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INFLATION & WELFARE ROBERT E. LUCAS 2
Introduction In a monetary economy, private interest is to hold not non-interest bearing cash. Individual efforts due to this incentive must cancel out, because someone must hold it all. Real Recourses are wasted on a task that should not have to performed at all. 3
Introduction Opportunity cost of holding cash is nominal interest rate. The time devoted to economizing on holding cash, is an increasing function of nominal rate and so inflation. Inflation should have an adverse effect on individual s welfare. 4
In this paper Research on the welfare cost of inflation is surveyed. The welfare cost of inflation for U.S. is estimated in a variety of ways. 5
It is shown that The gain of reducing nominal interest rate to about 0.1 percent is positive. Reducing inflation from 10 to 0 is equivalent to about 1 percent increase in income. Using aggregate evidence, the gain of zero nominal interest, may not be estimate reliably. 6
Money Demand & Consumer Surplus Money demand, as a function of nominal interest rate is estimated. (Meltzer 1963a) Welfare Cost is calculated, based on the estimated demand function. (Bailey 1956). There is no any theoretical interpretation. 7
Money Demand & Consumer Surplus M t P t = L r t, y t, L r, y = m r y 8
Money Demand & Consumer Surplus m r = Ar η 9
Money Demand & Consumer Surplus m r = Bexp( ξr) 10
Money Demand & Consumer Surplus Actual and Predicted Real Balances m r = Ar 0.5 11
Money Demand & Consumer Surplus Welfare Cost Definition: The area under the inverse demand curve between m(r) and m(0): The Lost Surplus. r nominal interest rate m(r) m(0) w r = ψ x dx m(r) money/output w r = m x dx 0 r r = ψ(m) rm(r) 12
Money Demand & Consumer Surplus Welfare Function: r w r = m x dx 0 rm(r) m r = Ar η w r = A η 1 η r1 η m r = Be ξr w r = B,1 (1 + ξ ξr)e ξr - 13
Money Demand & Consumer Surplus Welfare Cost function 14
Money Demand & Consumer Surplus Welfare Cost relative to 3% interest 15
Money Demand & Consumer Surplus Results: The two curves are similar above 3 nominal interest, which relates to about 0 inflation. The benefit of reducing Inflation from 10 to 0 is less than 1 percent. 16
Money Demand & Consumer Surplus Results: The welfare function below 3% interest is so different between two curves. Minimum cost is in zero nominal interest, which means deflation. (Friedman Rule-1969) Aggregate evidences is not sufficient. 17
Money Demand & Consumer Surplus Critiques: There is no any theoretical interpretation of this estimate. we need a model to see what changes in monetary policy might generate m(r) & w(r). 18
Money Demand & Consumer Surplus Critiques: Simply labeling the point in the figures demand function does not tell us what is our estimate. Giving colorful names to statistical relationships is not a substitute for economic theory. 19
The Sidrauski Framework Welfare cost is obtained based on a theory of deterministic general equilibrium (Sidrauski 1967a,b). Real Money demand is entered directly in utility as a proxy of transaction facility. It is shown that for the range of U.S. interest rates the solution of welfare cost is very close to the last results. There is no labor-leisure trade-off and Fiscal Policies is not entered explicitly. 20
The Sidrauski Framework Representative Household: Supplies one unit of Labor in each period with productivity y t = y t 1 (1 + γ) Gains utility in each period, from the consumption of one nondurable good: c Gains utility in each period, from holding real balances: z = M/P 21
