- Email: [email protected]
Budget constraints that are not soft
ELSEVIER
Japan and the World Economy 8 (1996) 225-228
ECONOMY
Budget constraints that are not soft Paul A. Samuelson The Center for Japan-US Business and Economic Studies, Stern School of Business, New York University, N Y I0012-1126, USA, and Department of Economics, MIT, E52-283-C, Cambridge, MA 02139-Y307, USA
Received 25 March 1994 The very definition of economics, as proposed by Lionel Robbins in his 1932 classic (see Palgrave), involves "the maximization of given ends under the constraints of scarce means with alternative uses". And today every school child in economics knows what Adam Smith didn't quite understand about the op-timality of his Invisible Hand. Not until my own day was that understood to be Pareto-optimality, wherein competition under favorable conditions contrives that the ordinal well-being of any one person is maximal subject to preassigned levels of well-being for all other persons. When a baby is born, and the Good Fairies ordain that she will be an economist, the first thing she is taught- after A, B, C, and D - is the nature and use of Lagrange multipliers. When the constraints are truly hard, she will learn in kindergarten that the Lagrangian multipliers are precisely the prices and marginal cost-utilities that arise out of, and support, the optimality equilibrium, And when the constraint is truly redundant, its Lagrangian price is truly zero. What is quite different, and as your discussions at this conference illuminate, a soft and fuzzy constraint entails conjugate prices of problematic ambiguity. Aside from the generality of constrained maximizations, there is a special mathematical problem that came to be of prime centrality after the 1870 Neoclassical Revolution of Marginalism and Subjective Value. Half of the problem of general equilibrium had been left out of the picture in the classical economics of Smith, Ricardo, Mill, and Marx-namely the analysis of consumer demand. Stanley Jevons, Carl Menger, L6on Walras-to say nothing of Marshall, Fisher, and Wicksell-defined the problem of a consumer with limited income or budget, and facing given market prices, who must reveal his/her demand functions for n goods.
S0922-1425/96/$15.00 Copyright© 1996ElsevierScienceB.V. All rights reserved PII S0922-1425(96)00008-4
226
P.A. Samuelson / Japan and the Worm Economy 8 (1996) 225 228
The problem is well framed, but not until 1915-my t i m e - d i d Slutsky manage to get the problem solved right. Even the great Pareto, a pretty good mathematician (whose Ph.D. thesis had been on what is today called the Laplace Transform), fell short of specifying a complete set of necessary and sufficient conditions. And no wonder. The Advanced calculus textbooks of even the present day stop short of presenting an adequately complete account of the subject. As a lunch-time digression, I have been asked to survey briefly the history of budget- constrained demand theory, from its 1870 beginnings to its completion by 1950. There is no pretence that soft-budgeting can learn from the softheadedness of pioneering contributors to this neoclassical subject. However, the minimax theory of constrained maxima can perhaps illuminate one advantage of the Lerner-Lange fairy-land of a socialist society organized by having decentralized bureaucrats play the game of using the competitive auction mechanism to generate Lagrangian shadow prices that sustain an efficient allocation of resources. The gradient differential equations work well for an unconstrained true maximum. For the hard-budget constrainment case, having the primal variables climb up to Lagrangian Max-Min-imand while the dual variables climb down it will also converge to the Lerner-Lange utopia. A different kind of Walrasian tattonement need not be sought. If time were longer, I could usefully survey the related history of the integrability problem in that same period. Tom Russell and I have been corresponding recently on the latest wrinkles in that problem. Here I must own up to a social faux pas. It is like the case of a guest invited to Isaac Newton's party, and who prattles about Leibniz's invention of the calculus. I should at this point have mentioned our host Ryuzo Sato's contribution to the integrability problem when he tied it up with the elegant theory of continuous Lie Groups. Mea culpa!! Having said only this much about the integrability problem, let me say epsilon more. You observe for a representative Hungarian in pre-Gorbachev days what she bought of (coffee, tea, cream) and the relative or real prices of them. I use this room as my three-good space, using that corner of the floor for its origin. I put at each point depicting its (coffee, tea, cream) a little pointed thumb tack with its point thrust away from the origin. I am not yet entitled to call this a little planar element of indifference. It is only revealed price-ratios data: (Ptea/Pcoffee, P ..... /Pcoffee) gives the tack its precise orientation. We can fill the room-space with thumb tacks everywhere. The integrability problem is this: Can they be joined up to form the bitsy tangent plans of a convex two-dimensional surface in our three-space? Are there nested kitchen bowls whose tangents generate the real price data? In Mises land, where all display perfect transitivity, we can test and affirm that the (q,p) data do satisfy integrability condition(s). For Stalin land, where the p's you pay are not the prices at which you could have bought a little more or a little less, it would be an
