Here is a brief discussion of the exponential and logarithmic functions. After this I have a short lecture on manufacturing.
The Exponential and Logarithmic Functions
The exponential function and the logarithmic function are perhaps the most important functions used in science. These functions are intimately related to the general concept of growth. In addition, if x is a complex number, it can be shown that the functions are related to cycles and waves used to describe many types of physical phenomena such as light and sound. This note uses very simple mathematical ideas to show the relation between the two functions and the enormous utility they provide.
Definition: Let . This constant is called Napier’s constant. It is a transcendental irrational number (i.e., it is not the root of any algebraic equation having rational coefficients) and 2.71828… Here is a table showing the rate of convergence.
n
------
1 2
10 2.5937
100 2.7048
1000 2.7169
10000 2.7181
etc
We can define ex as a function using the definition of e given above.
where n’ = nx
Therefore, we can write the exponential function as .
Theorem 1: If , then .
The proof of this is very easy.
Therefore, the exponential function has the remarkable property of being its own derivative. This means that if , then , or .
Definition: The natural logarithmic function of x is defined as
Area A = .
This idea is very easy to explain graphically. Consider the following diagram.
We can now see what happens if we change the value of x by a very small amount.
In the limit we have the following theorem:
Theorem 2: .
Proof: The proof of this theorem is motivated by the above graph and is a natural result of the Fundamental Theorem of Calculus.
Using the so-called “chain rule” of differentiation, we can write
Theorem 3: .
Proof: If we take a simple, non-rigorous approach, we can write
.
Now suppose that z = z(x), we can divide both sides by dx to get
which is the result we want.
If we set , then it is immediately clear that
and ln(z(x)) = x. It follows that
Theorem 4: .
The proof of this uses the above theorems 1 and 3.
We now see that the function is the inverse function of. We will find this theorem very useful indeed.
Theorem 5: ln(yx) = xln(y)
Proof: Let y = ez for some value of z which implies ln(y) = z by Theorem 4. Therefore,
ln(yx) = ln((ez)x) = ln(ezx) = xz = xln(y)
Theorem 6: ln(xy) = ln(x) + ln(y)
Proof: Let x = ea for some a and let y = eb for some b.
ln(xy) = ln(eaeb) = ln(ea+b) = a + b = ln(ea) + ln(eb) = ln(x) + ln(y)
Theorem 7: ln(1+z) z for |z| small.
Proof: We know that , but we can also write this as
which means that ez(1+z) for small z. Taking the natural log of both sides, it follows that ln(ez) = z ln(1+z).
Lecture on Manufacturing
We started this lecture by looking at a few stylized facts concerning manufacturing. As Rowthorn and Ramaswamy (1997) pointed out manufacturing is declining in ALL industrialized, advanced economies. It is not simply a US or European phenomenon. Manufacturing is declining in Japan as well. Rowthorn defines deindustrialization as the % employment manufacturing in total employment. Here is a graph of this for Taiwan using data 1978-2005:
By Rowthorn’s definition, Taiwan began deindustrializing in 1987 and this deindustrialization ended in 1996. Rowthorn discusses the deindustrialization of Taiwan in his article, but he could not discuss the fact that it apparently stopped in 1996 since the paper was written in 1997. Also, it is interesting to see that Taiwan is not being deindustrialized by China as many people predicted during the 1990’s.
Here is Rowthorn’s chart
Clearly all the countries are experiencing substantial drops in their manufacturing employment, while service employment is booming.
But, what is the definition of manufacturing?
Definition: Manufacturing of a product is the design, fabrication, packaging, and sale of a physical object to consumers or to other businesses. In some cases, some parts of this definition (excluding fabrication) may involve instead the use of the service industry, depending on the degree of vertical integration. Manufacturing involves the creation and sale of a tangible object – often components, value added raw materials or inputs used in the creation of other manufactured products; machines used by businesses; or appliances used by households. Manufacturing can be divided into light and heavy manufacturing industries. Textiles are an example of light manufactured products. Steel is an example of a heavy manufactured product. Manufacturing is often referred to as a secondary industry, as opposed to primary industry (agriculture) and tertiary industry (services). Industry is a broader term than manufacturing and encompasses three large categories, one of which is manufacturing. For the purposes of identification, tariff collection, and statistical analysis, manufactured products can be classified using a digit system.
