Production Possibilities

1. It introduces the concept of the production function which we will use throughout our study of microeconomics and macroeconomics.
2. It introduces the concept of opportunity cost which is perhaps the most valuable single concept of economics.
3. It introduces you to the concept of an economic model with reasonably simple ingredients, and uses those ingredients to come to a surprising conclusion -- that gains from trade can be reaped by both parties to international trade if they specialize along lines of comparative advantage.
4. The model can easily be modified to take into account more realistic production functions and to treat other issues such as the impact of international trade on the distribution of income. While the model of the introductory course is of course much simpler than the models economists use in practice -- just as the model of an atom in an introductory physics or chemistry course is much simpler than the models physicists or chemists use in practice -- the model of production possibilities and comparative advantage you will meet will introduce you to the key concepts which are applied in studies of the impact of NAFTA, the WTO or APEC.

where Qw is the quantity of wheat and Lw the amount of labor devoted to growing wheat.

You may be tempted to raise several issues, all of which are perfectly valid:

1. Where does the number 10,000 come from? Simple answer: the amount of wheat a farmer can grow depends on the quality of land available to him and on the amount of capital (plows and tractors) he has.
2. Won't the number differ between farms, or at least between states? Answer: yes, of course. While we will not be too interested in the difference between farms, we will be very interested in the difference between countries -- the Ukraine has much better land for growing wheat than Saudi Arabia, and the number should be higher for the Ukraine.
3. But are you really justified in ignoring the differences within a country? South Dakota's Red River Valley has a lot better land for growing wheat than does the Ohio Valley. There are two answers here:
   First, we want to ignore relatively minor differences in order to bring out the major differences, and which differences are minor depends on the problem we are addressing. If we plan to apply our model to international economics, it makes sense to treat the differences within a country as less important than the differences between countries. Second, we admit that the differences are important for some issues, and we will be looking at these issues later in the course. Most industries -- and certainly agriculture -- run into the problem of diminishing returns. As you keep adding wheat farms, you eventually are forced to use lower quality land. But these issues can and will be addressed not by abandoning the model, but by extending it.

Production functions and opportunity cost

and

in mind will help explain the terminology.

The first concept is that of labor productivity -- how much can one typical worker produce? The answers, from the equations above, are simple:

• In the X industry, one worker produces 5 units of output in a given period of time.
• In the Y industry, one worker produces 10 units of output in a given period of time.

The second concept is that of activity requirements. Where the labor productivity coefficient measures the amount of output one worker produces, the activity requirement measures the amount of work needed to produce one unit of output. Rewriting the production function will show us the activity requirements:

since Qx = 5 Lx, Lx = (1/5) Qx, that is, the activity requirement in the X industry is 1/5 or 0.2 units of labor to produce one unit of output.

In the Y industry, the activity requirement is _____click here for answer_____ .

The connection with the idea of opportunity cost comes through the fact that, assuming conditions of full employment or at least "normal" rates of unemployment, the only way to get more of one good is to transfer a worker away from some other good. If you want another worker on the farm, you must take a worker away from the factory, and the increased agricultural output will have as its opportunity cost the decreased production of manufactured goods.

Let us apply this to our production functions and calculate the opportunity cost of good X. Since Qx = 5 Lx or Lx = (1/5) Qx, to produce one more unit of X requires 1/5 more unit of labor. The labor must, in full employment conditions, come from the Y industry. This means that the amount of labor in the Y industry will decline by 1/5 and output in the Y industry will decrease:

Qy = 10 (- 1/5) = -2

We are using the production function for good Y, Qy = 10 Ly. Note that this is not a subtraction problem, but a multiplication problem -- the parentheses are there to separate the two terms. If you have had calculus, you will note that we are talking about changes in Qy, and should more correctly write the equation as dQy = 10 dLy, where "d" stands for difference or change. Sometimes the Greek letter delta -- a small triangle -- is used in place of "d".

We therefore say that the opportunity cost of good X is two units of good Y.

Question: What is the opportunity cost of good Y? The answer may be found by ____ clicking here____ .

Production functions and the Production Possibility Frontier

we can construct the production possibility frontier for the economy in question. The fact that to move a worker into the X industry means that he moves out of the Y industry means that there will be a trade off between producing X and producing Y, a trade-off which will be shown by the negative slope of the production possibility frontier, and which tells us that to produce more of good X, we must produce less of good Y (and vice versa).

How can we draw a graph of the Production Possibility frontier or the PPF?

First, consider the maximum possible outputs of X and Y. Since we have assumed a labor force of 50 (fifty million would doubtless be more realistic if we are talking about countries, but would force us to put a lot of zeroes on all of our calculations), we could:

1. put everyone to work in the X industry, and therefore produce
   Qx = 5 Lx = 5(50) = 250

   the maximum number of units of X possible. Of course, this means that we are producing zero units of Y, and in translating this point to a graph would graph it at (250, 0) or the tick mark representing 250 on the X-axis.