The Sidrauski Framework Representative Household: U c t, z t = 1 1 σ cφ(z) 1 σ c There is no long run trend in the real balance income ratio. The constant Risk aversion is consistent with balanced growth path. 22
The Sidrauski Framework Representative Household: Maximize the total utility over his lifetime: V = t=0 1/ 1 + ρ t U(c t, z t ) Subjected to his constraint by choosing c t, z t 23
The Sidrauski Framework Household Constraint: t M t+1 = M t H t + P t y t P t c t H t : Lump sum tax m t z t y t = M t P t y t, ω t c t y t, v t H t P t y t, 1 + π t = P t P t 1 (1 + γ) 1 + π t+1 m t+1 = m t v t + 1 ω t 24
The Sidrauski Framework Household Behavior: Household begins in period 1 with balance M and real wage y. t=0 V = V y, z = max 1/ 1 + ρ t U(c t, z t ) c 1, c 2, 25
The Sidrauski Framework Household Behavior: V = V y, z = max U c 1, z + max c 1 c 2, c 3, 1 1 + ρ t=1 1 1 + ρ t U c t+1, z t+1 V = V y, z = max *U c 1, z + 1 1 + ρ V(y, z )+ c 1 26
The Sidrauski Framework Household Behavior: 1 V = V y, z = max * 1 ς cφ(z c ) 1 σ 1 + 1 + ρ V(y(1 + γ), z )+ c z = z + y c 1 + π 27
The Sidrauski Framework Household Behavior: V = V y, m = max * y1 σ 1 σ 1 ς ωφ(m ω ) ω + 1 1 + ρ V(y(1 + γ), m )+ m = m v + 1 ω (1 + π)(1 + γ) 28
The Sidrauski Framework Household Behavior: V y, m = y 1 σ v(m) 1 1 σ v(m) = max * 1 ς ωφ(m ω ) ω + 1 + γ 1 σ 1 + ρ v(m )+ m = m v + 1 ω (1 + π)(1 + γ) 29
The Sidrauski Framework Household Behavior: F.O.C. : φ m ω σ φ m ω m ω φ m ω = 1 1+r v m 1 1 + r = 1 + γ σ 1 + ρ (1 + π) 30
The Sidrauski Framework Household Behavior: Envelope Condition : v m = φ m ω σ φ m ω + 1 1 + r v m 1 1 + r = 1 + γ σ 1 + ρ (1 + π) 31
The Sidrauski Framework Monetary and Fiscal Policy: M t = 1 + μ M t 1 H t = v P t y t = v t 32
The Sidrauski Framework Balanced Growth Path: (1 + γ) 1 + π t+1 m t+1 = m t v + 1 ω t ω t = c t /y t = ω m t = M t P t y t = m 1 + π t = 1 + π = (1 + μ)/(1 + γ) μm = v + 1 ω 33
The Sidrauski Framework Household Behavior: φ m ω φ m ω m ω φ m ω = r 1 1 + r = 1 + γ σ 1 + ρ (1 + π) 34
The Sidrauski Framework Solving the Model: M t & Y t is known, but P t is unknown, so m. ω is unknown. There is only one relation for m & ω from Household maximization. 35
The Sidrauski Framework Solving the Model: Market clearing in each time: c t = y t ω = 1 36
The Sidrauski Framework Solving the Model : φ m φ m m φ m = r 1 1 + r = 1 + γ σ 1 + ρ (1 + π) r is a function of economic growth, γ, which is taken exogenous. 37
The Sidrauski Framework Nominal interest rate: For small growth : r ρ + ςγ + π, π μ γ In a real economy with durable good, balanced growth is determined by the capital return. Real interest rate is : ρ + ςg. γ could be replaced by balanced growth in this economy. r is taken to be nominal interest rate. 38
The Sidrauski Framework Real balance output ratio: φ m(r) = r 1 + m(r)r φ m(r) Real balance output ratio is obtained as a function of nominal interest rate in this micro based theory model. 39
The Sidrauski Framework Real balance output ratio : φ m r = 1 + mr φ > 0, φ < 0 : m r < 0 φ m U c, z = U y, m r y is increasing function of z : U r < 0 Maximum Utility is obtained at zero nominal interest rate: Friedman Rule (1969) The best Policy Rule is deflation equal to real interest. 40
The Sidrauski Framework Welfare Cost: The percentage income compensation needed to leave the household indifferent between r and 0 U 1 + w r y, m r y = U,y, m 0 y- 1 + w r φ m r 1 + w r = φ,m(0)- 41
The Sidrauski Framework Welfare Cost: w r = ψ m r 1 + w r m (r) ψ is the inverse function of m(r) : r = ψ(m) For small w we have : w r = ψ m r m r w r = ψ m dm (Consumer Surplus) 42
