The Specific Factors Model – Article 1: The Production Possibility Frontier

oatsandsugar —  December 6, 2011 — 2 Comments

This series, in three parts, will seek to show how trade may be modelled using two production possibility frontiers derived from the specific factors model, explaining relative advantage based on factor endowments in the process.

In international economics, we were introduced to the Specific Factors Model (“SFM”), to allow us to model the differences between two simplified economy’s Production Possibility Frontiers.  This article will seek to explain this model, as it is quite powerful, simple, and useful.

Article 1 – the PPF:

Section 1 will discuss the SFM;

Section 2 will discuss Cobb-Douglas Production Functions;

Section 3 will construct a PPF for a single economy;

Article 2 – trade

Section 4 will introduce another economy into the mix;

Section5 will explore the effects of different factor endowments on trade; and,

Article 3 – Rybczynski’s Theorem

Section 6 will explore the effects of changing the factor endowments of an economy.

Section 1: The Specific Factors Model

In order to construct a PPF, we must first explore the SFM, most importantly, discussing its assumptions and limitations.  Modelling an economy using the specific factors model is quite simple, since there are only three (in this version of the model, at least) resources in the production process and there are only two sectors in the economy.

Labor is the first resource, and has a few assumptions attached to it.  Firstly, labor is considered homogenous.  Every unit of labor is considered to be identical to every other, and, thus, equally suitable for utilization in either sector of the economy.  Further, labor is perfectly substitutable: that is, a unit of labor can more frictionless, and without loss of productivity, to either sector of the economy.  This is shown below, in Figure 1 and Equation 1.

Specific Factors Model - 1 - Labour is perfectly substitutable

Figure 1: perfectly substitutable labor - when a unit of labor is removed from food production, this model states that exactly the same amount will be added to manufactures production. That is, the amount of labor allocated to manufactures is directly inversely proportional to the amount of labor allocated to food.

Equation 1: Thus, an increase in Labor(Food) necessarily requires a decrease in Labor(Manufactures) since the total amount is constant and they are directly inversely proportional.

For the sake of simplicity, we will call the two sectors of the economy “food” and “manufactures”.  The second and third resources are tied to these sectors, that is, they are the specific factors of production only suitable for use in that specific sector.  Again, for the sake of simplicity, we will call the specific factor for food “Land” and the specific factor for manufactures “Capital”.  These are not substitutable, that is, in this model, you cannot sell Capital to buy Land, and vice-versa.  Furthermore, until we relax the assumption later in the article, the levels of these factors will be considered fixed.

In the production of Food, our relevant inputs are the labor allocated to food production – “Labor(Food)” – and Land; in the production of Manufactures, the relevant inputs are the labor allocated to the production of manufactures – “Labor(Manufactures)” – and Capital.    Since Land and Capital are considered fixed, the only factor that can change is the allocation of labor, and, as discussed above, these are intrinsically linked.

In order to continue, we must now model the effect of a change in labor on the production of each resource: we must model the Production Function for Manufactures, which brings us to our second section.

Section 2: The production function

There are many methods for constructing production functions, which most of you would have learned in introductory economics.  The SFM uses a Cobb-Douglas Production Function, since it displays many attributes which help to make this model more accurate.  Further, since there are two sectors in this economy, it is easiest to model each sector’s production function separately.

2.1 – The Production Function of Food

The production function for Food is described by the following equation, Equation 2.  The Cobb-Douglas production function allows us to create a set of production functions, one for Food and one for Manufactures, which will help us synthesize our PPF.

Equation 2: the Cobb-Douglas Production Function for Food Production. The Cobb-Douglas production function allows us to create a set of production functions, one for Food and one for Manufactures, which will help us synthesize our PPF

Equation 2: the Cobb-Douglas Production Function for Food Production.

First, as always, it is easiest to understand this equation if its elements are explained.

  • Output, here  is the amount of food produced at each input coördinate (each selection of labor and land).
  • Labor, here is Labor(Food).
  • Alpha is the weight of labor in food production, which is directly and inversely proportionate to the weight of Land in food production.  If Alpha is higher, food output is more influenced by Labor than by Land.
  • K represents the specific factor, here, land is the specific factor of production connected to food.
  • 1-Alpha is the weight of land in food production.

