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Addendum on the theoretical aspects of productivity…

For those who have the time and desire to delve more deeply into the theoretical aspects of productivity and how that is related to the economy’s behavior, we refer you to the following.

Now we turn to the microeconomic causes and effects of productivity change in such concepts as the law of variable proportions of which diminishing returns is a part.  The variations on this topic are endless.  In this analysis we will focus on how productivity related to the labor market. 

Productivity is the essence of many popular terms, usually not very well understood by many including those using the term.  Automation, labor intensive, comparative advantage and featherbedding are but a few of those terms into which we will now delve.

PRODUCTIVITY: a Microeconomic journey

Focusing on labor productivity, we can distinguish between several measures of the physical productivity of labor, namely total, marginal, and average physical product.  Each has different implications.  Just below, you'll find a picture or graph of what is called a production function, in this case for labor. It assumes that all other productive resources such as physical capital are fixed in quantity and that the only resource that can be varied in quantity employed is that of labor. On the vertical axis of this graph is measured the output of units of production resulting from the transformation process of production.  On the horizontal axis we measure the quantity or number of units of the variable input labor, e.g., person years.

See below the graphs related to the following paragraphs.    

Graph one is the labor market.  The supply of labor and the demand for labor determines the equilibrium real wage.  The use of the real wage or nominal wage divided by the price level assumes that there is no price illusion.  In the current literature, this is equivalent to the assumption in rational expectations, where the consumer reacts immediately to any change in prices in the market place.  “Money is a veil,” as noted by Jean-Baptiste Say, where he points to the idea that financial capital merely facilitates the purchase of goods and services.  

The quantity supplied of labor increases as the real wage rises.  This is so for two reasons.  As each worker works more hours, less leisure remains.  The scarcer is leisure, the more valuable is each remaining hour of leisure.  To overcome this, a higher real wage must be paid.  Secondly, as the real wage rises, it overcomes the reservation price of those who are now willing to engage in work.  

The demand for labor is the marginal revenue product of labor.  If labor is subject to diminishing returns, as more labor is hired, the marginal physical product declines.  This is NOT because firms are scraping the bottom of the labor pool so to speak, they are all clones of each other.  It is due to the more intensive use of labor relative to a fixed amount of capital, etc.  To monetize the physical product of labor, we multiply labor’s physical product by the price it sells for in the firm’s product market (price X MPP or value of labor’s marginal physical product or by the marginal revenue (MR X MPP or labor’s marginal revenue product.  

If the firm employing labor is selling in a perfectly competitive market, the two are the same since price equals marginal revenue for a perfectly competitive firm since the demand curve facing a firm in a perfectly competitive market is perfectly elastic or horizontal at the level of the market price.   


1-Production Funtion.jpg

2-Level of Production and relationship of the productivity of labor and the quantity of labor employed.gif


In graph two, directly below the labor market graph, is the relationship of the output or physical product of labor as we increase the quantity of labor by one unit (e.g. person years) at a time.  It is called the production function.  Beneath it in graph three are the counterpart, marginal physical product of labor and average physical product of labor curves.  Since we are assuming a fixed stock of capital, constant technological state, etc., we are in the short-run; the physical product is not monetized.  A traditional production function is used in the following example.  It exhibits, first increasing physical returns to labor, maxes out and then exhibits diminishing physical returns to labor.  Once the point of diminishing physical returns is reached, there are three aspect of this phenomenon can be distinguished. 


They are diminishing marginal returns, diminishing average returns, and absolute or total diminishing returns.  The Marginal Physical Product (MPP) and Average Physical Product (APP) curves are drawn directly below the total Physical Product (TPP) curve (production function) so graphs two and three have identical X or quantity of labor axes.  In graph two, the slope of the tangent to each point on the TPP curve is the MPP.  In the same graph, the slope of a line from the origin to each point on the TPP curve is the APP curve in graph three.  Note there are three points of inflection in the TPP curve.  The first identifies the point of diminishing marginal returns, where MPP is maximized.  The second is the point of diminishing average returns, where APP is maximized.  The third is the point of diminishing total returns, where TPP is maximized. 


By extending the equilibrium real wage from graph one down to graph two, the level of total product produced by labor can be determined.  This is termed the natural level of production as produced by the natural level of labor employed (at the equilibrium real wage).  The natural quantity of labor employed in producing all goods and services (all markets cleared) divided by the labor force is the natural rate of employment.  That percent subtracted from 100% is the natural rate of unemployment. 


In the classical model, what assures that there will be sufficient demand for the natural level of production?  The answer to that question is found in the classical theory of interest rates, where prices of goods and services rise and fall to clear markets (where the price of a good or services is too high and results in increased supply, which is followed by a fall in prices to facilitate market clearing to eliminate the glut. 

Lord Keynes (John Maynard Keynes) disputed this vigorously and offered an alternative theory of interest rates called the Liquidity Preference theory of interest rates.  He also pointed out that there were other withdrawals besides saving: taxes and imports, like savings, resulted in unconsumed production.  They   were not determined solely by interest rates but rather by other things, especially the domestic level of production and income.  He also disputed the dominance of interest rates as the variable determining the behavior of saving.  Both saving and consumption were predominantly determined by the level of income and production, the other side of the income coin, so to speak.


Note that if no labor was used, looking at the total product curve, no output will be produced. But as we increase the quantity of labor employed by the firm in the production process, total output rises. We are portraying in this graph the more classical view of first experiencing increasing returns, then constant returns, and finally diminishing returns as we continue to increase the quantity of labor employed by a firm with a fixed amount of all other resources. As mentioned above, this is an example of the law variable proportions at work.

