Inflation & Welfare 1gsme.sharif.edu/~seminars-macro/OLD/files/9.pdf · Money Demand & Consumer...

1 Inflation & Welfare

INFLATION & WELFARE

ROBERT E. LUCAS

Inflation & Welfare 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.


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.


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.


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.


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.



𝑴𝒕𝑷𝒕= 𝑳 𝒓𝒕, 𝒚𝒕 , 𝑳 𝒓, 𝒚 = 𝒎 𝒓 𝒚



𝒎 𝒓 = 𝑨𝒓−𝜼



𝒎 𝒓 = 𝑩𝒆𝒙𝒑(−𝝃𝒓)



𝑚 𝑟 = 𝐴𝑟−0.5

𝑨𝒄𝒕𝒖𝒂𝒍 𝒂𝒏𝒅 𝑷𝒓𝒆𝒅𝒊𝒄𝒕𝒆𝒅 𝑹𝒆𝒂𝒍 𝑩𝒂𝒍𝒂𝒏𝒄𝒆𝒔



Welfare Cost Definition:

• The area under the inverse demand curve

between m(r) and m(0): The Lost Surplus.

nominal interest rate

money/output 𝑟

m(𝑟)

𝑤 𝑟 = 𝜓 𝑥 𝑑𝑥𝑚(0)

𝑚(𝑟)

𝑟 = 𝜓(𝑚)

𝑤 𝑟 = 𝑚 𝑥 𝑑𝑥𝑟

0

− 𝑟𝑚(𝑟)



Welfare Function:

• 𝑚 𝑟 = 𝐴𝑟−𝜂 ∶ 𝑤 𝑟 = 𝐴𝜂

1−𝜂𝑟1−𝜂

• 𝑚 𝑟 = 𝐵𝑒−𝜉𝑟 ∶ 𝑤 𝑟 =𝐵

𝜉,1 − (1 + 𝜉𝑟)𝑒−𝜉𝑟-

𝑤 𝑟 = 𝑚 𝑥 𝑑𝑥𝑟

0

− 𝑟𝑚(𝑟)


𝑾𝒆𝒍𝒇𝒂𝒓𝒆 𝑪𝒐𝒔𝒕 𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏




𝑾𝒆𝒍𝒇𝒂𝒓𝒆 𝑪𝒐𝒔𝒕 𝒓𝒆𝒍𝒂𝒕𝒊𝒗𝒆 𝒕𝒐 𝟑% 𝒊𝒏𝒕𝒆𝒓𝒆𝒔𝒕



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.



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.



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).



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.


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.



Representative Household:

• Supplies one unit of Labor in each period with productivity 𝑦𝑡 = 𝑦𝑡−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




• 𝑈 𝑐𝑡, 𝑧𝑡 =1

1−𝜎𝑐𝜑(𝑧

𝑐)1−𝜎

There is no long run trend in the real balance

income ratio.

The constant Risk aversion is consistent with

balanced growth path.




• Maximize the total utility over his lifetime: 𝑉 = 1/ 1 + 𝜌 −𝑡𝑈(𝑐𝑡, 𝑧𝑡)

∞𝑡=0

Subjected to his constraint by choosing 𝑐𝑡, 𝑧𝑡



Household Constraint:

∀𝑡 ∶ 𝑀𝑡+1 = 𝑀𝑡 −𝐻𝑡 + 𝑃𝑡𝑦𝑡 − 𝑃𝑡𝑐𝑡

𝐻𝑡: Lump sum tax

𝑚𝑡 ≔𝑧𝑡

𝑦𝑡=𝑀𝑡

𝑃𝑡𝑦𝑡 , 𝜔𝑡 ≔

𝑐𝑡

𝑦𝑡 , 𝑣𝑡 ≔

𝐻𝑡

𝑃𝑡𝑦𝑡 , 1 + 𝜋𝑡 =

𝑃𝑡

𝑃𝑡−1

(𝟏 + 𝜸) 𝟏 + 𝝅𝒕+𝟏 𝒎𝒕+𝟏 = 𝒎𝒕 − 𝒗𝒕 + 𝟏 −𝝎𝒕



Household Behavior:

• Household begins in period 1 with balance 𝑀 and real wage 𝑦.

