Experience, Mathematics and Theory (I)

                           Experience, Mathematics and Theory

                                                   -A Brief Discussion on the Methodology of Economics

Since the publication of Adam Smith’s masterpiece The Wealth of Nations in 1776, economics has started a process of gradually developing, enriching and deepening, while the publication of Marshall’s great work Principles of Economics in 1890 marked that modern economics has begun to form its own theoretical system, and its development speed has greatly accelerated, and it is a somewhat obvious basic fact that economics in the 20th century has undergone a many-sided flourishing development. However, with the rapid development of economics, due to its multiple branches (the research scope of every branch is also very broad, and meanwhile, these branches are also continuously interbreeding and mixing) and complex levels, and thus, to some extent, contemporary economics gives people a dazzling, difficult to grasp overall impression; therefore, systematically combing the internal structure and discipline level of modern economics becomes a quite meaningful thing (and this is also an issue which numerous economists have discussed and deeply interested in). Our general view is that modern economics is made up of experience, mathematics and theory these three major parts, and these three broad parts are also rapidly expanding and advancing, and they separately and blendedly constitutes the vast field of modern economics.

(I) The Experience Part in Modern Economics

For the economists mainly engaged in empirical research, they can often feel that the empirical range of modern economics is very broad. To begin with, take international economy as an example, the issues people often discuss about are quite a lot: for instance, many scholars will discuss currency issue, like the exchange rates of dollars to Euro, yen, rouble, RMB, etc (in today’s February, 2018, the exchange rates of dollars to the above currencies are roughly 0.81, 109, 58 and 6.4), and meanwhile, many economists will care about international oil price, and economic scholars will also heatedly discuss many widely concerned questions, like American economy, European economy and Chinese economy. It should be noted that the empirical research of economics should be based on a large amount of concrete data, and take the corporate bond for example, American corporate bond now is somewhat booming, and the reason is mainly due to America’s long-term low interest rate policy, and many companies like Goldman Sachs and Wal-Mart (Wal-Mart  issued 5 billion bonds in April, 2013) are issuing corporate bond, and Avago Technologies used 37 billion to buy Broadcom in May, 2015, also mainly relying on bond, in recent years, Apple and Broadcom have raised about 18 billion dollars capital from the corporate bond market, while Apple has issued 56 billion bond since April, 2013. Meanwhile, economists will also care about global stock and bond market, in them, after a surge of rising  in the global stock market, it is gradually stabilizing, while the profitability of global bond market is somewhat going down, for instance, the profitability of German national debt is under 1%, and the profitability of national debt in France, Japan and America is also very low, these all show that the price of debt is high and the bond market is hot, while the low-priced bond in emerging market countries is also hot, like Belarus and Egypt, and thus has a possible default risk. To sum up, the empirical range of global economy is very broad, and meanwhile, due to the complexity of global economy, leading to a somewhat superficial understanding of many scholars, without solid data and a large number of varied case studies.

Secondly, we want to discuss about some issues of Chinese economy. To begin with, about the reasons of Chinese economy’s development in recent 20 years, we think three are 3 major points: firstly, the process of urbanization results in the need of buying houses for many migrant workers, and when decorating houses, people also need to buy large goods, like TV, air conditioner, car and refrigerator, which leads to an increase of domestic demand; secondly, the wages of our ordinary people have a certain growth, and therefore, the national consumption has a certain improvement, which stimulates the development of many fields, like restaurant, tourism, consumer goods and electronic products; thirdly, our air conditioner, TV, refrigerator and smart phone are exporting in large scales (in them, the total exports of Chinese electronic products are over 1000 billion dollars last year, and thus, is one of the major sectors for export), but the major reason is low price, low content of technology, and has a price advantage; a large portion of our manufacture is assembling, and the core components of electrical appliances like refrigerator are also not our own, we mainly do the assembly part, and in robotic field we also mainly make assemblies. About the state of our country’s tourism and catering, many people think that our country’s tourism is fastly developing, which is a good phenomenon, however, the deeper reason is easy to understand and alert: those middle-aged and old people around us are somewhat frugal, but the young generation is not so restrained and is fond of seeking pleasure, and therefore, our tourism and catering are naturally thriving, most of our middle-aged and old people are frugal, while only few of the young generation are thrifty, for instance, in the 130 million tourists for foreign travel in China in 2017, young people born after 1980 take up about 70%, in a word, we need to have a rational and sober awareness about the development situation of our country’s catering and tourism. (of course, we need to admit that with the certain increase of our people’s income, the number of trips of all ages and various places has gradually increased, which is certainly one important reason for the development of China’s tourism) About the current financial condition of our local government at different levels, on the whole, is somewhat tight (the reasons include long-term infrastructure investment and civil servants’ wages), and the debt is somewhat heavy, and the local government debt has some solutions, for instance, the banks can use local government debt as mortgage, whose term is 2-3 years, interest rate is 4.4%, to change into bonds like SLF and MLF of central bank, whose interest rate is 3.5% and has a 1% interest margin, and the term changes into 10 years, namely, using long debt to replace short debt, namely, about local government debt issue, we also cannot just have hollow views, but need to know a lot of data and concrete facts. About the overall situation of our manufacturing industry, we have to say that we are still relatively backward, take medical instruments as an example, we are familiar with some American companies like Baxter、Tyco and Medtronic, and some German company like Fresenius, and they all have a wide market, but we perhaps donot have domestic companies whose name we know. To sum up, various aspects of our country’s economy and technology are somewhat backward, and the reasons probably are: 1 we donot profit much in advanced manufacture fields, take medical instruments for example, producing one CT and NMR can get high profits, and meanwhile, in pharmaceutical field, we also donot have pharmaceutical companies like Merck, Pfizer, GlaxoSmithKline, Roche and Johnson (the gap between our pharmaceutical companies and foreign companies is large), and what our  pharmaceutical companies can produce are just some simple drugs, and mostly are generic medicines and donot have original drugs, and most high-tech, expensive drugs are imported from abroad, and take big machinery as another example, many of our soil shifters and loaders are also imported, in brief, only these high-tech products with core technology can really have lucrative profits. What our companies have are mostly some simple, peripheral technologies, not some difficult, sophisticated core technology, and thus, only few of our companies get high profits, and mostly are small profits and slightly profitable. 2 Our suits, make-ups and movies also cannot export to foreign countries, and the added values of these products are very high, but they mostly need art creation and conception design, and require high-level intellectual qualities, which our country currently relatively lack. To sum up, we at least lack the above two kinds of companies, and thus, our companies with high profits are relatively few, and the small profits of companies naturally lead to the low salary of workers, namely, the lack of profitable companies has a profound influence to us.

Through the above simple analysis about international economy and Chinese economy, we can get two basic conclusions: firstly, if we just have some vague overall views about various issues, then our analysis cannot be very meaningful, when many of our scholars discuss economic affairs, they can just give some general viewpoints, like ‘urbanization’, ‘social inequality’, ‘inflation’, ‘joining WTO’, ‘government regulation’, etc, and they perhaps do not know some concrete information, like Ctrip, T+1 in finance, some tools like credit asset mortgage reloan of central bank, the currency of Australian dollar in recent two years, etc, and thus, their analysis are not very persuasive, namely, the empirical research of economics should be based on two solid foundations : 1 a large amount of data, 2 many concrete facts. Secondly, the range of empirical research of economics is very broad, as shown before, we briefly analyze some issues like macro-economy, corporate bond, local government bond, currency, tourism, debt market and manufacture, and they are naturally just a very small part of empirical research, and thus, to do good empirical research of economics, economists should remember tens of thousands of numbers and concrete facts in mind. However, we probably need to add that, for myself, I do not like to talk about concrete economic facts, because I spend a long time and through many channels to get these valuable information and ideas, and thus, the facts mentioned above mostly are issues which I have discussed at other places, and we donot add too many new things, in summary, I just want to talk about some general understandings, not about concrete economic facts.

About the empirical research of economics, one important characteristic is that different scholars’ understandings about the same issues are often different, for instance, about local debt, some scholars think it is very serious, while some others think it is not so serious, and about China’s manufacturing industry, some scholars’ view is very optimistic, while some others are relatively pessimistic, about the currency of RMB, some people think we should appreciate, and others think that we should depreciate, and about social financing, different people’s opinions also differ. The reason why economists have different views on many issues (or rather, most issues) is naturally somewhat complex, and probably are the below 2 points: firstly, the breadth and precision of different people’ s knowledge and experience is different, some scholars’ knowledge structure is more theoretical, and thus, they do not know too much concrete information, while some scholars’ knowledge structure focuses on experience, and thus, they know more concrete information than the former, and thus, their judgements about many issues are more accurate, and meanwhile, even for two scholars who mainly do empirical research, due to the difference of experience, knowledge and ability, also leading to a big gap in the quantity and quality of their thoughts, and therefore, more capable individuals’ judgements are more objective and accurate. Secondly, even for two scholars whose knowledge and experience capacity is roughly equal, the specific issues they concern are also different, for instance, one of them will pay attention to stock market and tourism, while the other cares more about international trade, express industry and entertainment, namely, the study subjects they specialize are different, which will also result in the difference of their judgements about many issues.

