Walras' Law and the Problem of Money Price Determinacy Rainer Maurer Paper presented at the Annual London Conference on “Money, Economy and Management” July Imperial College, South Kensington

Prof. Dr. Rainer Maurer -2- Overview 1. The Problem of Walras 2. The Neoclassical Dichotomy Approach & Patinkin’s Criticism 3. Patinkin’s Solution: A Real Wealth Effect 4. Weil’s Criticism: Money Is Not Net Wealth in Ricardian Economies 5. Benassy’s Solution: Non-Ricardian Economies Provide a Wealth Effect! 6. Alternative Solution: Institutionally Correct Modelling of Money Supply 6.1. Example: A Textbook Macromodel 6.2. Example: The Case with N Goods 7. Conclusions 8. Appendix: The Untenability of the Neoclassical Dichotomy 9. Literature Detailed Paper “Walras' Law and the Problem of Money Price Determinacy“ available at:

Prof. Dr. Rainer Maurer The Problem of Walras ■Since Walras (1874) has shown that the keeping of the budget constraints implies an equilibrium on the n th market, if n-1 markets are in equilibrium, there is one equation missing to unambiguously determine the money prices of goods. ■Some researchers have therefore given up the idea that money prices are well determined: ● In his voluminous book “Interest and Prices” Woodford (2003, p. 34) cites Wicksell (1898, pp ), “who compares relative prices to a pendulum that always returns to the same equilibrium position when perturbed, while the money prices of goods in general are compared to a cylinder resting on a horizontal plane, which can remain equally well in any location on the plane to which it may happen to be moved”. ■Contrary to this view, I will show in the following: If money supply is modelled in an institutionally correct way, there will always be “an equation left” to allow for money price determinacy: Hence money prices too have the properties of a pendulum!

Prof. Dr. Rainer Maurer The Problem of Walras ■In an economy with N goods markets, only N-1 prices can be determined: ● Household budget with N goods: ● Adding up the budgets of all H households: ■Rearranging the sums: ≡ Lange (1942, p.50): “Walras’ Law”

Prof. Dr. Rainer Maurer The Problem of Walras ● Therefore, if N-1 markets are in equilibrium: ● The N th market too must be in equilibrium, as a subtraction of (2) from (1) shows: ≡ Patinkin (1965, p.35) : “Walras’ Law”

Prof. Dr. Rainer Maurer The Problem of Walras ■Consequently, if all households keep their budgets, only N-1 independent equations exist. => Even if the “counting criterion” holds, => it is only possible to determine N-1 relative prices in terms of the numéraire. The price of the numéraire is set equal to 1. ■Solution following the “Neoclassical Dichotomy Approach”: ● To determine the N money prices of all goods we can simply “add a money market equation” to determine the money price of the numéraire: N-1 relative prices determined by the N-1 independent market equilibrium conditions “Money price” of the numéraire “If the equation system is linear and the coefficient matrix of the linear equations is non-singular, the equality of the number of equations and the number of unknowns is sufficient for the existence of a unique solution.”

Prof. Dr. Rainer Maurer -7- ■Patinkin’s criticism of the “Neoclassical Dichotomy Approach”: ● It leads to a logical contradiction: ♦ If there is a general market equilibrium on all N markets plus the money market, ♦ a duplication λ = 2 of all money prices will leave the N goods markets in equilibrium, since it does not change the relative prices: ♦ The money market equation however will display excess demand: ♦ However, by Walras’ Law this is not possible, since the money market must be in equilibrium, if all other markets are in equilibrium. ♦ Therefore, following Patinkin, the Neoclassical Dichotomy Approach leads to a logical contradiction! 2. The Neoclassical Dichotomy Approach & Patinkin’s Criticism

Prof. Dr. Rainer Maurer Patinkin’s Solution: A Real Wealth Effect ■Patinkin’s solution: The Real-Balance-Effect ● Patinkin (1949) proposed the introduction of a wealth-effect by adding the real value of money holdings as a positively valued argument in the demand functions for goods: ● such that an increase of the price level leads to a decrease of the real value of money wealth and hence the emergence of excess supply of goods, ● which by Walras’ Law is consistent with excess demand for money:

Prof. Dr. Rainer Maurer Weil’s Criticism: Money Is No Net Wealth ■Weil (1991) criticism: Money is no net wealth! ● Following Barro’s “Ricardian Equivalence” Weil shows that in a standard Ricardian (infinitely-lived representative agent) economy, even outside money holdings cannot be net wealth. ● The basic argument for the simplified case of a constant interest rate i = i t and an infinite time horizon, t = 1,2,.. ∞: Present value of the opportunity costs of holding money Value of money holdings

