About the Income Effect

Among the hundreds of terms in the microeconomics, income effect confused me most. According to the textbooks' conception, when price of a certain goods, say X, raises, its effect on the Marshallian demand quantity can be split into two sections: substitution effect and income effect. Relationship of them three can be described by the Slutsky equation:

 (1)

where the two terms on the right hand side are the Hicksian substitution effect and the product of marginal Marshallian demand change with respect to welfare (can be represented by income) and the current demand quantity based on utility level U0, which is the income effect.  If let i = j, the whole equation can be used to describe the behavior of good i's demand with the control from parameters p, w, and u. 

Now lets take a close look at the income effect. If writing Slutsky equation by the language of elasticity, the income effect will be the product of good i's elasticity of income and its spending share, which is the ratio of spending on good i and the total income. Lots of literatures (Hicks; Marshall) have recognized the role of the spending share and concluded that a small proportion of spending share will help us to ignore the income effect. The small spending share is consistent with Marshall's basic assumption on Marshallian's demand when he used it to calculate the welfare by consumer surplus: the marginal utility of income is consistent. Why? First of all, think of the intuition of income effect from equation (1). Besides the negative sign, it shows a direct linear augment (or reduce) of i's current quantity demand, the second part of the income effect, while the degree of magnification (or shrink) is the marginal sensitivity of the quantify demand to the welfare income. This means, income tries to constrain the scale of consumption. When the marginal demand of good i is sensitive to income, the fluctuation of income will have a deeper control of the scale of consumption.  Here I use density as a pictorial description of the partial derivative of the quantity demand with respect to income, which reflects how much the consumption relies on the absolute level of income. Now back to our question on marginal utility of income. When income increase by 1, the scale of the whole bundle of consumption, assuming including n goods, tends to be augmented, which means every goods needs to be consumed more. As we all know, at equilibrium, the marginal utility of every goods are equal and they are also all equal to the marginal utility of income. When the spending share of good i or its demand's sensitivity to income is small, good i's consumption scale won't change much, which leads to a little change of its marginal utility given the usual diminishing marginal utility assumption. But if we want to hold the result - marginal utility of every goods are equal - to be true, we cannot rely on a "small demand sensitivity to income" any more since the income must change someone's consumption big within the bundle or this 1 unit gain will be wasted, which will ensure at least one good's elasticity of income to be large. However, spending share does not have such problem. As long as n is large enough, spending share of every good could be very small. Then, a change of income won't affect the quantity demand of any of them much, and thus hold marginal utility of every good, as well as the marginal utility of income, to be approximately constant. Vice versa, constant marginal utility of income must be guaranteed by small spending share.  

All of these analysis actually gives us the fundamental explanation of one question: where does the income effect definitely come from? The answer is the change of marginal utility of income and the equilibrium requirement. This seems to be the end of the interpretation of income effect. But I have at least two questions left that I hope to least here. First one is that preference order information is not included in this interpretation. Second, this theory is within a n goods bundle; what if as income increase, the bundle's scale increase? Will there be a discontinuity problem? Suppose those goods on the waiting list be capital goods which may violate the small spending share assumption, which is intuitive, how would we update our fundamental interpretation of income effect? 

Reference: 

Wikipedia, income effect

Some classic economics literatures on demand theory

Notes for [The Grand Design]. In Chinese.

每读完一本书总是觉得思想有了进步，但又慨叹细节没记住什么。所以，Notes是非常重要的。这次的Notes是关于Prof. Stephen Hawking 2010年出版的书[The Grand Design]，中文译名[大设计]。

我对科学历史的阅读一直很浅显，在脑海中搜索“科学史”的话，飘出的大部分碎片都是关于人类如何通过科学地解释某个现象而摆脱了“世界被神控制”的“邪说”云云。其中最典型的莫过于哥白尼日心说vs托勒密地心说的故事。我应该算是Dr. Hawking在书中所说的“科学决定论”者。科学决定论是在牛顿发现运动定律和万有引力定律之后被提出的。牛顿定律优雅而简洁的方程是它的主要魅力之一，它让人们猜想，科学决定世界的方式，是否也是一个或一族简洁的定律。随着地球上的现象不断地被表观定律所解释，科学决定论者甚至渐渐形成了这样的思想：“必须存在定律的完备集合，只要给定宇宙在某一时刻的状态，它就指明宇宙从哪个时刻往前如何发展；这些定律应当在任何地方、任何时刻都成立；否则的话他们就不是定律；不可能有例外或奇迹；神祗或魔鬼都不能干涉宇宙的运行。[1]” “初始条件 + 大一统定律 = 现在的世界”，一切都是客观的，且很可能是唯一的（I say maybe）。