In the empirical research of economics, famous economist Friedman is a good example, and in the magnum opus A Monetary History of the United States, 1867-1960 which he co-authored with Schwartz, they collect, organize and analyze very many economic facts, and we can quote one passage from this book: “From January 1867 to February 1879, the money stock at nominal value rose at the rate of 1.3 percent per year; the price index fell at the rate of 5.4  percent per year. Part of this contrast probably reflects statistical defects in our estimates. We have seen that our money figures overstate the rise in the money stock by failing to allow for the excess of market over nominal value of gold. But this would have only a minor effect in bridging the gap between the changes in money and in prices, at most reducing the estimate of the rate of rise in money from 1.3 to 1.1 percent per year. We shall assume that the low figure measures the true rate of growth of the money stock over this period.”[1]  In this passage, Friedman and Schwartz mainly discuss the relationship between money stock and price index, in them we can feel that Friedman and Schwartz have a very delicate and systematic understanding of economic experience, and A Monetary History of the United States can be seen as a classic work of empirical research. From this passage, we can also get another basic conclusion, namely, many economists think that they know quite a bit of economic experience, but, in fact, what they know are just some macro facts, like education, government policy, economic development and social security, and these facts are perhaps not practical economic experience which the business and industry world care about, and thus, what these economists know are mostly some indirect, general economic facts. 

We need to point out that the empirical study of economics should not be just restricted in the economic affairs, and it should also be based on extensive life experience, because CEO trained by business school should be responsible for the overall operation of a company, then what he knows should not be limited to business management ideas, and to maintain the operation and development of one company's various aspects, he must be familiar with much life experience. Here, we want to use the great investor Buffett as an example, when Buffett explains the business philosophy of his company, he says: “A restaurant could seek a given clientele-patrons of fast foods, elegant dining, Oriental food, etc-and eventually obtain an appropriate group of devotees. If the job were expertly done, that clientele, pleased with the service, menu and price level offered, would return consistently. But the restaurant could not change its character constantly and end up with a happy and stable clientele. If the business vacillated between French cuisine and take-out chicken, the result would be a revolving door of confused and dissatisfied customers.” [2]   Namely, Buffett uses the restaurant as an example to explain that Berkshire should keep stable operation mode, and should not pursue high liquidity and active trading of its stocks. At another place, he uses the competition skills of car racing to explain the reason why Berkshire rarely borrows money, he says: “This conservative approach is not good for our operating performance, but considering our fiduciary duties to policy-holders, creditors and many stockholders who give most of their net assets to us to manage, this is the only way for us to feel comfortable. As one of the Indianapolis 500’s winner said: ‘To finish first you must first finish.’”[3]  From these two examples, we can preliminarily feel that Buffett has a strong interest in food (like cooking and Coca Cola) and sport (as is well known, Buffett is quite interested in sports like baseball and golf), while Buffett is naturally an entrepreneur who does business, and obviously, the range of life experience is very broad, one people can know little about economic empirical research, but he must be familiar with rich life experience, and for people who are in real life, if he does not have broad, systematic, mature and delicate daily experience, then it will be a horrible thing, and correspondingly, economic research without the foundation of life experience is mostly unilateral and vulnerable. To sum up, economic empirical research and daily life experience are both necessary for good economic empirical research. (Here, Adam Smith is also a good example, it is well known that Smith discusses a lot of life experience in the Wealth of Nations, and also makes a wide range of economic analysis in theoretical level.)

Now, we want to do some general discussions about the empirical research in economics, as we know that, quite a number of scholars in the economical world mainly do empirical research, involved in related aspects like stock market, local debt, bank, real estate, monetary policy, and this research method is quite normal and reasonable, because if most economists just do purely theoretical research, then the economical world will lose broad and active connections with real life and the business world, then economic research will lose most of its empirical basis and lose valuable direct information like lots of actual experience and data, and this kind of research method will make economic research into ivory-tower pure theory, then economics will become increasingly closed, and economic research will become water without a source, and thus, the overall effect will be disastrous. In a word, we think many scholars study some practical issues, like the price trend of real estate market in Australia in 2017, the change of currency at the international level between 2015 and 2018,  the bank rate in Mexico in February, 2018, the trend of international oil price in 2017 and the unemployment rate of Europe in recent 6 months, is not only beneficial to the society, and also beneficial to the overall healthy development of economics. But, conversely, it is also not appropriate if no one in the economical world does foundational theoretical research, because the thought system, analytical concept and theoretical perspective provided by theoretical research are the driving force for the continuous development and expansion of economics. To sum up, empirical research is an important component of economics, and takes up a large area of economic science, and it is the cradle of complex economic experience, and also has a useful interaction with theoretical research-much economic theoretical research originates from the direct, strong stimulus of practical experience, and the research of economic theory will also have multi-faceted fundamental impact to empirical research.

(II)The Mathematics Part in Modern Economics

      It is widely known that mathematics has an all-round, deep and strong influence over various fields of modern economics, like macroeconomics, microeconomics and international economics, and to better feel the complex influence of mathematics to modern economics, we want to discuss some models and theories in economics below, and using them to discuss about the internal relationship  between economic model and mathematics. 

1The technology part in microeconomics

As we know, modern microeconomics has formed a profuse theory system, and in it, technology has been systematically and detailedly analyzed, here, we want to do some inspections on several concepts of technology.

(a) Regular technologies. Firstly, we want to briefly discuss one common kind of technology-regular technologies, and its topological definition is: for any y≧0, V(y) is a non-empty closed set, here, V(y) represents the input requirement set, namely, the set of input sequences which can at least produce y unit of output.

One meaning of the assumption that V(y) is a closed set is : assume we have a sequence of input bundle xi , and each one of them can produce y unit of output, and assume that this sequence converges to input bundle x0,  in other words, the input bundles in this sequence can be infinitely close to x0. If V(y) is a closed set, then this determines that the input bundle x0 must produce y unit of output. It is easy to understand that most technologies we are familiar with are all regular technologies. 

(b) Technology Substitution Rate (TRS) problem. TRS is also a common concept on the analysis of technology in microeconomics, and it is provoked by the following question: assume that we have a given technology which can be described  by one smooth production function, and we are producing at one certain point y*=f(x_1^*,x_2^*),  if we want to increase the amount of input 1 and also reduce the amount of input 2, to keep an unchanged output level, then how can we determine the technology substitution rate between these two factors? 

After identifying the basic intension of this concept, now we want to solve it. Let x2(x1) be the implicit function of how many units of x2 we need to produce y units of output if we are using x1  units of other input. Then, by definition, the function x2(x1) must satisfy the identity 

f(x1,x2(x1))≡ y

Now, we can figure out of  ∂x2(x1*)/∂x1.  Differentiate the above identity, and we can get :

  ∂f/(∂x_1 )+∂f/(∂x_2 )  (∂x_2 (x_1^(*)))/(∂x_1 )=0

or

   (∂x_2 (x_1^(*)))/(∂x_1 )=-(∂f/(∂x_1 ))/(∂f/(∂x_2 ))

Then this exactly gives a definite form of TRS.

Below we want to solve one typical example: the TRS of Cobb-Douglas technology

Given f(x1,x2)=x_1^a x_2^(1-a),  and compute the partial derivative, we get

     ∂f/(∂x_1 )=ax_1^(a-1) x_2^(1-a)=a(x_1/x_2 )a-1 

 ∂f/(∂x_2 )=(1-a)x_1^a x_2^(-a)=(1-a)(x_1/x_2 )a

 From this, we can get 

              (∂x_2 (x_(1)))/(∂x_1 )  =-(∂f/(∂x_1 ))/(∂f/(∂x_2 ))=-a/(1-a)  x_2/x_1 

This is naturally a typical problem to solve TRS, and from it, we can feel that it has a very clear method to solve this kind of problem, and meanwhile, TRS problem itself also has very clear economic and mathematical intension.

(c) Scale return and Scale elasticity problem. Scale return is a concept we are familiar with, and using strict mathematical language, one technology shows constant scale return means one of the following conditions is satisfied: 

(1) For any t ≧0, y is in Y implies ty is in Y,

(2) x is in V(y) implies for any y≧0, tx is in V(ty)

(3) For any  t≧0 , f(tx)=tf(x), namely, the production function f(x) is linear homogeneous. 

For constant scale return problem, the reason why we give three interrelated but different definitions above is that constant scale return is often a reasonable hypothesis about technology, but, in some cases, it seems not to be a reasonable hypothesis.  

Because scale return is essentially global, and therefore, it is likely that one technology shows increasing scale return for some values of x, but shows decreasing scale return at some other values of x, and thus, in many cases, the local measurement of scale return is useful, and scale elasticity is exactly the concept to locally measure the change of scale return, and what it measures is due to the scaling change of all the input factors-namely, the change of production scale, the percentage change of output divides the percentage change  of production scale.