P.A. Samuelson / Japan and the World Economy 8 (1996) 225 228
227
unbelievable miracle if the (q,p) data passed integrability tests like the Houthakker-Samuelson-Strong Axiom. Why know this esoteric math garbage? It may prepare you for the following: the best Bergson-Kravis calculations might show 1987 East Germany with 80% of the per capita real income of West Germany. And these subtle statisticians maybe did their job well. However, once the Wall came down, it appears that maybe 33% would be a better guess than 80%. Why this bias? It is there because of the soft-budget and other characteristics of the command bureaucracy. In sum the (price, quantity) data observed from Eastern European command economies would not in that soft-budget environment be expected to satisfy transitivity axioms of weak and strong preference nor, in the smooth continuous case, admit of "integrating factors" that permit identifying exact differentials of existent ordinal-utility functions. Statistician emptor: China's present 10% growth may turn out to be 10% - X genuine rate. The first pioneers Gossen, Jevons, Walras, Edgeworth, early-Fisher-spoke prose without knowing it. Their analysis of comparative statics, of how a change in a pj or income would change q~ or qk, could be unself-conscious in their Santa Claus case where each good had independent utility, with marginal utility that was a declining function of solely its own consumption amount. Believing in the law of declining marginal utility, they could perceive intuitively that all goods had positive income elasticity; that a rise in pj had to decrease qj and had to increase or decrease all other qk'S depending upon whether q~ was of own-inelastic or own-elastic demand. Pareto sought to go farther. He (and Edgeworth and Fisher) realized that utility might be a joint function of two or more goods which made pairs of them complements or rivals. Giffen, somewhere, had alerted economists to the admissible phenomenon of inferior goods on the Emerald Isle, where higher pj for needed potatoes evoked higher rather than lower q/ Pareto, after fifteen years of toil, never could arrive at a complete and correct catalog of comparative statics of admissible price elasticities, own and cross elasticities, and of income-elasticity signs. Why? The answer is purely technical, not related to any McCloskey or Mirowski rhetoric. Pareto simply could not know how to handle the bordered matrices and determinants of his hard-budget problem. Why didn't he go to the vast thermodynamics literature of Gibbs and company? Maybe he didn't know it. But that was just as well since that literature never reached the level of mathematical sophistication of the Slutsky-to-Hotelling economic literature. You know, because I tell you so, that Weierstrass in his German seminar lectures had worked out before 1890 the requisite complete theory of maxima subject to equality constraints. I know this because a young American mathematician sat in Weierstrass's classroom and took careful notes on his exposition. Later when Hancock published his Maxima and Minima treatise, he
228
P.A. Samuelson / Japan and the World Economy 8 (1996) 225 228
acknowledged that he had benefitted from Weierstrass's lectures. But he did not say his book was a word-for-word crib from them. How do I know that? Because when rambling through the mathematical stacks of Harvard's Widener Library, I once stumbled on the typescript copy of those lectures that Hancock must have deposited there. Even the Greek letters denoting the variables in Hancock were faithful copies of those Weierstrass happened to use on Mondays, Wednesdays, and Fridays. What doesn't get published in effect does not exist. Slutsky had to work the results out for himself. By a lucky trick or insight, while Roy Allen and John Hicks were still in primary school and Lionel McKenzie was not yet born, Slutsky perceived that iso-utility demand reactions provided the symmetry and negative semi-definiteness of comparative-statics matrices that closed the important open questions. Slutsky did publish. But in Italian and in the middle of World War I when his work was sure to go without notice. So Allen and Hicks, and Hotelling had later to independently arrive at the problem's complete solution- except for the finite-inequality approach of revealed preferences, which Pareto might have got hints of if he had read Gibbs's mid-1870s gems on equilibrium of heterogeneous substances. Lunch time, alas, is all too short. But the story of which I have sung is a splendid example of cooperative public science in the making. We all add our brick to the Cathedral. And peer review, ex ante and ex post, Darwinianly weeds out the false steps and unaesthetic detours. Working scientists, when you tell them about Tom Kuhn's notions that it is all a game of mutuallyback-scratching coteries, proceed to forget your words and go back to the drawing boards eager to shape their brick and get it in first.