The Sidrauski Framework Real balance output ratio : φ m φ m m φ m = r Note that φ is the utility of household over m. What is it for the American household? 43
The Sidrauski Framework Results: If the m(r) takes the form of m r best estimate for U.S. data: φ m = 1 + A2 m 1 = A/ r as is the 44
The Sidrauski Framework Results: Welfare Cost based on this theoretical model is obtained as: A r w r = 1 A r For A = 0.05, r < 10%(U.S. data), the difference between this relation and the formula based on consumer surplus is less than 2 percent. 45
The Sidrauski Framework Results: Based on this theoretical model, Curves m(r) and w(r) are tracing out Steady States of Deterministic economies in balanced growth path, Subjected to different constant rates of money growth. 46
The Sidrauski Framework Importance of Assumptions: In deterministic framework, the costs related to price and inflation variability is dismissed. Based on Cooley & Hansen (1989), the effect of introducing stochastic events is negligible. There is no labor-leisure trade-off and fiscal policies is not interred directly in this model. In the next model, labor-leisure trade-off and fiscal considerations are introduced in this model. 47
Fiscal Considerations Welfare cost is obtained based on a theory of general equilibrium (Sidrauski 1967a,b). Real Money demand is entered directly in utility as a proxy of transaction facility. Labor-leisure trade-off is considered. Fiscal policy and government consumption is entered directly to the model. It is shown that above very small interest rates, estimated welfare cost is the same as the last model. 48
Fiscal Considerations Fiscal Constraint: v = μm(r) r = (ρ + γς) + (μ γ) δ r μ v = δ r m(r) 49
Fiscal Considerations Fiscal Constraint: v = δ r m(r) m r = A/ r In the optimal interest (r = 0), m, so v Lump sum tax takes infinite value! 50
Fiscal Considerations Fiscal Constraint: The Policy r = 0 is not feasible. The Friedman Rule requires qualification. 51
Fiscal Considerations Representative Household: Gains utility in each period, from leisure share of its time : x Supplies 1 x unit of Labor in each period with productivity y t = y t 1 (1 + γ) Gains utility in each period, from the consumption of one nondurable good: c Gains utility in each period, from holding real balances: z = M/P 52
Fiscal Considerations Representative Household: U c t, z t = 1 1 σ 1 σ cφ(z ) (x) c There is no long run trend in the real balance income ratio. The constant Risk aversion is consistent with balanced growth path. There is no long run trend in the share of working. 53
Fiscal Considerations Representative Household: Maximize the total utility over his lifetime: V = t=0 1/ 1 + ρ t U(c t, z t, x t ) Subjected to his constraint by choosing c t, z t, x t 54
Fiscal Considerations Monetary and Fiscal Policy: M t = 1 + μ M t 1 Government purchase in each time: G t = g t y t Government collect tax from household income with rate of τ. 55
Fiscal Considerations Household Constraint: t M t+1 = M t + P t (1 τ)(1 x t )y t P t c t m t z t y t = M t P t y t, ω t c t y t, 1 + π t = P t P t 1 1 + γ 1 + π t+1 m t+1 = m t + (1 τ)(1 x t ) ω t 56
Fiscal Considerations Household Behavior: Household begins in period 1 with balance M and productivity y. t=0 V = V y, z = max 1/ 1 + ρ t U(c t, z t, x t ) c 1, c 2, x 1, x 2, 57
Fiscal Considerations Household Behavior: V y, m = y 1 σ v(m) 1 1 σ v(m) = max * 1 ς ωφ(m ω ) (x) ω, x + 1 + γ 1 σ 1 + ρ v(m )+ m = m + (1 τ)(1 x) ω (1 + π)(1 + γ) 58
Fiscal Considerations Balanced Growth Path: 1 + γ 1 + π t+1 m t+1 = m t + (1 x t )(1 τ) ω t ω t = c t y t = ω x t = x m t = M t P t y t = m 1 + π t = 1 + π = (1 + μ)/(1 + γ) μm = (1 x)(1 τ) ω 59