We use y=f(x) to express production function, and let t be a positive scale, and consider function y(t)=f(tx), if t=1, we keep the current production scale, and if t>1, we upward adjust all the input, and if t<1, we downward adjust all the input. 

Now, we introduce the computation formula of scale elasticity

e(x)=((dy(t))/(y(t)))/(dt/t)，

and the above formula is the percentage change of output divides the percentage change of production scale, and because what scale elasticity measures is instantaneous change, at this point the change of production scale compared with the original scale is small, and thus, t changes around 1, and thus, we need to compute the value of the above formula when t=1. Rearrange this , we have 

e(x)=(dy(t))/dt t/(y(t))|t=1=(df(tx))/dt  t/(f(tx))|t=1

Note that we must figure out the value of this formula when t=1, and compute the scale elasticity at x. When e(x) is greater, equal, or smaller than 1, we say that this technology shows locally increasing, constant or decreasing of scale return.

Example: Scale return, scale elasticity and Douglas technology.

Assume y=x_1^a x_2^b , then f(tx1,tx2)=(tx1)a(tx2)b=ta+bx_1^a x_2^b=ta+bf(x1,x2). And thus, if and only if a+b=1, f(tx1,tx2)=tf(x1,x2), and at this point, scale return is fixed. Similarly, a+b>1 means scale return increases and a+b<1 means scale return decreases. 

In fact, the scale elasticity of Cobb-Douglas technology is exactly a+b. We use the definition

(d〖(tx_1)〗^a 〖(tx_2)〗^b)/dt =(dt^(a+b) x_1^a x_2^b)/dt=(a+b)ta+b-1x_1^a x_2^b

Compute the value of this derivative at t=1 and divides f(x1,x2)= x_1^a x_2^b , then we can get this result.  [4]

(d) Some discussions. From the above analysis about technology in microeconomics, we can feel that many concepts, ideas and methods in it all have clear economic intension, and also have clear mathematical computation approach, in summary, the discussions about technology in microeconomics are systematic and strict, and these different contents also have close internal connections and form a coherent, organic whole, not just some unconnected knowledge fragments. It is obvious that technology is only a small part of microeconomics, while if we want to make good innovations in the future, then we need to lay a solid foundation in various aspects of microeconomics and need to systematically and deeply absorb a lot of ideas, concepts and approaches in it, because they have many useful small parts and these small parts are often indispensible ingredients for the future important innovations.

2 Samuelson’s general equilibrium model of public goods and other theories

As we know, Samuelson built many important economic models, and here, we firstly discuss about the the general equilibrium model he built in public goods field.

This clear and broad model has the following 5 basic hypotheses: 

(1)There are only two consumers in the society: consumer A and consumer B;

(2)Social products can be strictly divided into private products and public products, and in them, the total amount of private products is X1,  and the quantity consumer A and B respectively consume is X_1^1 〖 and X〗_1^2, and they satisfy the quantity relationship X1= X_1^1+X_1^2; the total amount of public goods is X2,   and the quantity consumer A consumes is X_2^1, the quantity consumer B consumes is X_2^2, and they satisfy the quantity relationship X2= X_2^1+X_2^2;

(3)The utility function of two consumers respectively is: uA(X_1^1, X_2^1) and uB(X_1^2,X_2^2);

(4)There exists one production function (production possibility curve/ cost function), and let it be F(X1,X2), namely, X1  and X2  should satisfy the internal constraints of this production condition; 

(5)There exists one social welfare function, used to evaluate the size of social utility and consumption, and let it be U(uA,uB).

Meanwhile, we need to add two concepts: marginal rate of transformation and marginal rate of substitution, and the former one represents with a given social resources, the output of commodity 2 we need to give up to increase one unit of commodity 1, and the marginal rate of transformation of production 1 to production 2 is often expressed as MRT1,2, and its computation formula is MRT1,2=-(dx_2)/(dx_1 ), and the later one represents in the condition to keep total utility unchanged, the consumption quantity of commodity 2 the consumer need to give up to increase one unit of commodity 1, and the marginal rate of substitution of production 1 to production2 is often expressed as MRS1,2,  and its computation formula is MRS1,2=(MU_2)/(MU_1 ) .

With the above theoretical assumptions and concept preparations, now, we can deduce Samuelson’s general equilibrium model of public goods, and the objective function of this model is to maximize social welfare function U(uA,uB), and the constraint condition is cost function (production function/production possibility curve), namely, F(X1,X2)=C. Therefore, the supply of pure public goods needs to solve the following problem:

max      U(uA,uB)       s.t.   F(X1,X2)=C                                                          (1)

{ X_1^1, X_1^2,X2}

This is naturally a typical conditional extremum question, and we already have very mature mathematical approaches to solve this kind of problem, namely, Lagrange multiplier approach. In this question, we first need to construct the corresponding Lagrangian function L= U(uA,uB)- λ(F(X1,X2)-C), and after it, we need to figure out the following four partial derivatives: 

∂L/(∂X_1^1 )=0   →  ∂U/(∂U_A )  (∂U_A)/(∂X_1 )  〖∂X〗_1/(∂X_1^1 )  -λ ∂F/(∂X_1 )  〖∂X〗_1/(∂X_1^1 ) =0 → ∂U/(∂U_A )  (∂U_A)/(∂X_1 )-λ ∂F/(∂X_1 )=0                                                   (2)

∂L/(∂X_1^2 )=0   →   ∂U/(∂U_B )  (∂U_B)/(∂X_1 )  〖∂X〗_1/(∂X_1^2 ) -λ ∂F/(∂X_1 )  〖∂X〗_1/(∂X_1^2 )=0 →     ∂U/( ∂U_B )  (∂U_B)/(∂X_1 )-λ ∂F/(∂X_1 )=0                                                   (3)

∂L/(∂X_2 ) =0  →   ∂U/(∂U_A )  (∂U_A)/(∂X_2 )+∂U/(∂U_B )  (∂U_B)/(∂X_2 )  - λ ∂F/(∂X_2 )=0                                                                                   (4)

∂L/∂λ=0  →  F(X1,X2) =C                                                                                                          (5)

From (2) and (3), we can get: 

{█(∂U/(∂U_A )=λ ∂F/(∂X_1 )/MU_1^A@∂U/(∂U_B )=λ ∂F/(∂X_1 )/MU_1^B )┤                                                                                                              (6)

Bring the two equations of (6) into (4), we can get the following identity:

λ ∂F/(∂X_1 )  (MU_2^A)/(MU_1^A )+λ ∂F/(∂X_1 )  (MU_2^B)/(MU_1^B )-λ ∂F/(∂X_2 )=0                                                                                                   (7)

Arrange (7), we can get the following identity: (   MU_2^A)/(MU_1^A ) +(MU_2^B)/(MU_1^B )-=F_2/F_1                                              (8)

Generalize the conclusion of equation (8) to the situation of multiple consumers, we can get the general form of Samuelson’s condition:

〖 ∑〗_(i=1)^s (MU_G^i)/(MU_X^i )=F_G/F_X          (Here, G is the total amount of public goods, and X is the total amount of private goods)                                                                                                                              (9)

Equation (9) is also: 〖 ∑〗_(i=1)^s MRS_(X,G)^i = MRT_(X,G)  

Equation (9) indicates that the sum of marginal rate of substitution of all individuals’ public goods and private goods equals the marginal rate of transformation between public goods and private goods. To sum up, in this model, Samuelson assumes that there is an omniscient planner in life, and he knows everyone’s willingness to pay for public goods, and thus can figure out the quantity of public goods the government needs to supply. Here, Samuelson uses an assumption of an omniscient planner to overcome the difficulty of individual’s  display about public goods preference, and also overcome the difficulty of adding up the individual preference to construct social welfare function, and obviously, this model has a somewhat strong government regulation 

color.

      In his long academic career, Samuelson also built many other important models, like the Stolper-Samuelson Theorem in international trade. This theorem indicates that, in the long run, after conducting international trade, the relative price of one country’s output products will rise, and the relative price of input products will drop; due to the increase of output products’ price, the labor and capital return used by output industries will both increase, and conversely, with the decrease of input products’ price, the labor and capital return used by input competitive industries will both decrease. In summary, in international trade, the return of the production factors intensively used by output products production (namely, the abundant production factors of one country)  will increase, while the return of the production factors intensively used by input products production (namely, the scarce production factors of one country) will decrease, and whatever industries these production factors are used by are all so. Obviously, this theorem has a gap with the basic idea in free market theory that international trade will benefit all the industries, and it also shows the necessity of government regulation in economic development.

     Samuelson’s other theoretical contributions also include Heckscher-Ohlin-Samuelson Theorem, which shows that if there exist differences of product price, the two countries will continue to conduct trade, and the final result is an equal price of these two countries’ two products, and the price of production factors is also equal, at this moment, if other conditions keep unchanged, then the trade will end, and thus, this theorem is also called “Theorem of Equalization of Factor Price”. In addition, there is also the well-known Balassa-Samuelson Hypothesis, which is the most influential theory in the study of the relationship between economic growth and real exchange rate, and also an important foundational proposition in contemporary international economics.