Focusing on the total product curve, as we increase the quantity of labor employed from zero to a positive number, note that total product rises at an increasing rate, or more than proportional to the increase in labor, then begins to increase at a decreasing rate or less than proportional to the increase the quantity of labor. We have passed through this inflection point where for an instant the productivity of labor has gone from one of increasing returns to diminishing returns.

But as we continue to increase the quantity of labor employed with the fixed amount of all other resources note, that the total product curve continues to increase at a decreasing rate and finally reaches a maximum of total output or total product. Further increases in the quantity of labor employed cause total product to fall meaning that we have reached the point of absolute diminishing returns.  If we were to continue adding more labor to the same amount of fixed amount of all other resources, we would eventually have the total product curve become negative as it crosses the quantity of labor or X axis.

If we were to take a line straight line from the origin that is just tangent to the total product curve, that is to say it just touches it in one place, and take the trace or the locus of those tangency points, we would generate what is called a marginal physical product curve which is seen in the second graph just below the first or upper graph.  Note that when the first inflection point is reached in the upper graph, increasing returns changes to diminishing returns, we have reached the maximum marginal physical product and by juxtaposition going up and down between two graphs, you can see that the two correspond to each other at that first point of inflection and the maximum marginal physical product.  If we continue to increase the quantity of labor employed with the given amount of all other resources, the marginal physical product will gradually turn negative.  If we look at that point in the lower graph and go straight up to the upper graph you see that's were total physical product is reached the maximum quantity.   Any further increase in the quantity of labor will bring us into the negative marginal physical product range or the range where the total physical product begins to turn downward.  Unless you subsidize the employment of that resource called labor, it would be economically irrational to use labor past this point or this quantity.  Furthermore, once we recognize that you have to reward labor and pay them an income, it becomes economically irrational even when labor's marginal physical product is positive at least over some of the range of diminishing marginal physical product of labor curve.

If we were to take a straight line from the origin of the upper graph and trace the locus of where that straight line intersects the total product curve in the upper graph we would generate an average physical product curve as is seen in the lower graph.  Note that when the straight line emanating from the origin is just tangent to the total product curve the average physical product curve is at a maximum and given this classical drawing of the total product curve that would be to the right or were a greater amount of labor is employed with the fixed amount of all other resources to the right of the maximum marginal physical product, in other words, we would first encountered the point of diminishing marginal returns and then the point of diminishing average returns or diminishing average physical product of labor. Finally as we continue to add more labor, we would reach the point of absolute or total diminishing returns which is the uppermost part of the total product curve in the upper graph.

This relationship is sometimes called the average marginal relationships and many instructors in economics would explain all this as it relates to batting averages.  If the player comes into the game with a 0.250 batting average or one for four over the season, he must hit on the marginal or incremental day better than 0.250 or better than one for four in order to improve his average.  Should he go one for 4, it will not change his season’s average and should he end up zero for four his season average will fall.  These relationships are very general relationships which occur in many areas of the measurement. 

In our case where labor is concerned, the MPP intersects the APP from above at the maximum average physical product.  As we monetize these physical product curves we will generate the cost curves of a firm.  We will see that the marginal cost curve intersects the average cost from the bottom at the minimum average cost.  

So far we have been in physical space, the relationship of output to the inputs that are used in the transformation process called production that produces the goods or services.  

These physical relationships are only part of the problem and part of the solution.  What we have to do is to monetize the physical product curves.  We do this in order to arrive at the cost curves for total cost, average cost, and marginal cost for the firm.

This is done by dividing the physical product of measures of total, average, and marginal into their related costs.  Just below is a chart showing these relationships and how we monetize the third graph by converting the physical product curves, given the cost of various resources, into the cost curves of a firm’s total, marginal and average cost curves. 


– Explaining how changes in labor compensation change unit labor costs –

Unit labor costs rise when labor compensation increases at a rate greater than the rate of increase in labor productivity and fall when compensation increases at a rate less than the increase in the labor productivity rate of increase.

3-Example of productivity the more productive labor holding compensation constant the lower the unit labor cost.gif


By making these calculations, the cost curves become U-shaped as the product curves are inverted.  Again, they are U-shaped as they are related to the physical product curves as cost curves are partly derived from the physical product curves and once we monetize a physical product curves we have arrived at the cost curves of the firm.

4-Showing the relationship of how adding labor as an input affects the output (production) until diminishing returns sets in.jpg

Where Average Physical Product is at a maximum, Marginal Physical Product is equal to it.

MPP > APP, APP is rising

MPP = APP, APP is at a maximum

MPP < APP, APP is falling

5-Monetizing the Cost Curves Decreasing costs to rising costs as more units are produced.gif

Where Average Variable Cost is at a minimum, Marginal Cost is equal to it.

MC < AVC, AVC is falling

MC = AVC, AVC is at a minimum

How does the principle of diminishing returns generally result in a U-shaped, or bow-shaped average and marginal cost curve?

At low levels of production where variable inputs such as labor are used less intensively relative to capital such as plant and equipment, physical productivity of labor is relatively low.  As more efficient combinations of labor and capital is achieved by increasing production, productivity of a variable input, such as labor, increases over a range of output or production.  In this range of output, increasing ret urns to labor result.  But eventually that physical productivity maxes out so to speak, and diminishing returns to labor occur.  Labor is being used in physical or engineering sense, too intensively.  

6-showing the relationship of units produced to Average Total Cost - Average Fixed Cost - Average Variable Cost.gif