Prof. Dr. Rainer Maurer Weil’s Criticism: Money Is No Net Wealth ■Weil (1991) criticism: Money is no net wealth! ■Mathematically equivalent alternative statement of the argument: ● Barro’s (1974) proof that government bonds are no net wealth under such circumstances: If the government increases its consumption and finances this by issuing government bonds, the representative household receives, on one hand, additional interest payments from these bonds plus the face value at the end of maturity. On the other hand, the present value of these payments equals exactly the additional future taxes, which the household has to pay to finance these interest payments plus redemption. Consequently the net present value of holding these bonds is zero for the household. ● Analogously, if the government increases its consumption and finances this by paying with banknotes, the representative household does, on one hand, not have to pay additional future taxes to finance any interest payments or the redemption, but receives, on the other hand, no interest payments and no repayment of the face value from holding these banknotes. Consequently the net present value of holding these banknotes in a Ricardian economy is for the same reasons zero as the net present value of holding government bonds. If instead of government consumption lump sum transfers to households are assumed, the argument does not change. The only difference in this case is that the disposable income of households stays constant at the end of the day, while in the case of government consumption, the disposable income is reduced.

Prof. Dr. Rainer Maurer Benassy’s Solution: Non-Ricardian Economies Provide a Wealth Effect! ■Benassy’s (2007) solution: In Non-Ricardian Economies money is net wealth! ● These are economies where the utility of future generations is less appreciated by current generations than their own. ■Unattractive properties of this approach: 1. Money price determinacy depends on the non-altruism between old and new generations. 2. Only in economies with a growing population money price determinacy is ensured by a positive relationship between money supply and money prices, while in economies with shrinking populations the relationship between money supply and money prices becomes negative. 3. However, the perhaps most unsatisfying aspect of a wealth-effect- based money price determinacy is its dependency on outside money. Many modern central banks offer most of their money as a credit to the private sector, i.e. as inside money.

Prof. Dr. Rainer Maurer Alternative Solution: Institutionally Correct Modelling of Money Supply: A Textbook Macromodel ■The following calculations show based on the standard three market textbook macromodel that ● if money supply and demand is modeled ● in a realistic, institutionally correct way, there is always an equation left, which can be used to determine the money prices of goods – even in an Ricardian economy, where money is no net wealth.

Prof. Dr. Rainer Maurer Alternative Solution: Institutionally Correct Modelling of Money Supply: A Textbook Macromodel The Three Market Neoclassical Macromodel The Outside Money CaseThe Inside Money Case Markets: Household Budget: Government Budget: Firm Budget Household Budget: Government Budget: Firm Budget

Prof. Dr. Rainer Maurer Alternative Solution: Institutionally Correct Modelling of Money Supply: A Textbook Macromodel The Three Market Neoclassical Macromodel The Outside Money CaseThe Inside Money Case Given the budgets constraints and an equi- librium on the labor and capital market: There will not necessarily be an equi- librium on the goods market:

Prof. Dr. Rainer Maurer -15- The Three Market Neoclassical Macromodel The Outside Money CaseThe Inside Money Case Given the budgets constraints and an equi- librium on the labor and capital market: There will not necessarily be an equi- librium on the goods market: 6. Alternative Solution: Institutionally Correct Modelling of Money Supply: A Textbook Macromodel There will not necessarily be an equi- librium on the goods market:

Prof. Dr. Rainer Maurer -16- The Three Market Neoclassical Macromodel The Outside Money CaseThe Inside Money Case Given the budgets constraints and an equi- librium on the labor and capital market: 6.1. Alternative Solution: Institutionally Correct Modelling of Money Supply: A Textbook Macromodel

Prof. Dr. Rainer Maurer -17- The Three Market Neoclassical Macromodel The Outside Money CaseThe Inside Money Case Given the budgets constraints and an equi- librium on the labor and capital market: 6.1. Alternative Solution: Institutionally Correct Modelling of Money Supply: A Textbook Macromodel

Prof. Dr. Rainer Maurer -18- The Three Market Neoclassical Macromodel The Outside Money CaseThe Inside Money Case Given the budgets constraints and an equi- librium on the labor and capital market: 6.1. Alternative Solution: Institutionally Correct Modelling of Money Supply: A Textbook Macromodel

Prof. Dr. Rainer Maurer -19- The Three Market Neoclassical Macromodel The Outside Money CaseThe Inside Money Case Given the budgets constraints and an equi- librium on the labor and capital market: 6.1. Alternative Solution: Institutionally Correct Modelling of Money Supply: A Textbook Macromodel Only if money demand equals money supply, the goods market is in equilibrium!