     As described above, we briefly discuss Samuelson’s partial academic contributions, while all of  his contributions are naturally much broader, and through the above discussions, we want to make two summaries : firstly, many models Samuelson builds have a strong tendency of government regulation theory, and it is true for his international trade theory, taxation thoughts and welfare economics, which is naturally inseparable with his overall economic views. Secondly, when facing many models Samuelson builds, people cannot help asking: why can he make so many brilliant contributions? In fact, it indeed is not a very difficult thing for Samuelson to build new economic models, which naturally stems from his broad and deep mathematical background and economic theory background, and is an organic combination of mathematics and economic theory, on one hand, Samuelson’s mathematical (and economic theory) background is deep and has enough depth, which is easy to understand, on the other hand, his mathematical (and economic theory) background is also broad and has enough breadth, thus, he is able to build many different economic models with different approaches in many fields, namely, the breadth of mathematical (and economic theory) background is also an indispensible important premise for good economic modeling. Broadly speaking, many models Samuelson builds are all clear and complete, and also have definite and wide economic intension, when our mathematical (and economic theory) foundation is not solid enough, we will marvel at these models, but when our mathematical (and economic theory) foundation is deep and broad enough, we will find that his models are actually traceable and the deductions behind many notations all have definite and orderly motivations and mathematical background.

3 General Equilibrium Model and related analysis

In microeconomics, general equilibrium model has attracted a great deal of attention, and after many scholars’ repeated perfection, this theory already has a somewhat mature form. We want to give some simple introductions about it below and make some related analysis.

(I)The assumptions of this model

Firstly, general equilibrium model has two elements:

(1)One price function P, its independent variables are a group of commodities, which are  transformed into a set of real numbers reflecting their prices, and normally, the price function is linear.

(2)One distribution matrix X, for every i∈1,2,•••,n, Xi  is the commodity vector allocated to agent i.

These elements should satisfy the following requirements:

The preference degree of every agent to a group of commodities can be reflected through price function, namely,

∀i∈ 1,2,•••,n, if P(Y)≤ P(Xi), then Y〖≼ 〗_iXi

Normally, there is an initial endowment matrix E: for ∀i∈ 1,2,••• ,n, Ei  is the initial endowment of every agent.

With these assumptions, general equilibrium model should satisfy the following 3 conditions: 

(1) Demand equals supply, namely, ∑_(i=1)^n?X_i =∑_(i=1)^n?E_i 

(2) Individual Rationality: the conditions of all the agents are better than before the exchange, namely, ∀i ∈1,2,••• n: Xi ≽_iEi

 (3)Budget balance: all the agents can afford their allocated commodities based on their endowment.

(II) The proof of general equilibrium model

Arrow-Debreu model indicates that in every exchange economy which satisfies the following two conditions, the competitive equilibrium always exists: 

(1) every agent  is strictly convex preference,

(2) all the commodities are desirable, which means that if one commodity is freely supplied, namely (Pj=0), then every agent wants to get as many as possible from that commodity.

The proof of this well-known theorem can be broken down into the following 5 steps:

A Assume that there exist n agents and k distributable commodities, and normalize the price vector, we get :∑_(j=1)^k?p_j =1. Thus, all the possible price space forms a k-1 dimensional simplex.

B Let z be the excess demand function, when the initial endowment is constant, this is a function of one price vector p:

z(p)= (p,p.Ei)-Ei

While we know that  when the agent has strictly convex preference, Marshallian demand function is continuous, thus, z is a continuous function of price P

C Define the following function which maps the price simplex to itself, 

gi (p)=(p_i+max(0,z_i (p)))/(1+∑_(j=1)^k?〖max(0,〗,z_i (p)))   ,                                        ∀i∈ 1,••• k

This is a continuous function, and thus, according to Brower fixed point theorem, we can get that there exists a price vector p*, satisfying p*=g(p*)

Therefore, p_i^* =(p_i+max(0,z_i (p)))/(1+∑_(j=1)^k?〖max(0,〗,z_i (p))),                          ∀i∈ 1,••• k

D Using Walras’s law [5]   and some algebra computations, we can get that in this price function, for any commodity, there is no more excess demand, namely, 

zj (p*)≤ 0,                      ∀  j ∈ 1,••• k

E Demandable hypothesis can deduce that all the products now have prices strictly greater than 0, namely, 

pj>0,                                 ∀  j∈1,••• k

By Walras’s law, p*• z(p*)=0. But this can deduce that the above inequality must be an identity, namely, 

zj(p*)=0,                         ∀ j∈ 1,••• k

This naturally indicates that p* is a price vector with competitive equilibrium. 

(III) Related analysis.

From the above analysis, we can see that general equilibrium model has a strict proof, and is very beautiful, but does this model have any practical value? It is easy to see that this model does not give quantitative results and is just a qualitative conclusion, thus, so far, this model does not find many applicable occasions. To sum up, the major value of this model is naturally theoretical, namely, demonstrating the validity of free exchange economy. 

Another related question is, does general equilibrium model have any inherent defect? About it, scholars have raised many queries, here, we think that one of the major defects is that it is a static model, not a dynamic model, and thus, results in two problems: firstly, in the above model, one important assumption is the initial endowment Ei is given and fixed, but in actual life, economic activities are naturally ongoing and developing, and an appropriate example is that global GDP is growing every year, so, Ei can not be a fixed value; secondly, about the conclusion, when Ei  keeps changing, we probably cannot get an equilibrium outcome, but can just get an middle state between equilibrium and chaos, in other words, we are hard to get one economic equilibrium, but can just get an order of some degree (the proof of this qualitative conclusion probably needs methods in dynamical system). In summary, the assumptions and basic conclusions of general equilibrium model both have some problems, and they all deserve our deep thinking.

4 Some analysis of Solow Model

In the above sections, we investigate 3 models of microeconomics, now, we want to investigate one important model in macroeconomics, namely, Solow Model, as is well-known, Solow Model is the major model in economic growth theory, and is also a foundational model in macroeconomics, in this model, it mainly focuses on four basic variables: output Y, capital K, labor L and technology A, and the total production function of one economy can be written as: 

Y(t)=F(K(t),A(t)L(t))                                                                                                 (1)

The assumptions of this model are: 

(1) Producing technology assumption: we use capital K(t) and labor L(t) these two factors in production, and these two factors can be mutually replaceable, and the production function Y(t) has continuous first and second derivatives. The marginal output of various factors is greater than 0, and their marginal revenues decrease. The scale return of production is unchanged, namely, F(cK,cAL)=cF(K,AL), for c≧ 0.

(2) The production function satisfies “Inada condition”.

(3) The capital depreciation rate is δ,and δ>0.

(4) The initial levels of capital, labor and technology can be seen as fixed, and they are K(0)，L(0) and A(0) , and meanwhile, labor L(t) and technology A(t) grows with a constant speed, namely:  (L(t)) ̇/(L(t)) =ρ, (A(t)) ̇/(A(t))=g, and thus, the changes of labor and technology are both exogenous, which indicates that in all the input factors, only the growth of capital input is endogenous, and thus, the capital growth equation becomes the most critical equation of this model. 

(5) The market is fully competitive. 

 (I) The growth in steady state

Now, we want to analyze the basic intension of this model; firstly, it holistically examines the dynamic feature of economic growth, namely, the economic growth will always tend to a steady state. To do this, we investigate with the change of every unit of effective labor capital stock as the cut-in point; as is well known, saving rate determines the change or growth of capital stock in one economy, and Solow Model assumes that saving rate is exogenous. Assume that all the savings are used for investment, we have: 

I=S=sY• Y(t)                                                                         (2)

Here, we need to consider capital depreciation issue, and at this moment, every year’s increased capital after the deduction of depreciation is the net capital growth:

(                                    k(t)) ̇=(dk(t))/dt=S-δ K(t)=sY•Y(t)-δ K(t)                                            (3)

     Meanwhile, we need to examine the capital growth issue after further introducing population growth and technical progress, here, we define the capital per capita of effective labor as: 

     k(t)=K(t)/A(t)L(t)

     Totally differentiate the above equation, we can get the growth of capital per capita of effective labor, namely: 

(k(t)) ̇=(k(t)) ̇/(A(t)L(t)) –(K(t))/([A〖(t)L(t)]〗^2 )[A(t)(L(t)) ̇ +(A(t)) ̇L(t)]

=(k(t)) ̇/(A(t)L(t))- (K(t))/A(t)L(t)   (L(t)) ̇/(L(t))-(K(t))/A(t)L(t)   (A(t)) ̇/(A(t))

=(•S_Y Y(t)-δ K(t) )/(A(t)L(t))-kρ -kg

=sYf(k)-(δ+ρ+g)k                                                                                                         (4)

Namely, k ̇=sYf(k)-(δ+ρ+g)k                                                                                           (4)