2.2 – Marginal Product of Labor in Food production

In this model, Land is fixed, thus, we must model what happens to food output as the only dynamic factor, labor, changes.  Whilst this may be done by inputting different values for labor into Equation 2 (which will give a certain curve for given values of land and alpha), this method is clumsy, since it only describes one production function.  By deriving the production function with respect to labor (Equation 3) we get the Marginal Product of Labor (“MPL”), which allows us to prove certain traits about any production function.

The Marginal Product of Labor

The MPL is reached by deriving the production function with respect to labor, and describes the increase in food output for each unit increase in labor.

Equation 3: The MPL is reached by deriving the production function with respect to labor, and describes the increase in food output for each unit increase in labor.

This equation allows us to ascertain the following:

  • Firstly, the marginal product of labor is always positive (since MPL>0), that is, for each unit increase of labor, there will always be some increase in food output.
  • Secondly, the marginal product of labor is always decreasing (for any positive value of K and L, MPL is positive), that is, each unit of labor has a subsequently decreasing effect on Food output.
  • Finally, that the MPL is directly related to the ratio of the specific factor (K, which here represents Land) to labour.

This allows us to sketch the production function for food: Figure 2.

Specific Factors Model - 2 - Food Production

Figure 2: The Production Function of Food

Why the decreasing marginal product?  Basically, this can be explained by Equation 3: we see that the marginal product of labor is specifically determined by the ratio of labor to land.  Where there is no labor, a single unit increase will be very effective in increasing food production.  This worker will have lots of land to till, and will not be limited by land.  However, where there are more and more workers, the Land will become saturated, and each unit increase of labor will have decreasing additional productivity.

Whilst some models allow for a negative marginal product to reflect the extreme of inefficiency which can occur when a factor is saturated, in using Cobb-Douglas Production Functions, this is not possible.

2.3 The marginal product of land in Food

This model assumes that land is fixed.  However, if it were to be modified, its marginal product would be expressed by deriving the production function with respect to land (equation 4).

Equation 4.2 the Marginal Product of the Specific Factor

Equation 4.2 the Marginal Product of the Specific Factor, here "K" represents land

2.4 The production function for manufactures

In expressing the production function of manufactures with a Cobb-Douglas function (equation 5), the function (Figure 3) and the marginal products (Equation 6, 7) will be almost identical to the functions discussed in sub-s 2.1-2.3, above, substituting what we define K as (above, it represented Land, here, it represents Capital).

Figure 3: the production function of Manufactures

Figure 3: the production function of Manufactures

Part 3: Constructing a PPF for a single economy

Thus, we now have the labor function (derived in s 1), and the production functions of manufactures and food (derived in s 2) with respect to changes in labor.  These three diagrams allow us to construct a production function for this economy: that is, what are the maximum production points available in the economy for food and manufacture outputs, and what are the points in between which utilize all resources.

So, first, lets sketch out what the final axes will look like:

Axes of the SFM

Figure 4: Axes of the Specific Factors Model

So, if you look closely at the axis above, and at the figures we have created in the sections above, you can see that we can already fill in three quadrants from what we have derived above, with some flipping of course:

  • Figure 1 (Labour is Substitutable) fits into the bottom left quadrant;
  • Figure 2 (Production function of Food) fits into the bottom right quadrant; and,
  • Figure 3 (Production function of Manufactures) fits into the top left quadrant.

Thus, only by flipping what we have above, we can create the following:

Figure 5: What we have already

Figure 5: What we have already

But we need to populate the top right quadrant: the production of food v the production of manufactures in the particular economy, since this graph internalises the use of all specific factors,and represents the PPF of the economy.  But how do we do this?  Well, in the same method that we are used to from creating the IS/LM curve in macro-economics (if you haven’t, try follow along anyway, it is quite intuitive).

So, the first thing we do is choose a point on the the labor spectrum, and we trace it to both production functions, and see where they intersect in the PPF’s plane (basically, we create the dotted line squares you see below).

Figure 6: Creating the PPF

Figure 6: Creating the PPF

Thus, now we have a Production Possibility Frontier that is not invented, but, rather, constructed based on an economy’s access to labor, capital and land.  This article has shown you how to build a PPF from information about factors of production in the Specific Factors Model.  The next article will introduce a second economy and show the effects on trade of different factor endowments.

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oatsandsugar

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LLM candidate at Cornell Tech. Consultant for King & Wood Mallesons and Project Evangelist for Legalese.

2 responses to The Specific Factors Model – Article 1: The Production Possibility Frontier

  1. 

    Hi Motek

    You deviated from the expected. thanks for the easy reading.

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