Fiscal Considerations Household Behavior: There are 2 First Order and 1 Envelope Conditions r φ m ω m ω φ m ω = φ m ω φ m ω m ω φ m ω φ x 1 τ = ω φ m ω φ (x ) 60
Fiscal Considerations Market Clearing & Budget Constraint: There are 2 relations because of Market Clearing and Budget Constraint: c t + G t = y t 1 x t ω + g + x = 1 μm = 1 τ 1 x ω 61
Fiscal Considerations Solving the Model: There are 4 unknown variables: ω, x, τ, m The policy rule of g and μ is given, so r ρ + ςγ + μ γ = δ + μ For any given policy rules, μ, g, the four equations can be solved for ω, x, τ, m, as a function of r, g. 62
Fiscal Considerations Functional form of φ, φ φ m = 1/(1 + 1/(km)), k is constant. m(r) must take the form of m r = A/ r. k can be solved as a function of A φ x = x β, β is a constant. 63
Fiscal Considerations Welfare Cost: The percentage income compensation needed to leave the household indifferent between r and 0 U 1 + w r c r, m r, x(r) = U,c δ, m δ, x(δ)- 64
Fiscal Considerations Welfare Cost δ = 0.02 1 g = 0.35 β 1 = 0.0001 β 2 = 0.3 β 3 = 0.6 β 4 = 0.9 65
Fiscal Considerations Results: Based on this theoretical model, Curves m(r) and w(r) are tracing out Steady States of Deterministic economies in balanced growth path, Subjected to different constant rates of money growth and different constant size of government. 66
Fiscal Considerations Results: Deviation of Optimal r from 0 is positive for β > 0 but it is too small (0.1%) Friedman Rule needs qualification but with small magnitude! 67
Fiscal Considerations Results: Difference in the welfare cost with respect to the last models is small (0.1%): Labor-Leisure Trade off is not important! Fiscal Considerations is not important! 68
Fiscal Considerations Results: 69
Fiscal Considerations Critiques: Why real balances should increase the utility? What people do exactly with their money holdings? Holding money does not increase utility itself It is the ease of transaction, and the less time devoted to it that makes someone better off. 70
The McCallum-Goodfriend Framework Welfare cost is obtained based on a theory of general equilibrium (McCallum & Goodfriend 1987). Transaction behavior is directly interred to the model as labor-transaction trade-off. There is no labor-leisure trade-off and Fiscal Policies is not entered explicitly. Another theoretical justification of welfare cost formula in step 2 is provided. 71
The McCallum-Goodfriend Framework Representative Household: Devote the fraction s of its time in each period to carry out transactions. Supplies 1 s unit of Labor in each period with productivity y t = y t 1 (1 + γ) Gains utility in each period, from the consumption of one nondurable good: c 72
The McCallum-Goodfriend Framework Representative Household: U c t = 1 1 σ c t 1 σ c t = z t f(s t ) f is the transaction technology : f > 0 and f 0 = 0 73
The McCallum-Goodfriend Framework Representative Household: Maximize the total utility over his lifetime: V = t=0 1/ 1 + ρ t U(c t ) Subjected to his constraint by choosing c t, s t. 74
The McCallum-Goodfriend Framework Monetary and Fiscal Policy: M t = 1 + μ M t 1 Government take the lump sum tax H t in each period 75
The McCallum-Goodfriend Framework Household Budget Constraint : t M t+1 = M t H t + P t (1 s t )y t P t c t m t z t y t = M t P t y t, ω t c t y t, 1 + π t = P t P t 1, v t = H t /y t 1 + γ 1 + π t+1 m t+1 = m t v t + (1 s t ) ω t 76
The McCallum-Goodfriend Framework Household Transaction Constraint: c t = z t f(s t ) ω t = m t f(s t ) 77
The McCallum-Goodfriend Framework Household Behavior: Household begins in period 1 with balance M and productivity y. t=0 V = V y, z = max 1/ 1 + ρ t U(c t ) c 1, c 2, s 1, s 2, 78