2. alternatively, we could put everyone to work in the Y industry, produce the maximum amount of Y possible, which would be:
   Qy = 10 Ly = 10 (50) = 500

   Again, this would mean that we were producing zero units of X, so the point would be (0, 500) or the tick mark representing 500 on the Y-axis of our graph.

Note that we are calculating here possibilities rather than what actually will happen. We can therefore imagine ourselves as the supreme dictators of the country, and simply allocate labor any way we see fit. An explanation of how labor actually is allocated between tasks requires a study of the operation of both goods markets and labor markets later in the course.

Even if we were the supreme dictators of the country, we could not escape the fundamental fact of opportunity costs -- the fact that if we decide to put a worker in the X industry, we are incurring an opportunity cost of the foregone Y industry output.

Let us consider some possible ways to allocate labor.

1. Put 10 workers in the X industry, and the remaining 40 in the Y industry. Output can be calculated by using the production functions:

   Qx = 5 Lx = 5 (10) = 50 and Qy = 10 Ly = 10 (40) = 400.

   Graph this point as (50, 400) on the graph of the PPF.

2. Put 20 workers in the X industry and the remaining 30 in the Y industry. Output will be:

   Qx = 5 (20) = 100 and Qy = 10 (30) = 300.

   The resulting output point will be the point (100, 300) on the PPF graph.

3. Put 33 workers in the X industry and the remaining 17 in the Y industry. Output will be:

   Qx = 5 (33) = 165 and Qy = 10 (17) = 170.

   The output point will graph as (165, 170).

Connect all the points.

You will find that you have drawn a straight line between the maximum amount of X that could be produced (250 units) and the maximum amount of Y that could be produced (500 units).

You have constructed a production possibility frontier. (For an alternative derivation of our PPF, using a bit more algebra but a bit less calculation, click here ).

What does the PPF show?

1. Opportunity cost is shown by the slope of the PPF. Our PPF goes up to point (0, 500) on the Y-axis, then slopes down to point (250, 0) on the X-axis. The slope of this line is (500 - 0) / (0 = 250) = -2. To produce one more unit of X, we must give up two units of Y.
2. Efficiency means, graphically, being on the line defining the production possibility frontier. To illustrate inefficiency, plot the point that would result from assigning 10 workers to the X industry and 10 workers to the Y industry -- and leaving the other 30 workers unemployed. Output would be Qx = 5Lx = 5(10) = 50 and Qy = 10 Ly = 10 (10) = 100, which would translate graphically into the output point (50, 100) Note that this point would be well inside the production possibility frontier. Points inside the PPF may result from unemployment, but they may also result from other forms of inefficiency -- for example, not using technology as effective as the production functions inducate is available, as well as the more obvious inefficiency of not supervising or motivating your workers as well as you might. This graphical illustration of inefficiency will play a role in our arguments about gains from trade, so be sure you understand it thoroughly.

What is the connection between production functions and prices ?

The connection between wages, the production function and goods prices is especially straightforward in the Ricardian model. The minimum price for which anyone could sell a good and stay is business is the cost of production of that good; Ricardo showed that a profit markup would not really change the results we will find by assuming the minimum price is the actual price of the good.

Let us suppose that the money wage of labor is $ 15 per unit of time. We can find the prices (or at least the minimum possible prices) of goods X and Y in our examples by looking at the activity requirement of those goods and multiplying by the wage.

Since good X requires 1/5 hour to produce, its cost of production -- and therefore its minimum price -- will be

The result should be fairly obvious: good X takes twice as long to make as good Y, and the labor costs are therefore twice as high, and the price of good X is twice as high as that of good Y.

Notice that in our model economy,

RELATIVE PRICES REFLECT OPPORTUNITY COSTS

Relative prices is the term used for the ratio of the price of good X to the price of good Y, here:

$3/$1.50 = 2

Note that the dollar sign cancels out -- a very useful feature if you want to make international comparisons, with some prices in dollars and others in francs or rubles or yen.

Relative prices are often the sort of prices we must think about if we want to think at all about international or historical issues. A pound of butter cost 50 cents in the late 19th century -- was that a lot or a litte? To answer the question, you might want to think about the price of butter relative to the price of a restaurant meal. If you find that a good restaurant meal -- the sort that would cost $ 30 today -- cost $ 1.50 in the late 19th century, you would conclude that the relative price of a pound of butter was 1/3 the price of a restaurant meal or roughly $ 10 in terms of modern money.

One other example of the usefulness of relative prices: we know that the price of a good horse was 50 sous in France in the 11th century, but we have no real idea of what a sou was worth.

Source: Georges Duby, La société aux XIe et XIIe siècles dans la region mâconnaise (Paris, 1971), p. 197n

We however do not get as good a sense of prices from the money price of the horse as from its relative price -- 5 times as much as a cow, half as much as a good house, and half as much as a suit of armor. (The fact that two horses -- it would be a good idea to have a spare on campaign -- and a suit of armor cost as much as two good houses goes some way toward explaining why there were so few knights around).