Saturday, October 27, 2012
|An Unsurveyable Rule For Generating A Real Number In Binary Format|
Noah Smith offers a definition: "Mathematics is the manipulation of the symbols of a language according to explicit, syntactical rules." ("Unlearning Economics" has also recently written on mathematics in economics). To me, the manipulation of meaningless symbols is a powerful form of reasoning. Taking this definition as is, I think two questions can be raised here:
- What is the interest that mathematicians find in these rules and these symbols in the historical circumstances current at the time?
- What does it mean to follow a rule?
Ludwig Wittgenstein is the philosopher most known, I think, for raising the question of what it means to follow a rule. Any summary of his views will be controversial, but I suppose one can fairly say that he adopted an anthropological point of view, at least for some purposes. Describing how to follow a rule by another rule raises the prospect of an infinite regression. Rather, one might show how people do actually follow a rule, how these uses and practices work pragmatically in some form of life. I find it difficult to see how such description conveys the logical must, so to speak, of many rules. But Wittgenstein was alive to this difficulty. He notes that a judge does not seem to treat a statute book as a manual of anthropology.
Furthermore, Wittgenstein spent quite some time in elaborating how these ideas relate to the philosophy of mathematics. His views on the foundations of mathematics seems to have been constructivist and included questioning whether mathematics needs a foundation. Wittgenstein has frequently been labeled an anti-foundationalist. From this viewpoint, one might question whether existence proofs that do not specify how to construct the relevant object can be reformulated. And one even ends up doubting the meaningfulness of defining the real numbers as, say, any set isomorphic to a set of certain equivalence classes of Cauchy-convergent sequences of rational numbers. The use of the notion of infinity remains, I guess, as a standard topic in the philosophy of mathematics.
It seems one of my favorite economists, Piero Sraffa, was an important stimulus in Wittgenstein's development of these views. Sraffa has been said to have led Wittgenstein to see the importance of an anthropological point of view. Sraffa's masterpiece, The Production of Commodities by Means of Commodities: A Prelude to a Critique of Economic Theory, is written in a unique style, not less in the presentation of the mathematics underlying the economics in the book. Sraffa frequently provides outlines of algorithms for constructive existence proofs, maybe most famously for the Standard Commodity. So Sraffa and Wittgenstein might be said to have shared a certain attitude to the philosophy of mathematics, although I do not expect to ever see oral discussions on this topic to be well documented. Sraffa's book can also be said to address only a limited range of topics in economics. An earlier statement of his seems to suggest that he thought room should exist in economics for non-formal treatment of some topics:
"The causes of the preference shown by any group of buyers for a particular firm are of the most diverse nature, and may range from long custom, personal acquaintance, confidence in the quality of the product, proximity, knowledge of particular requirements and the possibility of obtaining credit, to the reputation of a trademark, or sign, or a name with high traditions, or to such special features of modelling or design in the product as - without constituting it a distinct commodity intended for the satisfaction of particular needs - have for their principal purpose that of distinguishing it from the products of other firms. What these and the many other possible reasons for preference have in common is that they are expressed in a willingness (which may frequently be dictated by necessity) on the part of the group of buyers who constitute a firm's clientele to pay, if necessary, something extra in order to obtain the goods from a particular firm rather than from any other." -- Piero Sraffa (1926). "The Laws of Returns Under Competitive Conditions", Economic Journal (Dec.): pp. 544-545.
Whatever you think of the speculations in this post, I think some conclusions are nearly inarguable. Advocates and opponents of the use of mathematics in economics do not neatly divide between mainstream and non-mainstream economists. In particular, one important non-mainstream economist, Piero Sraffa, demonstrated one approach to mathematical economics, while still being aware of the limits to formalism in economics. Furthermore, any comprehensive scholarly study of the philosophy of mathematics will necessarily look at his work as long as Wittgenstein's later views are considered germane to such scholarship.