Japan and the World Economy 8 (1996) 225-228
ECONOMY
Budget constraints that are not soft Paul A. Samuelson The Center for Japan-US Business and Economic Studies, Stern School of Business, New York University, N Y I0012-1126, USA, and Department of Economics, MIT, E52-283-C, Cambridge, MA 02139-Y307, USA
Received 25 March 1994 The very definition of economics, as proposed by Lionel Robbins in his 1932 classic (see Palgrave), involves "the maximization of given ends under the constraints of scarce means with alternative uses". And today every school child in economics knows what Adam Smith didn't quite understand about the op-timality of his Invisible Hand. Not until my own day was that understood to be Pareto-optimality, wherein competition under favorable conditions contrives that the ordinal well-being of any one person is maximal subject to preassigned levels of well-being for all other persons. When a baby is born, and the Good Fairies ordain that she will be an economist, the first thing she is taught- after A, B, C, and D - is the nature and use of Lagrange multipliers. When the constraints are truly hard, she will learn in kindergarten that the Lagrangian multipliers are precisely the prices and marginal cost-utilities that arise out of, and support, the optimality equilibrium, And when the constraint is truly redundant, its Lagrangian price is truly zero. What is quite different, and as your discussions at this conference illuminate, a soft and fuzzy constraint entails conjugate prices of problematic ambiguity. Aside from the generality of constrained maximizations, there is a special mathematical problem that came to be of prime centrality after the 1870 Neoclassical Revolution of Marginalism and Subjective Value. Half of the problem of general equilibrium had been left out of the picture in the classical economics of Smith, Ricardo, Mill, and Marx-namely the analysis of consumer demand. Stanley Jevons, Carl Menger, L6on Walras-to say nothing of Marshall, Fisher, and Wicksell-defined the problem of a consumer with limited income or budget, and facing given market prices, who must reveal his/her demand functions for n goods.
S0922-1425/96/$15.00 Copyright© 1996ElsevierScienceB.V. All rights reserved PII S0922-1425(96)00008-4
226
P.A. Samuelson / Japan and the Worm Economy 8 (1996) 225 228
The problem is well framed, but not until 1915-my t i m e - d i d Slutsky manage to get the problem solved right. Even the great Pareto, a pretty good mathematician (whose Ph.D. thesis had been on what is today called the Laplace Transform), fell short of specifying a complete set of necessary and sufficient conditions. And no wonder. The Advanced calculus textbooks of even the present day stop short of presenting an adequately complete account of the subject. As a lunch-time digression, I have been asked to survey briefly the history of budget- constrained demand theory, from its 1870 beginnings to its completion by 1950. There is no pretence that soft-budgeting can learn from the softheadedness of pioneering contributors to this neoclassical subject. However, the minimax theory of constrained maxima can perhaps illuminate one advantage of the Lerner-Lange fairy-land of a socialist society organized by having decentralized bureaucrats play the game of using the competitive auction mechanism to generate Lagrangian shadow prices that sustain an efficient allocation of resources. The gradient differential equations work well for an unconstrained true maximum. For the hard-budget constrainment case, having the primal variables climb up to Lagrangian Max-Min-imand while the dual variables climb down it will also converge to the Lerner-Lange utopia. A different kind of Walrasian tattonement need not be sought. If time were longer, I could usefully survey the related history of the integrability problem in that same period. Tom Russell and I have been corresponding recently on the latest wrinkles in that problem. Here I must own up to a social faux pas. It is like the case of a guest invited to Isaac Newton's party, and who prattles about Leibniz's invention of the calculus. I should at this point have mentioned our host Ryuzo Sato's contribution to the integrability problem when he tied it up with the elegant theory of continuous Lie Groups. Mea culpa!! Having said only this much about the integrability problem, let me say epsilon more. You observe for a representative Hungarian in pre-Gorbachev days what she bought of (coffee, tea, cream) and the relative or real prices of them. I use this room as my three-good space, using that corner of the floor for its origin. I put at each point depicting its (coffee, tea, cream) a little pointed thumb tack with its point thrust away from the origin. I am not yet entitled to call this a little planar element of indifference. It is only revealed price-ratios data: (Ptea/Pcoffee, P ..... /Pcoffee) gives the tack its precise orientation. We can fill the room-space with thumb tacks everywhere. The integrability problem is this: Can they be joined up to form the bitsy tangent plans of a convex two-dimensional surface in our three-space? Are there nested kitchen bowls whose tangents generate the real price data? In Mises land, where all display perfect transitivity, we can test and affirm that the (q,p) data do satisfy integrability condition(s). For Stalin land, where the p's you pay are not the prices at which you could have bought a little more or a little less, it would be an