这曾经一度也是我的世界观。但事实证明，把三观建立在一个猜想之上是不牢靠的。例如我一直不能理解，为何顶级科学家中仍然有许多信仰上帝，难道他们不觉得在一个无法被干涉的定律世界中，上帝的存在感那么苍白吗？《大设计》在第一章的开始对这个问题给了我一个反问式的解答：自然的定律告诉我们宇宙怎样行为，但它能够回答我们它“为什么”这样行为吗？如果不能解答“为什么”，那么对这些定律的创造和运行它们的初始推动力便给了上帝生存的合适理由。我的资料库中没有对这个问题的解答。但幸运的是，Prof. S. Hawking给出了一个与我的三观不违背的答案：宇宙可以自己创生，不需要神来推动。然而，对这个答案的理解是我那些建立在“哥白尼层次”科学决定论上的世界观所做不到的。因此有必要做一个更新三观的导引。

若想知道定律能否为我们解决终极问题，首先需要对“定律”有个合理的定义。今天的大多数科学家和科学爱好者会说，定律必须基于“观察”或其可被预言的类似情形，因此每一组被认为合理的定律的诞生都必须有一个“被某某试验观测所验证”的环节，即“眼见为实”。但若两组猜想都可被验证我们该去哪个好呢？人类目前所描述的大部分定律都有一个不可获取的部分叫做“约束条件”，只有给定那些条件，该定律才可为我们预言结果。有些定律的约束条件后来被证明可被放宽，旧定律成为新定律系统中的一个特殊情况；有些定律则与其他定律共同解释一个现象且精确度无差，这时候，我们选择不对它们的优劣或大小做比较。它们属于两个不同的用来解释世界的系统，而在系统交界处，它们给出相同的预言。多系统serve一个结论是否是一种冗余的节奏呢？Dr. Hawking认为，这个世界的真实性基于我们对它的理解。这与唯心主义是两码事。唯心主义认为世界的存在基于意识，将生命体放在了至高无上的地位，而霍金认同的思维却有意无意地鄙视了人的能力，认为人的思维方式限制了人一次性看清宇宙全貌。宇宙的全貌需要我们从所有的角度看宇宙：若存在理想中的描述世界的统一理论，它应当是一族不同的理论，而不是一个没有“约束条件”的老大。这可以类比于我们用二维地图描述三维地球的情形，只有用相互交叠的多张图才能概览全貌，而在图与图的交界处，它们展示相同的图景（交叠很多时候被称为对偶性）。但我们只有一双眼睛，只能一次性识别一个三维的空间，所以我们看不到的东西实在太多。解释那些东西需要抽象思维，而想要用观测验证那些东西（我们偏偏就爱好眼见为实），就需要把他们transfer到我们可以看到的空间里，然后给出一个预言。这个大一统理论在书中被称为“M-theory”。

在上一部分中我们提到了“眼见为实”，并把它作为判断一个定律是否可被保存的依据。但其实这其中有一个严肃的问题没有明晰：眼是谁的眼？如果是生活在鱼缸中的金鱼的眼，那一条“这个世界是被透镜扭曲下的样子”的定律就是正确的。其实，古人说地球是平坦的，托勒密用大圆套小圆来描述行星的运动，都只是视角问题，无从说谁对谁错。既然定律的对错论被剔除了，那么怎样回答本段开始的问题呢？[大设计]的观点是：不存在与图像或理论无关的“实在”概念。Theory们是一个个勾勒世界图像并将其元素与观测连接的规则，而我们之所以更愿意接受哥白尼而不是托勒密，是因为哥白尼模型的运动方程具有更“简洁”的优势（至少在某些尺度下）。Prof. Hawking对哥白尼跳出地平线看世界，事情瞬间简单明晰了。此类“分层原理”的应用很多，经典的例子还包括哥德尔的“不完备性定理”等。但即使有些“圈外”模型在解释一些问题时更简单，也并不表明其他模型一无是处，模型被“复活”的例子更是不胜枚举，典型的例如牛顿“光是粒子”的假说在量子力学中被重新思考。总之，没有谁比谁更真实，只是我们的大脑产生的直觉不够牛逼（只有3维，还很窄），所以只能在所考虑的范围内，那个模型更方便就用那个来看世界。可见我们一开始说的，描述世界需要一族而不是一个理论是合理的。We never accept a theory, we just fail to reject it.