Saturday, October 20, 2012
More than two decades ago, I took a course in intermediate microeconomics. The textbook was R. Robert Russell and Maurice Wilkinson's Microeconomics: A Synthesis of Modern and Neoclassical Theory. "Modern", in this case, refers to the use of set theory terminology, linear programming, and proofs like those in an introductory real analysis class. In contrast, "Neoclassical" refers to the use of continuously differentiable functions. In any case, the substance of the theory - which is only one possible theory - is unaffected. (I would not have been clear on this at the time.)
One day, our professor was teaching us about oligopoly and the theory of the kinked demand curve. And, in response to a question, the professor said something like, "This is a theory I actually believe". Yet, in the rest of the classes, when he was teaching us to manipulate utility functions or production functions or to take Lagrangians or whatever, he never expressed an opinion of the empirical applicability of what he was teaching us.
I also recall that our professor made a special effort to teach us input-output analysis one week. This topic was not in the textbook, if I recall correctly. But Leontief was coming to give a lecture (not to our class, but in a big lecture hall, that is, CC308). And our professor wanted us prepared. As it was, Leontief's lecture did not concern the details of input-output analysis, but the complaint that most of then contemporary economics was unconcerned with empirical results. Most economists did not even cast their theory in a form where it could be connected up to empirical data that one might collect.References
- Wassily Leontief (1982). Academic Economics, Science, New Series, V. 217, N0. 4555 (9 July): pp. 104-107.
- Wassily Leonteif (1983). Academic Economics Continued, Science, New Series, V. 219, No. 4587 (25 February): 904.
- R. Robert Russell and Maurice Wilkinson (1979). Microeconomics: A Synthesis of Modern and Neoclassical Theory, John Wiley & Sons.
Sunday, October 07, 2012
|Figure 1: Less Plentiful Supply of Capital Lowers the Interest Rate|
I claim that capital reversing can be a source of instability and interesting dynamics in neoclassical models. I am interested in, for example, the convergence or not of equilibrium paths in models of intertemporal and temporary equilibrium to steady states, but not in tâtonnement dynamics. The ill-behaved nature of many neoclassical models is a challenge in demonstrating this claim.
This post is a start on revisiting these issues. I here outline a simple model of overlapping generations with a simple production model that cannot exhibit reswitching, capital reversing, or even price Wicksell effects. Yet, in this model, a greater willingness among the households to save is associated with a higher interest rate. This is inconsistent with the supposedly intuitive stories told in outdated and exploded neoclassical textbooks.2.0 The Model
The model describes an economy in which a single commodity, corn, is produced. In this model, corn functions as both the consumption good and as the only capital good. In production, all (seed) corn is used up in producing the harvest; that is, all capital is circulating capital. For my purposes in this post, I want to consider an economy in a stationary state.
The point of these assumptions is not to describe any actually existing capitalist economy. Rather, the point is to demonstrate that neoclassical theory does not justify conclusions commonly made. I suppose you can say that these types of models raise the following empirical question: why do mainstream economists continue to teach, both in the classroom and in policy work, conclusions long exposed as nonsense by their own theory?2.1 Utility-Maximizing Agents
Suppose the population consists of overlapping generations, as in Figure 2. Each generation lives for two years. In a given year, all members of the generation born at the start of that year work a full year. They are paid their wages at the end of the year. Out of their wages, they consume some and they save the remainder at the going interest rate. They are retired during the second year of their life. At the end of their second year, they consume the remainder of their income and die.
|Figure 2: Lifespans of Overlapping Generations|
Furthermore, assume that each generation consists of a single individual, also known as an agent. Furthermore, suppose all generations are identically characterized by the following Cobb-Douglas utility function:
U(c0, c1) = (c0)γ(c1)(1 - γ)
where c0 is the bushels corn the agent consumes at the end of the first year of their life, c1 is the bushels corn consumed at the end of the second year, and
0 < γ < 1
A higher value of γ indicates a lesser willingness to defer consumption and a smaller supply of savings. Let w be the wage, and r the interest rate. Under these assumptions, the agent born in each generation solves the following utility-maximization problem:
Given w, r
Choose c0, c1
To Maximize U(c0, c1)
Such that c0(1 + r) + c1 = w(1 + r)
ci ≥ 0; i = 0, 1.
