ . The answer would have been nearly identical to Example 2, but would mean something different because of how a function is defined. The function to the example question would have been this: ${\displaystyle H(Q_{2})=-\left({2 \over 3}\right)Q_{2}+(4)}$  ${\displaystyle H(Q_{2})=-{2 \over 3}Q_{2}+4}$

Using the function ${\displaystyle H(Q_{2})}$ , you can find any ordered pair ${\displaystyle (x,y)=(Q_{2},Q_{1})}$ . Substitute any rational value for ${\displaystyle Q_{2}}$  and evaluate from there. Let's try ${\displaystyle Q_{2}=2}$ .

 Example 4: Find the ordered pair of ${\displaystyle H(Q_{2})}$  for when ${\displaystyle Q_{2}=2}$  ${\displaystyle H(2)=-{2 \over 3}(2)+4}$  ${\displaystyle H(2)=-{4 \over 3}+4}$  ${\displaystyle H(2)=-{4 \over 3}Q_{2}+{12 \over 3}}$  ${\displaystyle H(2)={8 \over 3}}$  The value we see above tells us that when ${\displaystyle x=Q_{2}=2}$ , ${\displaystyle y=Q_{1}={8 \over 3}}$ .

If you don't have a graph, finding a function is great for finding any output ${\displaystyle y}$  from an input ${\displaystyle x}$ .

#### Calculating Opportunity Cost: Method 2

Before we get to the calculation, we need to know the definition of unitary opportunity cost. The unitary opportunity cost is the amount of an object we lose for every other one object. Take this example: if we want to choose between having three scoops of chocolate ice cream to two scoops of vanilla ice cream, we would say that the opportunity cost of having three chocolate ice cream scoops is two scoops of vanilla ice cream. However, this does not tell us how much chocolate ice cream scoops we waste per vanilla ice cream scoops. To do this, we need to compare. This is the fundamental reason why we have the unitary opportunity cost. This is usually useful whenever working with production.

One of the useful establishments of the PPF is that the PPF represents the opportunity cost of production. Any time that you can either make, for example, 3 ${\displaystyle x}$  or 5 ${\displaystyle y}$ , a PPF can represent the in-between production, making it also useful for finding the unitary opportunity cost of production.

 Example 5: Find the unitary opportunity cost of boxes to glasses. Remember that we are looking for the number of ${\displaystyle Q_{2}}$  lost for every ${\displaystyle Q_{1}}$  made, meaning that the numerator (top number of the fraction) must be a value from the ${\displaystyle Q_{2}}$  horizontal line, and the denominator (bottom number of the fraction) must be a value from the ${\displaystyle Q_{1}}$  vertical line. According to Harry's production line, Harry can make either 4 glasses or 6 boxes. Let's use those values to help find the unitary opportunity cost. First, set up a comparison between ${\displaystyle {Q_{2} \over Q_{1}}={x \over 1}}$ , where ${\displaystyle x}$  represents the number of ${\displaystyle Q_{2}}$  lost for every ${\displaystyle 1\,Q_{1}}$  made. ${\displaystyle {6 \over 4}={x \over 1}}$  In accordance to mathematical axioms, any number ${\displaystyle x}$  divided by 1 is equal to ${\displaystyle x}$ . As such, all that is needed is to divide ${\displaystyle {6 \over 4}}$ . ${\displaystyle {6 \over 4}={3 \over 2}=1.5=x}$

Note that the final answer is similar to when you use the slope of the PPF to represent the opportunity cost. Here is a calculation to help you find this out:

First, notice that the fraction ${\displaystyle m_{r,h}}$  is ${\displaystyle -{Q_{2} \over Q_{1}}}$ , which let us find ${\displaystyle x}$  in this equation: ${\displaystyle -{Q_{2} \over Q_{1}}={x \over 1}}$  where ${\displaystyle 1}$  represents ${\displaystyle 1\,Q_{1}}$ .