Prof. Dr. Rainer Maurer -20- The Three Market Neoclassical Macromodel The Outside Money CaseThe Inside Money Case Given the budgets constraints and an equi- librium on the labor and capital market: Consequently, to make sure that the goods market is in equilibrium, it is necessary to assume that the budget constraints hold, the labor and capital market are in equilibrium and that money demand is equal to money supply! 6.1. Alternative Solution: Institutionally Correct Modelling of Money Supply: A Textbook Macromodel

Prof. Dr. Rainer Maurer Alternative Solution: Institutionally Correct Modelling of Money Supply: A Textbook Macromodel ■If money supply is larger than money demand, there will be excess demand for goods, which will cause the price level for goods to increase: ■If real money demand depends (as usual) on the real transaction volume divided by the money velocity, this increase of the price level will cause real demand supply to decrease so that the excess supply of money and – simultaneously – the excess demand for goods disappears: <= =>

Prof. Dr. Rainer Maurer Alternative Solution: Institutionally Correct Modelling of Money Supply: A Textbook Macromodel Digression: Consequently, if money supply is institutionally correctly modelled, the above three market macromodel can be easily transformed into the standard textbook form: If the assumption is made that the budget constraints hold, three market equilibrium conditions are sufficient to guarantee a general market equilibrium: Either Labor market equilibrium: Capital market equilibrium: Money market equilibrium: Or Labor market equilibrium: Goods market equilibrium: Money market equilibrium: Consequently, the standard textbook system of market equilibrium conditions is compatible with the above described “outside money” case, as well as with the “inside money” case. The standard textbook model has in fact two monetary interpretations. The same holds of course for the Keynesian fixed price version of this model. => Goods market equilibrium Capital market equilibrium

Prof. Dr. Rainer Maurer Alternative Solution: Institutionally Correct Modelling of Money Supply: A Textbook Macromodel ■As this text book macromodel shows, if money supply differs from money demand, a “transaction volume effect” will cause the price level to adjust and restore an equilibrium on the goods market. ■Consequently it is a “transaction volume effect” the determines the price level – there is no need for Patinkin’s “wealth effect” (or “real balance effect” or “Pigou effect”). ■The following calculations show that the same holds of course for the case of an economy with N goods and H households:

Prof. Dr. Rainer Maurer Alternative Solution: Institutionally Correct Modelling of Money Supply: The Case with N Goods ■Starting point: An economy with H households, N goods markets and money supplied by the government: ● Household budget with N goods plus money demand: ● Government budget with N goods plus money supply: ● Adding up the budgets of all H households and the government:

Prof. Dr. Rainer Maurer Alternative Solution: Institutionally Correct Modelling of Money Supply: The Case with N Goods ■If N-1 markets for goods are in equilibrium: ● The N th market will not necessarily be in equilibrium, as subtracting of (4) from (3) shows: ● Consequently, to make sure that the Nth goods market is in equilibrium, it is necessary to assume that the budget constraints hold, the N-1 markets are in equilibrium and that money demand is equal to money supply!

Prof. Dr. Rainer Maurer -26- ■If money supply is larger than money demand, there will, as equation (3) shows, generally be excess demand for goods: ■Following the standard assumption, nominal money demand of a household depends on the nominal transaction volume of the household: 6.2. Alternative Solution: Institutionally Correct Modelling of Money Supply: The Case with N Goods

Prof. Dr. Rainer Maurer -27- ■Under this assumption, an increase of the price level will cause money demand to grow so that the excess supply of money and – simultaneously – the excess demand for goods disappears: ■As this model with N goods and H Households shows, if money supply differs from money demand, it is again a “transaction volume effect”, which will cause the price level to adjust and restore an equilibrium on the goods market. ■Consequently here too, it is a “transaction volume effect” the determines the price level – there is no need for Patinkin’s “wealth effect” Alternative Solution: Institutionally Correct Modelling of Money Supply: The Case with N Goods <= =>

Prof. Dr. Rainer Maurer Conclusions 1.If money supply and money demand is modelled based on a realistic institutional setup in the budget constraints and market equations, the resulting number of independent equations is always equal to the number of goods. ● Consequently, if the “counting criterion” holds, there are always enough equations to determine the money prices of all goods in an economy. 2.If money demand depends on the transaction volume of the economy, it will be a “transaction volume effect”, which restores the monetary equilibrium but not a “wealth effect”. In so far, Patinkin (1948) is wrong and the Neoclassical Dichotomy Approach is right.