The constraint states that the total value of consumption, evaluated at a single point in time, equals the income of the agent, also evaluated at the same point in time. The solution to this mathematical programming problem is:
c0 = γ w
c1 = (1 - γ) w(1 + r)
S = (1 - γ) w
where S is the bushels corn saved at the end of each year.2.2 Production
For simplicity, I assume a Leontief, fixed coefficients production function. Let L be the person-years of labor employed during the year, K be the bushels corn used as capital during the year, and q be the bushels corn produced during the year. The production function is:
q = min( L/a0, K/a1)
a0 > 0
0 < a1 < (1/2)
(Productivity has to exceed a certain threshold for an equilibrium to exist in this model.)
Only consider cases where both constraints bind. In a stationary state, the corn available at the end of the year is divided up into a1/a0 bushels to use as capital next year and (1 - a1)/a0 corn to consume, per person-year employed.
Given this technology, the wage-rate of profits frontier is easily expressed:
a1(1 + r) + a0w = 1
Hence, one can solve for the wage as a function of the interest rate and the coefficients of production:
w = [(1 - a1)/a0] - (a1/a0)r
When the interest rate is zero, the wage is (1 - a1)/a0, that is, the total surplus of corn, after subtracting the seed corn needed to sustain production at the same level. When the wage is zero, the interest rate is (1 - a1)/a1.2.3 Equilibrium
This model is completed by assuming that the households want to hold the capital stock at the end of every year. since only one generation is saving for retirement at the end of this year, this equilibrium condition is:
S = a1/a0
I might as well make an aside on marginal productivity. In models in which the firms choose the cost-minimizing technique, marginal productivity conditions are used to specify the coefficients of production. The price of each commodity used as a capital good is equal, in equilibrium, to the present value of the marginal product of that commodity. In models in which the technology is specified as a set of fixed-coefficient techniques, the value of marginal product, as I understand it, is an interval in which left-hand and right-hand derivatives enter. In any case, since prices and the quantities of capital goods are both found by solving the model, one cannot say that the (rental) price of a capital good is determined by its marginal product. Furthermore, wages are not determined by the marginal product of labor. A fortiori, the rate of profits is not determined by the marginal product of finance capital, even if one can concoct some equation involving the return on capital, some measure of the value of capital goods, and its marginal product.
Anyways, one can solve the above model to find the following closed-form expression for the interest rate in a stationary state:
r = [(1 - a1)/a1] - [1/(1 - γ)]
Figure 1 above graphs this function. And one can see that, in this model, a stationary state in which households are less willing to save is associated with a lower interest rate. If the interest rate were the price of capital and prices were indices of relative scarcities, this example could not be created. But equilibrium prices are not scarcity indices and neoclassical economics, as taught by most university professors, is nonsensical poppycock.3.0 Conclusion
This post has presented a simple neoclassical model, a limit point, in some sense, of the kind of model that neoclassical economists advocated as a resolution of the Cambridge Capital Controversies. And this simple model shows that much of mainstream teaching and policy work is theoretically unfounded, by their own logic.
Tuesday, October 02, 2012
"An increase in desired saving will only affect the rate of interest slowly, over time, as the greater flow of investment slowly increases the stock of capital and reduces MPK [Marginal Product of Capital]."
"If people want to save more, the rate of interest will fall, the price of capital goods will rise, and there will be a movement along the PPF as existing resources move away from producing consumption goods towards producing investment goods." -- Nick Rowe(Some have tried to explain.)