${\displaystyle -{3 \over 2}={x \over 1}}$

Since ${\displaystyle {x \over 1}=x}$  is true because of mathematical axioms and postulates, simply divide ${\displaystyle {3 \over 2}}$  to find the unitary opportunity cost of production for ${\displaystyle m_{r,h}}$ .

${\displaystyle -{3 \over 2}=-1.5}$

Because ${\displaystyle -1.5=x}$ , we are done, and can now interpret the unitary opportunity cost of boxes to glasses since the fraction to the right of the equal sign, ${\displaystyle {x \over 1}}$ , is a comparison of the slope ${\displaystyle m_{r,h}}$ .

${\displaystyle -{3 \over 2}={-1.5 \over 1}}$

The only difference is that the answer is negative for the slope but positive for the unitary opportunity cost. To fix this, simply take the absolute value of both to yield the same answer.

${\displaystyle \left|-{3 \over 2}\right\vert ={3 \over 2}=1.5=x}$

From this, we learned that the absolute value of the slope for any entity's PPF is identical to the unitary opportunity cost of any entity's production.

Before we move on to the next section, try to find Steven's opportunity cost of production. Hint: Look at the PPF at the beginning of the section.

1 What is Steven's opportunity cost of producing 1 glass? (2 marks.)

 ${\displaystyle 1}$  glass ${\displaystyle 1}$  box ${\displaystyle 2}$  boxes ${\displaystyle 1.5}$  boxes ${\displaystyle 0.{\bar {6}}}$  glasses.

2 Find Steven's slope of the PPF when producing the reciprocal, ${\displaystyle m_{r,s}}$ , where ${\displaystyle s}$  is used to deleniate Steven's line, to help the find the loss of ${\displaystyle Q_{2}}$  for every ${\displaystyle Q_{1}}$  gained. (1 mark.)

 ${\displaystyle m_{r,s}=-{2 \over 3}}$ ${\displaystyle m_{r,s}=-{3 \over 2}}$ ${\displaystyle m_{r,s}=-{2 \over 1}}$ ${\displaystyle m_{r,s}=-{1 \over 1}}$ ${\displaystyle m_{r,s}=-{1 \over 2}}$

The methods of calculation we learned now helps us answer the ultimate question here: should Harry make the trade? Since we know the slope of both Harry, ${\displaystyle m_{h}}$ , and Steven, ${\displaystyle m_{s}}$ , let's use those values.

Individual Slope of ${\displaystyle |m|}$  Slope of ${\displaystyle \left|m_{r}\right\vert }$
Harry ${\displaystyle \left|m_{h}\right\vert =0.{\bar {6}}}$  ${\displaystyle \left|m_{r,h}\right\vert =1.5}$
Steven ${\displaystyle \left|m_{s}\right\vert =1}$  ${\displaystyle \left|m_{r,s}\right\vert =1}$

Remember: the slope of ${\displaystyle |m|}$  represents the loss of ${\displaystyle Q_{1}}$  for every ${\displaystyle Q_{2}}$ , while the slope of ${\displaystyle \left|m_{r}\right\vert }$  represents the loss of ${\displaystyle Q_{2}}$  for every ${\displaystyle Q_{1}}$ . Let's now compare the opportunity cost of each individual.

• Harry can lose fewer glasses from gaining boxes than Steven can: ${\displaystyle 0.{\bar {6}}<1}$
• Steven can lose fewer boxes from gaining glasses than Harry can: ${\displaystyle 1<1.5}$

The phenomenon demonstrated above illustrates a special circumstance of economics that allows us to earn insight from it. In fact, this special circumstance has its own name: comparative advantage. The comparative advantage is the advantage of an individual in which the circumstances allow them to produce more of a good or service at a lower opportunity cost compared to another individual. This is the reason why we "gain from trade." By extension, this is where Harry gains from trade.