Dr. Hawking认为这个世界的之所以为现在的这个样子，是因为我们不断地在一个又一个的随机事件中得到了想要的结果，于是乎，宇宙有了一个可以适宜人类居住的角落，而角落里的我们在观测这个世界。其实，虽然感觉像是我们不断地在“上帝掷骰子”的赌局中赢着，但其实是环境造就了我们，而同时我们无法看到其他可能性维度的图景而已。对于这一种观点，拥有“人性至上”观点的人恐怕是难以接受的：难道我们的存在不是一个一个杰作所接续的成果，而仅仅是一个accident？我想答案恐怕是Yes，人类的存在没有特殊性，不过也没那么悲观，因为我们现在所观测的世界很可能是存在概率最大的一个。要解释这一点，回想一下高中学过的坑爹的“双缝实验”，尤其是那个严格的“一次只释放一个光子却发现还是出现了干涉条纹”的版本，是不是你的脑子里到现在还没法接受“光具有波粒二象性”这种坑爹的解释，是不是到现在你还在强迫自己记住“光的波动性不是一大群光子的行为，而是“单个粒子”的性质”。一个听着更靠谱的解释来源于Richard P. Feynman的历史求和理论，他认为一个粒子在“走路”的时候是有“分身”术的，到了地方之后所处的状态也是随机的，只是走有些路的概率大些。这样，它到了地方的样子（即它走到某一个点的概率幅度）需要通过将走所有的路的样子求和才可以得知。前述的干涉现象是光子的不同路径之间相互影响导致的结果，有些点的到达路径们相位求和后展现出比较高的概率，有的就黑线了。粒子的这种随机性为前面所述的Accident创造了可能性，我们称所有的那些可能为“可择历史”。

（我很想知道Feynman历史求和与Heisenberg测不准原理之间有没有关系，等以后知道了补在新日志里）

宇宙在宏观上可被认可的发展过程包括大爆炸、暴胀、微小的不均匀性创造恒星、恒星内部核聚变产生重粒子、超新星爆炸抛出重粒子、重粒子形成行星、未爆炸的恒星坍缩为白矮星或黑矮星或黑洞等环节。

我对科学史的阅读非常浅显，浅显到无以复加，浅显到读完这本书竟然只有surprise而给不出任何的critique，所以只能引述一些其他人的评论在这里。John Horgan in Scientific American批评了大设计对M-theory过分广泛的概括性的笃信，认为一个预言“一切皆有可能”的理论实际上根本就是废话。The Economist也说学界实际对亚原子的理论的意见还没有足够的一致性，因为尚有许多观测与理论有出入[2]。

越是读书越觉得世界可怕，因为挖的越深就越觉得自己是个浅显的人。


[1]援引自《大设计》第145页，作者Stephen Hawking, Leonard Mlodinow, 译者吴忠超，2010年。
[2]翻译自wikipedia [The Grand Design] 词条。原文地址为John Horgan;(2010-09-13). "Cosmic Clowning: Stephen Hawking's "new" theory of everything is the same old CRAP". Scientific American. Retrieved 2010-09-15.以及"Understanding the universe - Order of creation". The Economist. 2010-09-09. Retrieved 2010-09-18。

Information goods are "machines"?

Information goods are commonly noticed by economist because of its zero marginal cost (Varian, 1995), non-excludability, non-rivalry, and non-transparency (DeLong, 1999). Focused on these features of the product itself, researchers used the copyright system, market power with bundle pricing (Bakos and Brynjolfsson, 1999, Varian, Shaprio, 2004), or gift-exchanging models to solve the trouble brought by them so as to draw information goods back into the classical economic analysis. However, some facts in the modern information goods market cannot be ignored any more. First, copyright system is very costly, not only due to the obvious supervision cost, but also the high transaction cost from the failing to locate rights from occurring (Varian, 2007). Worse, with the piracy's unremitting interruption from all around the world, the copyright system is unstable. In developing countries such as China and Brazil, the music and software have been marked as "free"goods, which, interestingly enough, does not hurt the industry much or even promote it by bring competition (Anderson, 2010); developed countries also do not survive if you think of the world largest piracy's website "The Pirate Bay" who recently declared to move their server onto the moon to avoid any legal limitation. As DeLong(1999) noted, reforming the law to give sellers a property right in information about the prices that they charge appears extremely dangerous; there has never in the past been a legal right to exclude competitors from access to bulk pricing data. Piracy, undoubtedly, proved this well; but it brings another problem, which is showed in the gift-exchanging model that relies highly on the people's gratitude (George Akerlof, 1985): people gratitude is not always trustworthy, and, treating information goods as public goods also needs high transaction cost.

