Why is the production possibilities curve bowed out in shape?

The production possibilities curve is bowed in shape because of the law of increasing opportunity cost, which explains the idea that the more units of a product are produced, the less capability the economy has of producing other products.

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The Production Possibilities Curve represents the choice society faces regarding whether to invest resources (inputs) into producing one kind of product or service or another. The reason that this curve is bow-shaped is a direct result of the law of increasing opportunity cost. This law states that any time society...

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The Production Possibilities Curve represents the choice society faces regarding whether to invest resources (inputs) into producing one kind of product or service or another. The reason that this curve is bow-shaped is a direct result of the law of increasing opportunity cost. This law states that any time society decides to move along its production possibilities curve to produce a particular kind of good or service, the opportunity cost of making additional units of that good or service will increase. The reason for this increase in opportunity cost is that, when a certain amount of social or economic resources are used to produce a particular kind of product, they reduce the total amount of resources available to make everything else. The more resources that are used (corresponding to a higher number of units of the finished product), the less capable the economy is of sustaining other forms of production. At a certain point, it is no longer economically viable to continue to produce increasing numbers of a one particular good or service. The detriment it causes to the production of other things (the opportunity cost) is too great.

In modelling the production possibilities curve, we assume that the economy can produce two (or more) goods and that the technology and factors of production remain unchanged and equivalent for both. In general, the curve does not tell economists where a particularly production process will operate most efficiently (on the curve itself). Rather, it tells us what possibilities exist within a given economy.

I am going to replicate here an example provided in Timothy Tregarthen and Libby Rittenberg’s textbook, Economics, Second Edition, to illustrate this point (pgs. 45-46). Say we create a production possibilities curve to chart the production of guns vs the production of butter. The former is an example of a defense industry good, whereas the latter a civilian good. Say that guns are the y-axis, butter the x-axis, and that the curve bows outward as we move from left to right. As we move from left to right on the x-axis, we produce less guns and more butter. If we were to take any two points on that curve, say point A and B, with A being further to the left on the x-axis, the opportunity cost of producing less guns for more butter would be the number of guns at point A minus the number of guns at point B. Figure 2-6 in the textbook that I have linked below illustrates this situation well. The production possibilities curve is bow-shaped precisely because there reaches a critical point at which the produciton of less guns means the possibility for more butter, and vice versa. This feature cannot be represented via a linear model.

https://books.google.com/books?id=HveHgrID5GYC&pg=PA45&dq=productions+possibilities+curve&hl=en&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwi8m4uEn6_oAhWUQc0KHaD5Ak4Q6AEwAXoECAkQAg#v=onepage&q=productions%20possibilities%20curve&f=false

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The production possibility curve (PPC), also referred to as the production possibility frontier (PPF) or transformation curve, depicts the maximum output possibilities for two goods contingent upon a set of inputs (like resources and other factors) and operating under the assumption that all inputs are efficiently used. 

Where the curve lies on a graph is influenced by the resources available, which is in turn impacted by technology, capital, labor, and so on.

The curve is representative of the presence of opportunity cost when having limited resources causes an organization to choose between two options. 

The curve takes a bow or arc shape because of this opportunity cost; there is an increase in the opportunity cost of producing a good when more resources are dedicated to that good's production. This demonstrates that resources are not perfectly adaptable to good production and is referred to as the "law of increasing opportunity cost." The curve is downward sloping due to the scarcity of resources. One good has to be given up to produce more of a different good.

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Due to resources being scarce, an economy needs to determine which products or services can be produced. A production possibilities curve is developed to show which combination of products and services can be produced at the most efficient levels. This model assumes that only two products and/ or services can be produced at any given time. The model also assumes that the technology and quantity are held constant.

In an economy where two products (A and B) can be produced, different firms can produce A, B or a combination of both A and B. Due to the scarcity of resources, an economy producing only A would not have enough resources to produce B and vice versa, leading to varied production combinations of the two products in between the extremes.

The opportunity cost associated with producing more of B from a starting point of producing only A increases with each additional production of B, which affirms the law of increasing opportunity cost. While resource scarcity ensures that the production curve slopes downwards, it is the law of increasing opportunity cost that ensures the curve is bowed out.

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The reason for the shape of the Production Possibilities Curve (PPC) is something called the law of increasing opportunity costs.  Basically, what this means is that as an economy devotes more of its resources to one kind of product, it becomes less efficient.  This is why the PPC is flatter at its end points and more curved in the middle.

If an economy produces two kinds of goods, it stands to reason that some of its productive resources will be more efficient at making one kind of good and some will be better at making the other.  Some of them will be equally good at either.  For example, imagine that our economy can make capital goods or consumer goods and services.  Some people will be good at making either.  But there will be some people who are only good for making consumer goods or services (let’s say they are really good at dealing with people and are good salespeople) while others will only be good at making capital goods.

As the economy makes choices in the middle of the curve (making large amounts of both) it can use everyone very efficiently.  But let’s say it starts to make only capital goods.  The people who are only good for dealing with people will be practically worthless.  As they are employed to make capital goods, they will create less marginal product.  This means that the curve will flatten out as it gets toward its endpoints.

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