Equation (7) is the key equation of Solow model, and this equation indicates that the change of average capital stock of every unit of of effective labor is the difference between two items: the first item sYf(k) is the average real investment of every unit of effective labor: the average production of every unit of effective labor is f(k), and the proportion for investment of this production is sY. The second item (δ+ρ+g)k is the break-even investment, namely, with the conditions of capital depreciation, population increase and technical progress, the investment amount we need to keep k stay at the current level. This equation indicates that the growth of per capita capital has the following 3 different situations:

sYf(k)>(δ+ρ+g)k          At this moment, k rises, which means that the growth of physical capital is faster than the overall effects of population and technical growth

sYf(k)<(δ+ρ+g)k          At this moment, k decreases, which means that due to diminishing return, the growth of physical capital is slower than the overall effects of population and technical growth

sYf(k)=(δ+ρ+g) k         At this moment, k remains unchanged, which means that the growth of physical capital equals the overall effects of population and technical growth

From the above 3 situations and graph analysis, we will find: when k is small, the real investment is higher than the break-even investment, and thus, k rises, while when k is big, the real investment is lower than the break-even investment, and thus, k decreases, to sum up, the change of k will finally converge to the situation that the real investment equals the break-even investment, and at this moment, we have 

sYf(ke(t))=(δ+ρ +g)ke(t)                                                                                              (5)

Namely, now the per capita capital growth rate becomes constant, and further calculations show that many variables like the output growth rate and population growth rate now also become constant, and thus, economic growth realizes its steady state, while ke(t) is the per capita capital growth rate at the steady state. Solow model indicates that whatever the starting point of one economy is, it will converge to a balanced growth path, and in this path, the growth rate of every variable in this model is constant. 

 (II)The total capital growth rate, total output growth rate and output per person growth rate

In the above part, we get the basic conclusion that economic growth will tend to a steady state, now we want to further analyze the changes of total capital growth rate, total output growth rate and output per person growth rate when the economy is in steady state:

(a)The total capital growth rate. From k(t)=  (K(t))/(A(t)L(t)), we have: ln⁡〖k(t)=ln⁡〖K(t)-ln⁡〖A(t)-ln⁡L(t) 〗 〗 〗, differentiate it, we can get:

1/k  dk/dt=1/K  dK/dt-1/A  dA/dt-1/L  dL/dt=K ̇/K-(ρ+g)

The above equation can be rewritten as:     K ̇/K=ρ+g+k ̇/k,

This equation means that, in an economic society, the total capital growth rate at any time equals the sum of effective per capita capital growth rate and effective labor growth rate.

Therefore, when the economy enters into the steady state path (k^*(t) ,  y^*(t))((k(t) ) ̇=0), the capital growth rate will equal the sum of population growth rate and knowledge growth rate (technology progress rate), namely:  K ̇/K =ρ+g. This is exactly the total capital growth condition in steady state.

(b)The total output growth rate. Here, F_k  is the marginal output of capital, which represents the marginal production of capital, and F_AL is the marginal output of effective labor, which represents the marginal production of effective labor. We have:

(     Y(t)) ̇=  (dY(t))/dt  =F_k  dK(t)/dt +〖 F〗_AL(A(t)(dL(t))/dt  +  (dA(t))/dtL(t))

Bring 

K ̇/K=ρ+g+k ̇/k

into the above equation, we can get:  (Y ) ̇=〖 F〗_(k )K(ρ+g+k ̇/k)+F_ALAL(ρ+g)

Namely, Y ̇  =（F_kK+F_ALAL）(ρ+g )+F_kKk ̇/k=Y(ρ+g ) 〖+F〗_kKk ̇/k

Therefore, we have: Y ̇/Y  =ρ+g+α k ̇/k

In them,  α= F_k  K/Y  =  (∂F/F)/(∂K/K)   is the output elasticity of capital.

Therefore, when the economy enters into the steady state path (k^*(t) ,  y^*(t))((k(t) ) ̇=0), the total output growth rate will equal the sum of population growth rate and knowledge growth rate (technology progress rate), namely:  Y ̇/Y =ρ+g. This is exactly the total capital growth condition in steady state.

(c)The output per person growth rate. Finally, we also need to analyze the growth state of output per person (namely, labor productivity) v(t)=Y(t)/L(t) in the balanced growth path.

Since   ln⁡v(t)  =〖 ln〗⁡〖Y(t)〗/L(t) =  ln⁡〖Y(t)〗-ln⁡L(t),  differentiate it, and we get:

(v(t)) ̇/(v(t))=(Y(t)) ̇/(Y(t))-ρ=g+α k ̇/k

Thus, when the economy enters into the steady state path (k^*(t) ,  y^*(t))((k(t) ) ̇=0), the output per person growth rate v(t)=Y(t)/L(t) is g, namely, the output per person growth rate equals knowledge growth rate (technology progress rate), which shows that, the output per person in steady state is merely determined by technology progress.

Bring all the above systematic and complex analyses together, we can realize that, since economic growth will tend to the steady state, at this time, the output per worker growth rate merely depends on technology progress rate (from the analyses in part (c)), and thus, one basic conclusion from Solow model is: the only source of long-term output per person growth is technological progress. Meanwhile, from the quantitative analyses in part (b), we can know that, in the long run, the economic growth rate equals the sum of population growth rate and per capita output growth rate, while the per capita output growth rate is merely determined by technological progress, and thus, we can get an overall conclusion about economic growth: technological progress is the main factor to determine long-term economic growth.

(III) Some further discussions

In the above, we can see that, Solow builds a broad and elegant economic growth model, then does this model have realistic rationality? Firstly, we need to recognize that, in Solow model, it gives some quantitative analytical formulas about many factors, including capital, labor, output and population, not just some qualitative understandings, thus, based on various quantitative data, about checking the consistent degree between Solow model and empirical data, in this respect, there is already much research, and in them, according to Denison’s research, the notable feature of annual growth rate of actual output, growth rate of working hours and growth rate of capital stock in the US between 1909 and 1957 is their long-term stability; in the above part (II), the basic conclusion from Solow model is that the growth rate of capital and growth rate of output in the steady state are all constants, thus, Solow model fairly fits the historical facts of American economy. During this period, the growth rate of American population is : ρ=0.013, and the growth rate of total output and total capital respectively are 0.029 and 0.024, which are basically equal (though in growth theory, 0.5 percent is a relatively big difference). To sum up, many empirical studies have shown that Solow model partially explains the actual economic facts.

Meanwhile, we need to explain that, Solow model is naturally a highly simplified theory, and this model overlooks many important features of the real world: firstly, what it uses is just a production function with only three inputs, secondly, in this model, the saving rate, depreciation rate, population growth rate and technical progress rate are all invariants, and there are also some other factors which are simplified by Solow model, thus, it gets some false and even ridiculous conclusions, in summary, this model just has a partial explanatory power.

Finally, we need to ask a natural question: why can Solow build such an important model? The reason for this is naturally complex, firstly, the basic reason is that Solow has a deep foundation in both economic theory and mathematics: on one side, Solow has repeatedly considered many complex economic issues, including technology, capital, labor, capital depreciation, capital per capita, total production, saving rate, growth rate, etc, and the internal connections between them, on the other side, Solow’s mathematical foundation is also deep enough, thus, when he does the sophisticated deductions of this model, he can be skillful and proficient, to sum up, one economic scholar must have the above two basic conditions then he can build some important economic models. Secondly, what Solow explores is an important theme, namely, the influence of technology, capital and population to economic growth, and this theme has fundamental importance, and meanwhile, Solow’s treatment about it also displays great mathematical skill and theoretical analysis skill, while some economists’ research subjects are not important enough, and are just some peripheral and minor local issues, and meanwhile, even they select some important subjects, but due to that their mathematical and theoretical foundations are not deep enough, thus, they cannot fully reveal the rich and deep intension of the subject, thus, they naturally cannot build some important economic models. Thirdly, one important feature of Solow model is its richness, and our introduction to it above is only preliminary, in fact, Solow model has very rich intensions: 1 We can use the graph of Solow model to clearly analyze the influence of saving rate growth to investment, and also can easily analyze the influence of the increase of depreciation, population growth rate and technical progress rate to investment. 2 We can also use Solow model to analyze how the change of saving rate can influence many factors, including output, average output of effective labor, and consumption. 3 It can also quantitatively compute the speed of converging to the balanced path. In a word, Solow model can deduce rich conclusions, and it carefully and deeply analyze the interconnections of many concepts, including saving rate, capital, labor, investment, consumption, growth rate and per capita output, and in the complex process of building this model, Solow naturally repeatedly thinks about these issues, and thus, he can immerse into the bones and veins of the whole theory, and further extracts a broad, delicate and clear theoretical framework, while even though some economists also have good mathematical foundation, the intension of the models they build is not rich enough, just containing some shallow and simple ideas, and thus, they also cannot have a lasting impact. To sum up, it is a complex problem about how to build good economic models, and only by well combining many basic factors together can possibly achieve this, and meanwhile, considering that good economic models all at least have the above three basic features, and thus, it is naturally a difficult thing to build important economic models.