But wait, what on earth bring the tough problem to information goods. The zero marginal cost? Then who cause this low cost? The intangibility of the goods? Close but not exactly. My answer, the thing I want to emphasize in this article, is that: it is the world's easiest last step of the producing process of information goods, which fosters millions of consumers into the sellers' competitors, that drags information goods market into quagmire.

Why do I care much about the producing process? According to the Baumol (1982), the production directly determines the cost structure, which, will have significant influence on the market structure as well as market behavior of firms. Information goods has a high-fixed-cost / low-incremental-cost structure(Varian, 1995) due to its long and complicated R&D procedure but a "click-copy" manufacturing process. Varian had noticed the absolutely low-incremental cost that makes the information goods market into a monopolistic competitive one; whereas, he ignored the relatively large gap of fixed cost and variable cost. Actually, nearly everybody got lost in the miraculous zero marginal cost. But I must emphasize the word "relatively" because it will move people's eye from the small marginal cost to a new view, to one of the most common scene in the economics world, buildings, cars, CPU, iPhone, to what we call: durable goods. Durable goods, sometimes also called investment goods, are defined as "goods that yield utility over time rather than completely consumed at one time" (Sullivan, Sheffrin, 2003). I think this definition needs to be revised. As Thompson (1920) noted,  consumers' real target are the utilities in the goods, not the goods itself. This elaboration helps to clarify that the criterion that classified goods in to durable goods(investment goods), non-durable goods( fast-consuming goods), and services, are actually the quantity of utilities embodied in one unit of goods. That means, durable goods, with larger deluge of utilities set in, can not only release the utilities over time, but on space.  Under this definition, the word "durable" is not much proper; "investment goods" fits more. Then, information goods, which may wear out as soon as being released but has an infinite potential to feed the whole population's desire by unlimited costless copying, is incontrovertibly an investment good.

The advantage of treating information goods as investment goods is that we could jump out of the thinking of how to price the information goods when the marginal cost is 0 which then brings you into an infinite "Market Power vs. Social Welfare" debate. Investment goods, machines? Yes, information goods like machines, or more precisely, a "die". When firms sold an information good, they do not simply sell one consumption good, they sold them a die, with which and a computer, everybody can manufacture and share (because of non-rivalry) the good. This final production process is so simple, disperse, and technically hard prevented, that completely eradicated them is nearly impossible. So why don't IT firms change the mind to sell their goods only to consumers who want to put an investment on them? Or use the information goods as an investment of themselves and produce more service or non-durable goods? Lots of examples have been created by various markets of information industries: musicians put more attention on concerts and performance than on selling CDs; Google sends more efforts on producing people's effective attention than selling software; and so on.

Is this the bright future of firms in the world of piracy? Is this the hope of firms still suffers the "patent minefield" in the world of copyright and patent? I believe so.

Reference
Bakos,Yannis, Erik Brynjolfsson, Bundling Information Goods: Pricing, Profits, and Efficiency, Management Science, Vol. 45, No. 12, 1999, pp. 1613-1630
Baumol, William, Contestable Markets: An Uprising in the Theory of Industry Structure, American Economic Review, Vol. 72, No. 1, 1982, pp. 1-15.
DeLong, J. B., A. M. Froomkin, Speculative Microeconomics for Tomorrow's Economy, 1999, presented at Brian Kahin and Hal Varian's January 1997 Harvard Kennedy School conference.
Sullivan, Arthur; Steven M. Sheffrin. Economics: Principles in action. 2003, Upper Saddle River, New Jersey 07458: Pearson Prentice Hall. p.302. ISBN 0-13-063085-3.
Thompson, Charles. M., Elementary Economics, Chicago, B.H. Sanbom & Co., 1920
Varian, Hal R., Pricing information goods. Proc.Scholarship Nezv Infornm. Environment Sympos. Harvard Law School, Cambridge, MA.,1995, Differential pricing and efficiency, SIMS Working Paper, Berkeley, CA.
Varian, Hal R., J. Farrell, C. Shapiro, The Economics of Information Technology, 2004, ISBN: 0521844150
Varian, Hal R., Copyright Term Extension and Orphan Works, Industrial and Corporate Change, Vol. 15, No. 6, 2006, pp. 965-980


Sherry Xiao Wang
@COPYLEFT ALL WRONGS RESERVED