The McCallum-Goodfriend Framework Household Behavior: V y, m = y 1 σ v(m) 1 v(m) = max * 1 ς ω1 σ + ω, s 1 + γ 1 σ 1 + ρ v(m )+ m = m v+1 s ω (1+π)(1+γ), ω = mf(s) 79
The McCallum-Goodfriend Framework Balanced Growth Path: 1 + γ 1 + π t+1 m t+1 = m t v t + 1 s t ω t ω t = c t y t = ω v t = v m t = M t P t y t = m 1 + π t = 1 + π = (1 + μ)/(1 + γ) μm = v + 1 ω s 80
The McCallum-Goodfriend Framework Household Behavior: From The F.O.C and envelope condition: f s = rmf (s ) 81
The McCallum-Goodfriend Framework Market Clearing: c t = y t 1 s t 1 s = mf(s ) 82
The McCallum-Goodfriend Framework Solving the Model: Having the functional form of f(s), m(r) and s(r) could be solved as a function of r. 83
The McCallum-Goodfriend Framework Welfare Cost: s r = 0 = 0, s r > 0 s(r), the time spent for economizing on cash use, has the dimension of a percentage reduction in consumption for each nominal interest. s r is itself a direct measure of the welfare cost of inflation 84
The McCallum-Goodfriend Framework Welfare Cost : Without having the transaction functional form, s(r) can be solved as a function of m r : s r = rm (r)(1 s r ) 1 s r + rm(r) 85
The McCallum-Goodfriend Framework Welfare Cost 86
The McCallum-Goodfriend Framework Welfare Cost : For small r, s r 1, so : s r = rm r r, s r = ψ m dm This is the same formula based on consumer surplus of money demand! 87
The McCallum-Goodfriend Framework Functional form of f: Suppose f s = ks, in which k is a constant. For small r: m(r) takes the form of m r = A/ r with A = 1/ k. s r = r/k 88
The McCallum-Goodfriend Framework Results: In U.S. Economy with A = 0.05, s 1 % for r = 4% and s 2 % for r = 16% s r is 0. > 0, so the optimal nominal interest rate 89
The McCallum-Goodfriend Framework Results: Based on this theoretical model, Curves m(r) and w(r) are tracing out Steady States of Deterministic economies, in balanced growth path and constant transactional technology, Subjected to different constant rates of money growth. 90
Conclusions and Further Directions Fixed Costs of Asset Holding: There is a fix cost of holding positive amount of interest bearing securities (Mulligan & Salai-Martin -1996. In low interest rates, fewer households would be using resources to economize on cash holdings. About 59% of American households in 1989 hold no financial assets. The estimated welfare cost for small interest rates, may be overestimated. 91
Conclusions and Further Directions M1 as a Measure of Money Holding for Transactions: In this paper, M1 is taken to be a measure of noninterest bearing cash used in transactions. Other interest bearing assets may serve as means of payment. The estimated money demand do very badly in the 1990s: M1 is too narrow an aggregate for this period. The estimated welfare cost, may be overestimated. 92
Conclusions and Further Directions The Best nominal interest rate: The estimated gain of reducing inflation is positive, starting from any interest rate above 0.1% The Optimal Monetary Policy Entails a deflation with interest rate at or near zero (Friedman Rule) 93
Conclusions and Further Directions The Cost of Inflation: Based on theoretical models, reducing Interest rate from 14% to 3% (zero inflation), would yield a benefit equivalent about 0.8% of real income. This estimate is not at all sensitive to assumptions about Fiscal Policy Used to effect the interest rate reduction Adding realistic productivity or money supply shock o The theory of these models is not adequate for estimation of costs near zero interest rates. 94
Conclusions and Further Directions 95
Questions? 96