Note: the best way to find the comparative advantage is through the unitary opportunity cost. This is what we did in this section. Remember that the slope and the unitary cost answer are the same. We have proven that. This means that the opportunity cost of, for example, ${\displaystyle {2 \over 3}\,Q_{1}}$  is lost to make ${\displaystyle 1\,Q_{2}}$ . The slope is the unitary opportunity cost. That is what we proved last time, and this is why the slope or the unitary opportunity cost can be used in the table above and still prove the same thing we proved: Harry has a comparative advantage in making boxes and Steven has a comparative advantage in making glasses.

Remember what we learned at the beginning? By specializing people's talents, everyone as a whole can gain from the trade of talents. Here is no different, aside from the vernacular. Let's try to find the optimal trade.

 Example 6: Harry and Steven want to look like stand-out employees. Since they know they will gain from trading, they decide to trade a number of boxes made for glasses and vice-versa. If Harry decides to trade 1 box, how many glasses should he decide to gain such that the choice will benefit both parties? We know that Harry loses fewer glasses per each box made, so Harry should specialize in making boxes. If Harry wants to find the optimal unit of boxes for both him and Steven, he needs to find the number of units that will not go over either Steven's or Harry's opportunity cost of production. Remember: the opportunity cost of production is the slope of the PPF. Steven trading one box for ${\displaystyle {2 \over 3}\,Q_{2}}$  would not be optimal for Harry because Harry could just make that many himself. Steven trading one box for ${\displaystyle 1\,Q_{2}}$  would be optimal for Harry but not for Steven because he would go over his opportunity cost of production. Therefore, the number of ${\displaystyle Q_{2}}$  units needed to be made follows this relationship: ${\displaystyle {2 \over 3}  ${\displaystyle 0.{\bar {6}}  Remember: the number of units is per thousand, so those are the possible number of units to trade. The number of glasses to trade that would be optimal for both Harry and Steven could be ${\displaystyle {3 \over 4}}$  of one thousand units, or 750 glasses. Nevertheless, any number between ${\displaystyle {2 \over 3}  of one thousand units is optimal, so any integer answer between those ranges are necessarily the best for both parties.

Note: the same procedure done above will work when trading for glasses. The only difference will be the number of boxes traded for one 1 glass. Try out the next example problem to see if you understand. Review the previous sections so that you know the numbers to look for.

Harry and Steven want to look like stand-out employees. Since they know they will gain from trading, they decide to trade a number of boxes made for glasses and vice-versa. If Steven decides to trade 1 glass, how many boxes should he gain such that the choice will benefit both parties? (3 marks.)

### Conclusion

Although both Harry and Steven work the same job, they still benefit from trading. Keep in mind, though, that this only works if a comparative advantage exists for both parties. If one party in the trading relationship does not have a comparative advantage, then why trade at all? Work by yourself in that instance. However, this rarely happens, if at all.

The examples shown in the previous subsections are in no way intended to be realistic. After all, people don't barter, instead they get paid with income or money or some other form of monetary transaction. However, the lessons learned within will extend to both Microeconomics and Macroeconomics. Plus, the examples shown before amplify the rational fact of humans: if the opportunity cost of trade is more than not trading, people will not trade.

Finally, before ending with a few comprehension questions, let's realize one more fact: when markets extend in the skill or production of goods, the number of times specialization occurs will increase, and thus the gains from trade.

2 The ${\displaystyle y}$ -axis, denoted as ${\displaystyle Q_{y}}$ , represents the number of coconuts scavenged while the ${\displaystyle x}$ -axis, denoted as ${\displaystyle Q_{x}}$ , represents the number of fish caught. If Harry's slope ${\displaystyle m_{h}=-{1 \over 2}}$  while Steven's slope ${\displaystyle m_{s}=-3}$ , for the person who has a comparative advantage in catching fish, what is the opportunity cost of catching 2 fish? (3 marks.)