5 A brief introduction to social choice theory and some analysis

As is well-known, social choice theory has a deep study about voting and social welfare issues, and the major mathematical tool it uses is mathematical logic, below, we want to give some preliminary introductions and analysis about this theory.

(I) Binary relation

In social choice theory, binary relation is a basic concept, and its definition is, let xRy represents a binary relation between x and y, for instance, “x is at least as good as y” or “ x is greater than y”, and if this condition does not hold, for instance, “ x is not at least as good as y” or “ x  is not greater than y”, then we write it as ~(xRy).

In various problems, the types of binary relation are naturally diverse, and they may or may not satisfy some properties, for instance: 

(1) Reflexivity:  ∀ x∈S:xRx

(2) Transitivity:  ∀x ,y,z ∈S:(xRy & yRz)  → xRz

(3) Symmetry: ∀ x,y∈ S:xRy → yRx

(4) Completeness, Antisymmetry, Asymmetry

About this symbolic definition, we can give some appropriate examples, for instance, we consider to use relation “at least as quickly as” to compare the set of 10 swimmers whose results are recorded in 100 free stroke. This relation is reflexive, because any swimmer’s result is as quickly as himself, and it is also transitive, because if swimmer A is at least as quickly as swimmer B, swimmer B is at least as quickly as swimmer C, then swimmer A is at least as quickly as swimmer C, but it is not symmetric, because swimmer A is at least as quickly as swimmer B does not represent that swimmer B is at least as quickly as swimmer A.

(II) Maximal set and choice set

Now we want to explore the rich intension of social choice theory. Firstly, about binary relation of “weak preference” R (“at least as good as”), we can define “strict preference” relation P and “indifference” relation I 

Definition 1.1 xPy ↔ [xRy & ~(yRx)]

Definition 1.2 xIy ↔ [xRy &  yRx]

For a given set, if there exists one element who is not controlled by any other element in this set, then it is called the maximal element corresponding to the discussed binary relation in this set.

Definition 1.3 The element x in S is the maximal element corresponding to binary relation R in S if and only if 

~[∃ y: (y ∈ S &  yPx)]

The set of maximal elements in S is called maximal set, and recorded as M(S,R)

Except maximal elements, there also exist elements with stronger requirement-“best element”, and its definition is, element x is called the “best” (in size relation is the “biggest”) element, if corresponding to the related preference relation R, it is at least as good (big) as every other element in S 

Definition 1.4 The element x in S is the best element corresponding to binary relation R in S if and only if 

∀ y: (y ∈ S→ xRy)

The set of best elements in S is called its choice set, and recorded as C(S,R)

To clarify the difference of the above two concepts, we give two notes. Firstly, best element must be maximal element, but the converse is not true. If for all the y in S, we have xRy, then obviously there is no y in S such that yPx.

On the other hand, if xRy and yRx both donot hold, then x and y are both maximal elements in {x,y}, but they are both not best elements. Therefore, M(S,R)⊊ C(S,R)

Secondly, C(S,R) or M(S,R) can be empty set. For example, when xPy,yPz and zPx, there is no best element, because there is no element worse than other elements. If transitivity is true, when the set is infinite, M(S,R) may be empty. For instance, if x2Px1,x3Px2,••• , xnPxn-1,•••.  [6] On the other hand, if we have transitivity and finiteness, C(S,R) also can be empty. For instance, if  ~(xRy) and ~(yRx), then x and y are both elements in the maximal set of {x,y}, but they are both not elements in the choice set of {x,y}.

(III) A set of results about quasi-order

In mathematical logic, the types of orders are very diverse, and there are partial order, quasi-order, order, and strict partial order, here, we call the relation satisfying reflexivity and transitivity “quasi-order”. Below, we want to study some properties of quasi-order.

Lemma 1.a If R is a quasi-order, then for all x,y,z ∈S, we have 

(1)xIy & yIz →xIz,

(2)xPy & yIz → xPz,

 (3)xIy & yPz →xPz,

 (4)xPy & yPz →xPz.

Proof

(1) xIy & yIz →(xRy & yRz) & (yRx & zRy) → xRz & zRx → xIz.

(2) xPy & yIz →xRy &   yRz →xRz. Thus, only when zRx, namely, xIz, (2) does not hold. If xIz, then from (1), because xIz & yIz →xIy, we have xIy, but xIy does not hold.

(3) The proof is similar to (2)

(4) We can see that xPy & yPz →xRy & yRz →xRz, thus, only when zRx, namely, xIz, (4) does not hold. But, if xIz, then by (3) and xPy, we have zPy, but zPy does not hold.

Quasi-order has the following two basic results:

Lemma 1.b  Any finite quasi-order has at least one maximal element.

Proof Let the elements be x1,x2,•••, and let a1=x1.  Based on recursive rules: when xj+1Paj, let aj+1=xj+1,   or else, let aj+1=aj.  From the construction, an  must be maximal element.

We can see that this lemma is a simplified case of the well-known Zorn lemma in mathematics.

Lemma 1.c If R is reflexive, then xPy ↔{x}=C({x,y},R)

Proof Because R have reflexivity, we have: xRx, thus,

xPy →xRy & ~ (yRx)

→{x}=C({x,y},R).

From reflexivity, yRy, we have 

{x}=C({x,y},R) → xRy & ~(yRx) →xPy.

Therefore, x is the only element in the choice set of {x,y} if and only if x is strictly preferent to y.

For some cases, understanding the relation between maximal set and choice set is important. We already know that C(S,R) ⊊M(S,R), and we can further get the following result:

Lemma 1.d For a quasi-order R, if C(S,R) is non-empty, then C(S,R)=M(S,R)

Proof  x∈ C(S,R), then

z ∈ M(S,R) → ~(xPz)

Because xRz,

→xIz. 

From lemma 1.a and x ∈C(S,R)

→∀ y:[y ∈ S→ zRy] →z ∈ C(S,R)

Thus, M(S,R) ⊊C(S,R). Since we already know C(S,R)⊊ M(S,R), thus C(S,R)=M(S,R)

From the above discussions, we can realize that though the definitions of quasi-order and binary relation are simple, we can get very rich properties through various deductions.

(IV) Some further analysis

In the above, we have defined maximal set and choice set, and meanwhile, we can also introduce another set: suboptimal set L(S,R). 

Definition 1.5 Suboptimal is the set which are made up of elements belonging to maximal set but are not best elements.

L(S,R)=M(S,R)-C(S,R)

From lemma 1.d, we know that, L(S,R) only has the following two situations:

1 When C(S,R) is empty set, L(S,R)=M(S,R);

2 When C(S,R) is non-empty, L(S,R)= ∅.

Therefore, L(S,R) seems to be an ordinary set, but, in fact, the situation is not so, and the reason is that, in many cases, it is difficult to determine whether C(S,R) is non-empty, for instance, we have the following theorem:

Lemma 1.e For any quasi-order in a finite set S, we have 

∀x,y : [x,y∈ M(S,R) →xIy]↔ [C(S,R)=M(S,R)].

This lemma shows that, in many situations, it is not easy to determine whether C(S,R)=M(S,R) or whether C(S,R) is non-empty. Therefore, we think it is meaningful to introduce this set, and moreover, we can make a systematic analysis about this set.

Now we want to discuss about the basic property of this set. Here, we consider one specific issue, namely, the marriage issue of young people, and consider three individuals a,b,c who satisfy other conditions of marriage, and for these three people, we need to study two major conditions: 

1 Age, and if the age gap of two people is smaller than 1 year, we can regard them as “ as least as good as”;

2 Residence, here, local individual is better than remote one, while two people living in different places or two local individuals can be regarded as “at least as good as”.

With the above measurement criteria, we can consider the following two situations: 

Situation (1) The conditions of three individuals a, b, c respectively are: 27, remote; 28, local;

30, local. In this condition, we can easily see that, bPa and bPc, therefore, b is the best element, namely, now, C(S,R)=b, while L(S,R)= ∅

      Situation (2) The conditions of three individuals a, b, c respectively are: 27, remote; 28, remote; 30, local. In this kind of condition, we can easily see that (with appropriate condition): aIb,aIc and bIc, and thus, now, a, b, c are all maximal elements, and there is no best element, namely, now, L(S,R)=M(S,R)={a,b,c}, here, we can say that a, b, c these three elements are all suboptimal solutions of this problem, though not best elements.