P.A. Samuelson / Japan and the World Economy 8 (1996) 225 228
227
unbelievable miracle if the (q,p) data passed integrability tests like the Houthakker-Samuelson-Strong Axiom. Why know this esoteric math garbage? It may prepare you for the following: the best Bergson-Kravis calculations might show 1987 East Germany with 80% of the per capita real income of West Germany. And these subtle statisticians maybe did their job well. However, once the Wall came down, it appears that maybe 33% would be a better guess than 80%. Why this bias? It is there because of the soft-budget and other characteristics of the command bureaucracy. In sum the (price, quantity) data observed from Eastern European command economies would not in that soft-budget environment be expected to satisfy transitivity axioms of weak and strong preference nor, in the smooth continuous case, admit of "integrating factors" that permit identifying exact differentials of existent ordinal-utility functions. Statistician emptor: China's present 10% growth may turn out to be 10% - X genuine rate. The first pioneers Gossen, Jevons, Walras, Edgeworth, early-Fisher-spoke prose without knowing it. Their analysis of comparative statics, of how a change in a pj or income would change q~ or qk, could be unself-conscious in their Santa Claus case where each good had independent utility, with marginal utility that was a declining function of solely its own consumption amount. Believing in the law of declining marginal utility, they could perceive intuitively that all goods had positive income elasticity; that a rise in pj had to decrease qj and had to increase or decrease all other qk'S depending upon whether q~ was of own-inelastic or own-elastic demand. Pareto sought to go farther. He (and Edgeworth and Fisher) realized that utility might be a joint function of two or more goods which made pairs of them complements or rivals. Giffen, somewhere, had alerted economists to the admissible phenomenon of inferior goods on the Emerald Isle, where higher pj for needed potatoes evoked higher rather than lower q/ Pareto, after fifteen years of toil, never could arrive at a complete and correct catalog of comparative statics of admissible price elasticities, own and cross elasticities, and of income-elasticity signs. Why? The answer is purely technical, not related to any McCloskey or Mirowski rhetoric. Pareto simply could not know how to handle the bordered matrices and determinants of his hard-budget problem. Why didn't he go to the vast thermodynamics literature of Gibbs and company? Maybe he didn't know it. But that was just as well since that literature never reached the level of mathematical sophistication of the Slutsky-to-Hotelling economic literature. You know, because I tell you so, that Weierstrass in his German seminar lectures had worked out before 1890 the requisite complete theory of maxima subject to equality constraints. I know this because a young American mathematician sat in Weierstrass's classroom and took careful notes on his exposition. Later when Hancock published his Maxima and Minima treatise, he
228
P.A. Samuelson / Japan and the World Economy 8 (1996) 225 228
acknowledged that he had benefitted from Weierstrass's lectures. But he did not say his book was a word-for-word crib from them. How do I know that? Because when rambling through the mathematical stacks of Harvard's Widener Library, I once stumbled on the typescript copy of those lectures that Hancock must have deposited there. Even the Greek letters denoting the variables in Hancock were faithful copies of those Weierstrass happened to use on Mondays, Wednesdays, and Fridays. What doesn't get published in effect does not exist. Slutsky had to work the results out for himself. By a lucky trick or insight, while Roy Allen and John Hicks were still in primary school and Lionel McKenzie was not yet born, Slutsky perceived that iso-utility demand reactions provided the symmetry and negative semi-definiteness of comparative-statics matrices that closed the important open questions. Slutsky did publish. But in Italian and in the middle of World War I when his work was sure to go without notice. So Allen and Hicks, and Hotelling had later to independently arrive at the problem's complete solution- except for the finite-inequality approach of revealed preferences, which Pareto might have got hints of if he had read Gibbs's mid-1870s gems on equilibrium of heterogeneous substances. Lunch time, alas, is all too short. But the story of which I have sung is a splendid example of cooperative public science in the making. We all add our brick to the Cathedral. And peer review, ex ante and ex post, Darwinianly weeds out the false steps and unaesthetic detours. Working scientists, when you tell them about Tom Kuhn's notions that it is all a game of mutuallyback-scratching coteries, proceed to forget your words and go back to the drawing boards eager to shape their brick and get it in first.