There are two ways of classifying manufacturing products: NAICS (pronounced nakes), is the North American Industry Classification System. NAICS is used by business and government to classify and measure economic activity inCanada, Mexico and the US. It is in the process of replacing the olderStandard Industrial Classification (SIC) codes.
There are many people in the popular media who feel that the decline of manufacturing is a very bad thing. Others claim it is only a natural change in the economy. Here are some views commonly seen.
Pro: Manufacturing must decline as we increase the demand for services in the economy.
Anti: Service jobs that are arising to replace manufacturing jobs are low paying jobs at McDonalds and Starbucks. Workers are hurt by deindustrialization.
Pro: Manufacturing is just another industry, not special, and it is often involved in polluting the environment. Therefore we should let it move to other countries.
Anti: Outsourcing and the loss of manufacturing jobs areoccurring because business is looking for the lowest wages possible and does not feel a responsibility to keep the jobs at home. Manufacturing uses skilled labor and therefore the loss of manufacturing is a loss of high paying jobs. Moreover, many imports are being subsidized by foreign governments. We must protect our manufacturing industry.
The types of comments above can be heard on almost a daily basis.
What determines the fate of the manufacturing sector? There are three basic determining factors (aside from the fact that some parts of the manufacturing process can be spun off to the services sector – like designing, packaging, and marketing)
(1)Changes in the Demand for Manufacturing:
(2)Changes in the Productivity of Labor in Manufacturing and Services
(3)Changes in Net Imports of Manufactured Goods
Let’s look at each of these factors separately:
(1)Changes in Demand for Manufacturing
The demands in some sectors of the economy grow more quickly than the demands in other sectors. A good example is agriculture. Since we can only eat so much, as our incomes grow, the demand for food does not grow as much. We say that the income elasticity for food is less than one. This implies a concave Engel Curve.
As income grows, food demand grows, but more slowly.
This implies that the ratio (Food / Income) must fall as income rises. Some sectors of the economy are not like this. They increase more than proportionately as income grows.
Taiwan has three major areas where demand is shifting as income grows.
(i)Health expenditures
(ii)Education and Recreation (or Entertainment)
(iii)Transportation
The first two (i) and (ii) are clearly service industries, while the last involves both service (the airlines and buses) and manufacturing (autos). Here is a table showing the changes over time
Sector 1974 1985 1990 2000 2004
Health 3.92 5.26 4.82 11.09 12.91
Educ. & Rec 6.38 9.52 13.34 13.51 13.31
Transport. 4.99 8.83 8.83 11.37 12.54
Unfortunately this table is missing an important service sector, namely the financial sector, and therefore misses the tremendous increases finance has made beginning in the mid-1980’s. It also ignores changes in prices.
Clearly demand has been changing towards services although these changes do not exactly line up with the time that the manufacturing sector declined for Taiwanin our graph above.
(2)Changes in Productivity for Manufacturing and Services
This is the main reason Rowthorn gives for the general decline in the percentage manufacturing employment. Manufacturing productivity gains are faster than those in services.
The subject of productivity growth is very confusing. On the one hand, an increase in labor productivity makes labor more attractive and therefore labor’s demand increases. On the other hand, a rise in productivity is equivalent to an increase in labor and therefore this serves to reduce the marginal product of labor. Moreover, a rise in productivity increases supply (without an increase in employment) and reduces product prices which in turn reduces the demand for labor.
In competitive product and labor markets we know
P(MPL) = money wage
And, therefore for fixed money wages, any reduction in price P lowers the quantity of labor demanded. However, a rise in MPL will raise the demand for labor. The electronics industry is a good example of how that productivity can lead to reduction in overall employment in electronics. The mechanization and technical breakthroughs in electronics raise the productivity of labor and this leads to an expansion in output in the industry. If substantial, this can be a powerful boost to employment. However, these reductions in cost, along with fierce competition at home and abroad, reduce prices; and this can lead to a reduction in employment if money wages do not similarly drop. For many countries this is difficult, so industry tends to move to lower wage areas. Note how that wages are falling because of the strong growth in productivity and the intense competition among firms. We may not like the idea of industry moving to low wage countries, but we all enjoy the low priced high quality products that are offered by such firms. Instead of spending $20,000 NT for a DVD player, we now pay only $3000 NT, and the residual $17,000 NT can be spent on something else, including DVDs! (a service industry).