      From this example, we can realize that, L(S,R) indeed gives some elements satisfying suboptimal conditions, thus, it has certain substantive connotation, namely, this definition is of certain significance, thus, it is somewhat valuable for us to systematically and deeply explore the properties of this set. Obviously, it is also a typical problem about multiple preferences, as we know, multiple preferences is an essential problem in social choice theory, and meanwhile, it is also naturally a complex problem, and in the above problem, we just consider three elements and two preferences, the generated situation is already very complex, and if we consider more elements and more preferences, then it will naturally give rise to highly complex situations. The essential reason in it is: it is difficult to define the three basic relationships R,  I, P in multiple preferences, and if the definition is too strict, then the solution of this problem will not have richness and does not fit the reality, while if we consider too many middle situations, then it will be difficult to control in mathematics (like the situation (2) in the above case, about two people b and c, in age, b is “better” than c, while in residence, c is “better” than b, thus, how can we compare b and c now? We probably have two choices: firstly, define them as “indifferent”; secondly, define them to be uncomparable), this also shows that, in many cases, it is not enough to just consider choice and sorting issues from mathematical perspective, because the mathematical treatment of many problems will be too complex and entangled, thus, in many situations, we need to consider this kind of problem from practical experience perspective. [7]

     In summary, we give some simple discussions about social choice theory in this part, and from these concepts and approaches, we can preliminarily feel the specific style of this kind of problem, obviously, the starting point of social choice problem is very clear, namely, to consider the ordering of many optional elements, for this, we need to introduce some concepts and build some analytical frameworks, and get some important conclusions, but the infiltration of multiple preferences and the interconnection with many other problems make the whole problem more complex, and correspondingly, also make social choice theory have rich intension.

6 Herdsman model and further examination

As we know, Herdsman model is a common model in institutional economics, and its major purpose is to analyze the outcome brought by the overuse of public resources if without administration.

  (a) The basic intension of this model. The assumption of this model is: one pasture has n herdsmen, and they jointly own a meadow, and each herdsman has the freedom to graze on the meadow. Every spring, the herdsmen need to decide how many sheep they want to raise. We denote xi  as the number of sheep which the ith herdsman raises, then xi∈[0,+∞)(i=1,2,•••, n). 

Let V represents the average value of each sheep, then we can see V as a function of total number of sheep

 X=∑_(i=1)^nxi

Namely, V=V(X). Because one sheep at least needs certain number of grass to not starve to death, thus, the number of sheep this meadow can raise on this meadow is limited. Let Xmax  be this maximum, obviously, when X0; while when X ≧Xmax, we can think V(X)=0. And meanwhile, it is easy to understand that with the increase of the total number of sheep, the value of sheep will decrease, and moreover, the faster the total grows, the faster the value decreases, thus, in this model, we can assume that   ( dV)/dX<0,(d^2 V)/(dX^2 ) <0.

In this model, we think each herdsman will choose the number he wants to raise to maximize his own profits. Assume that the value of buying one sheep is c, then the profit the ith herdsman will get is :

Pi(x1,x2•••,xn)=xiV(X)-xic=xiV(∑_(i=1)^n?x_i )-xic,              i=1,2•••,n

Therefore, to maximize his profit, the number of sheep must satisfy the following first order optimization condition, 

(*)                  (∂P_i)/(∂x_i )=V(X)+xiV＇(X)-c=0                            i=1,2,••• ,n 

 Namely, the number of sheep each herdsman needs to get the maximal profit (the optimal raising number) xi(i=1,2,••• ,n) must be the solution of these equations, which is called optimal solution. This equation shows: the optimal solution satisfies the condition when marginal revenue equals marginal cost, and also shows that it has both positive and negative effects to raise one more sheep, and the positive effect is the value of this sheep V(X), and the negative effect is the decreased value of other existing sheep due to this sheep (because xiV＇ (X)<0).

From the above first order optimal condition, we can also see that, the optimal raising number of the ith herdsman xi  is affected by the raising number of other herdsmen, which naturally fits the actual situation, therefore, we can think that xi is a function of xj(j=1,2,•••,n, j≠ i), namely, 

xi=xi(x1,••• ,xi-1,xi+1,•••,xn),

Usually, it is called the reaction function. Then differentiate xj (j≠ i) in the first order optimal condition, we can get

V＇ (X)(  (∂x_i)/(∂x_j ) +1)+V＇ (X)  (∂x_i)/(∂x_j ) +xiV＇＇ (X)( (∂x_i)/(∂x_j )+1)=0.

Therefore, 

                                         (∂x_i)/(∂x_j )=-(2V＇ (X)+x_i V＇＇(X))/(V＇ (X)+x_i V＇＇(X))<0

This shows that the optimal raising number of the 〖 i〗^th herdsman xi  will decrease with the increase of other herdsmen’s raising number.

Naturally, solving equations (*) can get the optimal raising number x_i^*,i=1,2,••• ,n of every herdsman. We need to note that all the above computations are based on xi,   in other words, the obtained optimal number x_i^* is obtained through the below situation: when every herdsman decides to increase raising number, though he considers the negative effect to the value of current sheep, he just considers the impact to his own sheep, not the overall impact to all the sheep. Thus, the sum of individual optimal raising number is :

X*=∑_(i=1)^n x_i^* 

which is not necessarily the overall optimal raising number of the whole pasture. In fact, the obtained maximal profit of the whole pasture is the maximum of the below function 

XV(X)-Xc

Its first order optimal condition is

V(X)+XV＇ (X)-c=0

Let X** be the number of raising sheep such that the whole pasture can get maximal profit, namely, the optimal raising number of the whole pasture, then 

V(X**)+X**V＇ (X**)-c=0                                                 (1)

     While we also can consider the sum of every herdsman’s optimal raising number, then, add up all the n equations in (*), we get 

                                            V(X*)+X^*/nV* (X*)-c=0                                                      (2)

     Compare the above two equations and use the monotonous decrease property of V(X) and V＇(X), we can get 

                             X*>X**

     Namely, the sum of individual optimal raising number is greater than the optimal raising number of the whole pasture. This result shows that the public pasture can be overused if without administration, and this is exactly the Tragedy of Commons about public resources without administration which scholars often talk about.

     (b) Further relevant analysis

     From the above model deduction, we can get some further conclusions. If we quantitatively compare the differences between X* and X**, by subtracting equation (2)  from equation (1), we can get 

     V(X**)-V(X*)+ X**V＇ (X**)-X^*/n V＇ (X*)=0

Now, if we assume n is big enough, then we can approximately get

V(X*)-V(X**)=X**V ＇(X**)

Using differential mean value theorem, we can get 

               (X*-X**)V＇ (ζ)=X**V＇ (X**)

At this time, if we consider the property of function V(X) ( we need to notice that, in different problems, like overfished ocean, excessive deforestation, etc, the specific forms of V(X) are different, thus, it needs to be determined based on specific issues), we can roughly get the quantitative relationship between X* and X**.

 To sum up, the intension of Herdsman model is very definite, and I think most people will agree with the fact that fishes in the ocean will be overfished if without administration, in it, the essential reason is that marine resources are limited, which is different from free market theory, because free market theory believes that if the government deregulates, then it will stimulate the creative vigor among individuals, and thus leads to the continuous increase of social wealth, thus, free competition is effective economic institution, namely, one of these two theories aims at the situation where resources are limited, but the other aims at the situation where products are growing, thus, the conclusions are different. But, meanwhile, we also need to ask another question: public resources will be overused if without administration, like the overgrazing of pasture, the excessive toxic gas emission, which is a simple fact people all understand and widely accept, and it is not a difficult thing to understand, moreover, people have also already accumulated a lot of relevant experience and cases in this aspect, do we need to use a mathematical model to prove such an apparent conclusion? Moreover, is this kind of mathematical proof really more persuasive than a lot of data and cases and people’s daily experience? This is also a problem worth our thinking.

7 An inquiry into price, productivity and currency

About many issues, like currency, price, commodity and money demand, in the economical world, there are already many complex debates, and about this problem, in the book Purchasing Power of Money published in 1911, famous economist Fisher once raised the well-known exchange equation : MV=PT, in this equation, M represents the average amount of money in a certain period, V represents velocity of currency circulation, P means the average price, and T means the exchange quantity of commodity and labor service. This equation has a central meaning in monetary theory, but we think that this equation also has certain inherent defects, namely, it just considers the relationship between price and money, but if we enclose price and monetary issues in the pure economical range of currency circulation velocity, business cycle and inflation, then it is difficult to explain many practical issues, and considering this, in many issues, we need to expand the analytical perspective. In particular, in currency and price issue, we need to consider the productivity factor, for this, we put forward an equation here: MMI=(M(t))/(G(t)), in it, MMI represents monetary moderate index, M stands for the total amount of money in one country (like the common M2, namely, broad money supply), G means the total amount of one country’s productivity. Below, we want to discuss about the meaning of this equation.

Firstly, it can measure the inflation degree in social scope, and if the monetary moderate index is too high, namely, the ratio of the total amount of currency to total productivity in one country is too high, it naturally shows that too much money is issued in social scope, and at this moment, one country’s inflation degree is overly high. Secondly, it can also measure the financial foaming degree, and if this index is too high, it shows that there is too much currency in the whole society, and because financial institutions usually dominate the issue and holding of money, thus, at this moment, financial institutions often absorb much money, namely, the wealth in social scope has a certain degree of tilt to financial institutions, which is somewhat harmful to the healthy development of society (of course, financial institutions and financial products, like stock market, stock, bond and bank, are also important for the healthy development of a society, on one side, they can keep the normal operation of the economy and society, on the other side, for the starting of a company and its further development, they also have fostering, promotion, boosting and improvement functions, namely, they have basic values in maintaining and improving for the economic operation.) In summary, we think that this equation at least has the above two values.