Prof. Dr. Rainer Maurer Conclusions 3.However, Patinkin (1948) is right and the Neoclassical Dichotomy Approach is wrong in another important point: ● Money price determinacy excludes the assumption of “zero degree homogeneity” in money prices of supply and demand functions: A realistic institutional setup of money supply and money demand excludes the assumption of “zero degree homogeneity”. As the following appendix shows, if money supply and demand are modelled in an institutionally correct way, the assumption of “zero degree homogeneity” in money prices, leads to a logical contradiction. 4.As a result of this all: The standard procedure used in many monetary models, to “eliminate one market by Walras Law” and add a “money market” is correct.

Prof. Dr. Rainer Maurer Appendix: The Untenability of the Neoclassical Dichotomy ■Under an institutionally correct setup of money supply, the classical assumption of degree 0 homogeneity in money prices of the demand and supply functions, ■leads to a logical contradiction as the following shows: Starting with a general market equilibrium so that following eq. (3):

Prof. Dr. Rainer Maurer Appendix: The Untenability of the Neoclassical Dichotomy ■A multiplication of all money prices by a factor λ ≠ 1 yields: ■Given the assumption of zero degree homogenty in money prices this equals ■what contradicts the assumption that λ ≠ 1. = 0 = M S

Prof. Dr. Rainer Maurer Literature ■Debreu (1959), Gerard Debreu, Theory of Value, Cowles Foundation, Monograph 17, Yale University Press, New Haven. ■Baumol (1952), William, The Transactions Demand for Cash: An Inventory Theoretic Approach, Quarterly Journal of Economics 66, p ■Barro, Robert (1974), Are Government Bonds Net Wealth?, Journal of Political Economy 82, pp ■Bénassy, Jean-Pascal (2007), Money, Interest, and Policy, The MIT Press, Cambridge, Massachusetts. ■Bullard, J., and Mitra, K. (2002), Learning About Monetary Policy Rules, Journal of Monetary Economics 49, pp ■Calvo, Guillermo (1983), Staggered Prices in a Utility-maximizing Framework, Journal of Monetary Economics 12, pp ■Cooley, T., and Hansen, G. (1989), The Inflation Tax in a Real Business Cycle Model, The American Economic Review, pp ■Fisher, Irvin (1911), The Purchasing Power of Money, new and revised edition 1913, New York. ■Gurley, J., and Shaw, E. (1960), Money in a Theory of Finance, Brookings Institution, Washington D.C. ■Lange, Oscar (1942), Say’s Law: A Restatement and Criticism, pp in: Studies in Mathematical Economics and Econometrics, Chicago. ■McCallum, Bennett (1986), Some Issues Concerning Interest Rate Pegging, Price Level Determinacy and the Real Bills Doctrine, Journal of Monetary Economics 17, pp ■Maurer (2008), The Increasing Leverage of Central Bank Cash in Transition to a Cashless Economy - A DSGEM Analysis, Discussion Paper, ■Patinkin, Don (1948), Price Flexibility and Full Employment, American Economic Review 38, pp ■Patinkin, Don (1949), The Indeterminacy of Absolute Prices in Classical Economic Theory, Econometrica 17, pp ■Patinkin, Don (1965), Money, Interest and Prices, Harper and Row, New York. ■Pigou, Arthur Cecil (1917), The Value of Money, Quarterly Journal of Economics 27, as reprinted in Readings in Monetary Theory, 1951, editors F.A. Lutz and L. W. Mints, Philadelphia, pp ■Tobin (1956), James, The Interest Elasticity of the Transactions Demand for Cash, Review of Economics and Statistics, p ■Walras, Léon (1874), Éléments d’Économie Politique Pure ou Théorie de la Richesse Sociale, Imprimerie L. Corbaz & Cie., Lausanne, Reprinted Paris: Economica, English translation referred to in this paper: Elements of Pure Economics or The Theory of Social Wealth, translated by William Jaffé, first published in 1954, American Economic Association and The Royal Economic Society, George Allen and Uwin LTD, London. ■Weil, Philippe (1991), Is Money Net Wealth?, International Economic Review 32, pp ■Wicksell, Knut (1898), Interest and Prices, English translation referred to in this paper by R. Kahn (1936), Macmillan, London. ■Woodford, Michael (2003), Interest and Prices, Princeton University Press, Princeton.