The general form for productivity gains can be written
Y = Ao F( BoL, CoK )
In some cases, it is possible to combine Ao, Bo, and Co. A simple example would be the Cobb-Douglas production function
where . In this case, it is easy to see that an increase in A, B, or C will simply shift the MPL = and increase the demand for labor. The subsequent decrease in product price due to the increase in supply may lower employment in the long run, though.
This is not always true.
Suppose our (microeconomic) production function can be written as
.
In this case, an increase in B will actually reduce the demand for labor. Note how that a rise in B causes the MPL to decrease. The term “BL” refers to labor measured in “efficiency units”. Obviously, for fixed L, an increase in B raises (Y/L), which is productivity.
So, at times, an increase in B can increase the demand for L, and at times, it can decrease the demand for B. The issue is ultimately an empirical matter. The effect of productivity on employment is not at all clear.
We can see the effect of B on the slope of the production function from the following graph:
It is not hard to draw this graph so that L increases instead.
The basic mathematics can be seen as,
and therefore
Now, FLL is negative, but FL is positive. Therefore, it is not clear whether B will increase of decrease the demand for labor, MPL.
(3)Changes in Manufacturing Net Imports
A large increase in imports of manufactured goods can be expected to reduce the demand for domestically produced manufactured goods, even if they are not the same.
In addition, it may be the case that some firms locate abroad and cut domestic production with an aim to import the goods back into the country they have just left. The shoe industry in Taiwan has almost completely left the island and relocated in China and Southeast Asia. Shoes sold in Taiwan now are largely from those two areas.
Here is a small and simple graphical analysis of an increase in manufactured imports.
Clearly the increase in imports reduces the domestic output and increases imports. Domestic demand increases also. The expansion of exports is probably not very great. The increase in exports is satisfied by part of the domestic output, so that a smaller part of domestic output is devoted to domestic demand. This simple diagram helps to explain two important and seemingly contradictory stylized facts concerning deindustrialization in the US; (i) domestic manufacturing output as a percentage of GDP is falling, and (ii) exports of manufacturing as a percentage of total exports are increasing.
(4)Which of the Three Factors is Most Important
We finished the lecture by looking at a short paper by Josh Bivens that tries to quantify the impacts of the three factors determining deindustrialization in the US.
Bivens begins with an identity involving productivity:
where Y = real output in manufacturing, y = productivity in manufacturing, and L = employment in manufacturing. If we define d = Y/YD as the ratio of output to demand for manufactures, then we can rewrite the identity as
Next, we take the natural logarithm of both side to get
If we then difference both sides of this identity we get
which implies
(*)
Now, we must consider the term d more closely.
Obviously domestic demand is satisfied by domestic output and net imports, where net imports, NI, are just (imports – exports). This means we can write
.
Dividing by Y we get
where . From this it is easy to show that equation (*) above can be rewritten as
This is the accounting relation that Bivens uses in his paper. He then gets data on all four terms and computes the changes as given. The relation is not exact because it uses an approximation and also because it uses data from different sources. The residual that occurs between left hand side and right hand side is distributed equally among the three right hand side terms so that the accounting relation is exact and adds up completely.
What were the results of Bivens analysis? Which of the three right hand side factors (i.e., demand, productivity, net imports) were most important? Here is the table given in the paper.
(See next page)
The green box highlights the data (only three years) while the red box highlights the results. Note in particular how that (1) productivity growth lowered employment considerably (just as Rowthorn claimed), (2) demand increased for manufactures but not enough to offset the increase in productivity, and (3) imports accounted for a very heavy weight (roughly 1/2 as big an effect as productivity). This is evidence that imports were important in accounting for the declines in manufacturing in the US during the period 1998 – 2003.