8 Re-examination of consumption function theory

The examination of consumption has an important position in economics, in it, Keynes’s well-known consumption function theory is raised in his The General Theory of Employment, Interest and Money, and he thinks that the total consumption is a function of total income, and by linear function, this view can be expressed as : C(t)=a+bY(t), here, C represents the total consumption, Y means the total income, parameter a stands for the necessary autonomous consumption, parameter b is the marginal propensity to consume (MPC), and its value is between 0 and 1, bY represents the consumption brought by income, namely, this formula thinks that the consumption equals the sum of autonomous consumption and induced consumption. The marginal propensity to consume shows that, the consumption will correspondingly increase with the increase of income, but the extent of consumption increase is lower than the extent of income increase, namely, the marginal propensity to consume will decrease with the increase of income. But, as we know, this hypothesis is too brief, and just considering the impact of income to consumption, but overlooking many other factors, thus, the error is relatively large when used to predict.

As another scholar who studies consumption function theory, Friedman refutes Keynes’s diminishing marginal propensity to consume law, and he thinks that people’s consumption desire is infinite, and when the old is satisfied, the new desire will emerge. In consumption function field, Friedman also puts forward the view of permanent income hypothesis, and in this hypothesis, he distinguishes permanent income and temporary income, and he thinks that consumers’ consumption expenditure is mainly determined by permanent income, namely, only permanent income can affect permanent consumption, while temporary income will just affect temporary consumption. Based on this theory, it is ineffective for the government to use policies by increasing or decreasing taxes to affect the total demand, because people’s increased income due to tax reduction is only temporary income, and will not be instantly used to consume. It is easy to understand that Friedman’s these views also have partial validity.

After a brief introduction to Keynes’s and Friedman’s views, now we want to make some new discussions about consumption function theory, and we think there are at least three issues worth our consideration. Firstly, Keynes’s consumption function mainly focuses on aggregate economy, but overlooks the individual level, in fact, the marginal propensity to consumption of different individuals is naturally different, and here, we can modify Keynes’s theory into: c(i)=a+b(i)y(i), and c(i) is the consumption of individual i, y(i) is the income of individual i, b(i) is the marginal propensity of consumption of individual i, and in individual level, we think Keynes’s pessimistic attitude about marginal consumption issue and Friedman’s optimistic attitude are both unilateral, because in real life, three are at least 3 different types of individuals : for the first type, with the increase of income, their consumption tendency is decreasing, for the second type, with the income increase, their consumption moderately improves, and for the third type, with income increase, their consumption tendency is also growing, to sum up, the marginal consumption tendency (MPC) among the crowd actually at least has three different types, and Keynes’s overly pessimism and Friedman’s overly optimism are both inappropriate, and considering the situations in many societies, we think that the second type are probably the majority of people, namely, we think with the increase of income, the consumption in society will moderately grow, and Keynes’s diminishing consumption law is not very consistent with the objective facts. Secondly, for different societies, three also exists different marginal propensity of consumption, for instance, for some societies like America, their MPC is relatively high (we need to note that, American society in the 18th  and 19th century is relatively thrifty, and the saving rate then is higher), while for some societies like South Korea, their marginal propensity of saving (MPS) is relatively high, in a word, for different societies, we need to differently explain Keynes’s formula. Thirdly, another big problem about Keynes’s consumption function formula is that, with the change of time, b is always a constant, we think this is not appropriate, and the real situation obviously is, with the change of income Y(t), b will also correspondingly change, namely, it is more reasonable to express Keynes’s formula as C(t)=a+b(t)Y(t), and thus, the relationship between total consumption and total income is not linear. To sum up, we think consumption function theory still has many contents for us to study, and it is a  complex issue, and we should not simply treat it like Keynes or Friedman.

9 Summary. After the discussions of the above 8 economic models, we want to do some summary. About the internal relationship between modern economics and mathematics, we want to discuss about four basic aspects: firstly, we need to clarify the specific process of mathematicization of economics, as is well-known, the tide of economics mathematicization starts with Marshall, but what Marshall then used is just the simple part of calculus, while economics mathematicization greatly accelerates after World War II, and in it, Samuelson can be viewed as a good example, Samuelson build many important economic models in various fields of economics, and the mathematical tools he uses, like the extremum of multivariate calculus and linear programming, are naturally much deeper than Marshall, and meanwhile, all the economical areas undergo a process of continuous mathematicization after World War II. Secondly, we need to be clear that economics cannot be viewed as mathematics, some mathematicians have a good mathematical background, but they cannot do good economical research, because what economical research needs is economical ideas and economical intuitions, on the other hand, for good economic modeling, mathematics is also a necessary foundation, and if we just have economical intuitions but without mathematical foundation, then we also cannot build good economic models (but at this time, scholars can do good economical research of other types), to sum up, good economic modeling must be an organic combination of mathematics and economical ideas. Thirdly, the mathematical tools used in economics is very broad and deep, in the above problems, many mathematical tools, like total differential, maximum of multivariate function, conditional extremum, fixed point theorem and mathematical logic, are used, and these tools are also used delicately and ingeniously, to sum up, the mathematics which modern economics uses is very broad, and thus, to do good economic modeling, one scholar must grasp broad and systematic mathematical knowledge. Fourthly, the rigour of the above economic models naturally has no problem, because their deductions are all based on rigorous logic, and are all rigorous mathematical deductions, therefore, their rigour is unquestionable. (On the other hand, rigour does not stand for the realistic rationality of these models, because the assumptions of these models often simplify the complex practical issues, thus, they just partially reflect the economic reality.)

A natural question is: why will most economic models be read by nobody after just 3 or 5 years (most economic theoretical papers are also similar, and nobody will read them after 5 years, and is less unlikely to be repeatedly read by people after 100 years) , but the above good economic models can withstand the check for several decades? The reason in it is both complex and simple. The major reason is that most of the above good models are built by brilliant economists, and these brilliant economists’ economic theoretical foundation and mathematical foundation are all above ordinary economists, thus, these models have long-term vitality, namely, the deep foundation of these brilliant economists are the main and essential reason for the lifeblood of these economic models, thus, the above phenomenon is actually easy to understand. And meanwhile, in a deeper sense, brilliant economists often have a deep understanding about many areas of economics, and they also have a solid foundation in many areas of mathematics, namely, on the surface, some economists seem to just build one or two good economic models, but these economists’ scope of thinking is actually far beyond these one or two economic models, and they actually carefully and deeply think about many complex, deep and principle problems, and if their scope of thinking is very narrow and just covers the knowledge relevant to these one or two economic models, then they are probably unable to build these good economic models. In  conclusion, the reason for this phenomenon has a simple side, but also has a profound side.

We also need to ask another important question: do these many economic models have practical value? This is naturally a complex question and thus difficult to answer. One of the answers is that if one model just gives some qualitative results, without quantitative analysis, then it probably will not have much potential in application, as the above analyzed general equilibrium model and Herdsman model, and another partial answer is that due to the vast range of economic models, thus, part of them probably have certain practical value. Of course, whether one economic model has practical value or not is just one aspect to measure it, and we also need to consider other values of these models, such as: elaborate the inherent principles of social operation, systematize people’s understandings of economic activities, etc.

[1]  See A Monetary History of the United States, 1867-1960, Chapter 2, Section 2.2, “Changes in Money, Income, Prices, and Velocity”, pp.32-33, Princeton University Press, 1971

[2]  To the Shareholders of Berkshire Hathaway Inc. 1979, “Financial Reporting” Section

[3] Indianapolis 500 is the oldest and most prestigious car race in the world.

[4] The discussion of this part can refer to Varian’sAdvanced Microeconomics, Chapter I

[5]  Walras’s law has the following intension: for every agent i, let Ei  be his initial endowment, xi  is his Marshallian demand function, given a price vector p, the income of consumer i is p· Ei,   and  thus, his demand is xi(p,p· Ei).

Now the excess demand function is  z(p)=  (p,p· Ei)-Ei                                                                                 Walras’s law can be expressed as p·z(p)=0

The proof is this: By definition, the excess demand is                                                                                                                                      p· z(p)= p· xi(p，p· Ei)-p·Ei                                                                         In the premise of budget constraints, Marshallian demand is the agent utility when a set of consumptions achieve their maximum, and thus, the budget constraints here satisfy：                                                                              p·xi=p·Ei.                                                                                                       Thus,  all the items in the above sum are 0, and thus, the item itself is also 0.

[6] While if the set is finite, M(S,R) must be non-empty, please refer to the below lemma 1.b

[7] The contents of this part can refer to Amartya Sen’sCollective Choice and Social Welfare, Chapter 1* ”Preference relationship”