• 𝑉 = 𝑉 𝑦, 𝑧 = 𝑚𝑎𝑥 1/ 1 + 𝜌 −𝑡𝑈(𝑐𝑡, 𝑧𝑡)∞𝑡=0

𝑐1, 𝑐2, …



Household Behavior:

𝑉 = 𝑉 𝑦, 𝑧 = max 𝑈 𝑐1, 𝑧 + max1

1 + 𝜌

1

1 + 𝜌 −𝑡𝑈 𝑐𝑡+1, 𝑧𝑡+1

∞

𝑡=1

𝑉 = 𝑉 𝑦, 𝑧 = max *𝑈 𝑐1, 𝑧 +1

1 + 𝜌𝑉(𝑦, 𝑧′)+

𝑐1

𝑐1


𝑐2, 𝑐3, …


Household Behavior:

𝑉 = 𝑉 𝑦, 𝑧 = max *1

1 − 𝜍𝑐𝜑(𝑧

𝑐)1−𝜎

+1

1 + 𝜌𝑉(𝑦(1 + 𝛾), 𝑧′)+

𝑧′ =𝑧 − 𝑕 + 𝑦 − 𝑐

1 + 𝜋

𝑐



Household Behavior:

𝑉 = 𝑉 𝑦,𝑚 = max *𝑦1−𝜎

1 − 𝜍𝜔𝜑(𝑚

𝜔)1−𝜎

+1

1 + 𝜌𝑉(𝑦(1 + 𝛾),𝑚′)+

𝑚′ =𝑚 − 𝑣 + 1 − 𝜔

(1 + 𝜋)(1 + 𝛾)

𝜔



Household Behavior:

𝑉 𝑦,𝑚 = 𝑦1−𝜎𝑣(𝑚)

𝑣(𝑚) = max *1


𝜔)1−𝜎

+1 + 𝛾 1−𝜎

1 + 𝜌𝑣(𝑚′)+

𝑚′ =𝑚 − 𝑣 + 1 − 𝜔

(1 + 𝜋)(1 + 𝛾)

𝜔



Household Behavior:

F.O.C. :

𝜑𝑚

𝜔∗

−𝜎𝜑𝑚

𝜔∗−𝑚

𝜔∗𝜑′𝑚

𝜔∗=1

1+𝑟𝑣′ 𝑚′

1

1 + 𝑟=1 + 𝛾 −𝜎

1 + 𝜌 (1 + 𝜋)



Household Behavior:

Envelope Condition :

𝑣′ 𝑚 = 𝜑𝑚

𝜔∗

−𝜎

𝜑′𝑚

𝜔∗+1

1 + 𝑟𝑣′ 𝑚′

1

1 + 𝑟=1 + 𝛾 −𝜎

1 + 𝜌 (1 + 𝜋)



Monetary and Fiscal Policy:

• 𝑀𝑡 = 1 + 𝜇 𝑀𝑡−1

•𝐻𝑡

𝑃𝑡𝑦𝑡= 𝑣𝑡 = 𝑣



Balanced Growth Path:

(1 + 𝛾) 1 + 𝜋𝑡+1 𝑚𝑡+1 = 𝑚𝑡 − 𝑣 + 1 − 𝜔𝑡

• 𝜔𝑡 = 𝑐𝑡/𝑦𝑡 = 𝜔

• 𝑚𝑡 =𝑀𝑡

𝑃𝑡𝑦𝑡= 𝑚

• 1 + 𝜋𝑡 = 1 + 𝜋 = (1 + 𝜇)/(1 + 𝛾)

• 𝜇𝑚 = −𝑣 + 1 − 𝜔



Household Behavior:

𝜑′𝑚𝜔∗

𝜑𝑚𝜔∗−𝑚𝜔∗ 𝜑′𝑚𝜔∗= 𝑟

1

1 + 𝑟=1 + 𝛾 −𝜎

1 + 𝜌 (1 + 𝜋)



Solving the Model:

• 𝑀𝑡 & 𝑌𝑡 is known, but 𝑃t is unknown, so 𝑚.

• 𝜔∗ is unknown.

• There is only one relation for 𝑚 & 𝜔∗ from

Household maximization.



Solving the Model:

• Market clearing in each time:

𝑐𝑡∗ = 𝑦𝑡 𝜔∗ = 1



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.



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.



Real balance output ratio:

𝜑′ 𝑚(𝑟) =𝑟

1 +𝑚(𝑟)𝑟𝜑 𝑚(𝑟)

• Real balance output ratio is obtained as a function of

nominal interest rate in this micro based theory model.



Real balance output ratio :

𝜑′ 𝑚 =𝑟

1 +𝑚𝑟𝜑 𝑚

• 𝜑′ > 0 , 𝜑′′ < 0 : 𝑚′ 𝑟 < 0

• 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.



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 𝜑𝑚 𝑟

1 + 𝑤 𝑟= 𝜑,𝑚(0)-



Welfare Cost:

𝑤′ 𝑟 = −𝜓𝑚 𝑟

1 + 𝑤 𝑟𝑚′(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)



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?



Results:

If the 𝑚(𝑟) takes the form of 𝑚 𝑟 = 𝐴/ 𝑟 as is the

best estimate for U.S. data:

𝜑 𝑚 = 1 +𝐴2

𝑚

−1



Results:

Welfare Cost based on this theoretical model is

obtained as:

𝑤 𝑟 =𝐴 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.



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.



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.


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.



Fiscal Constraint: 𝑣 = −𝜇𝑚(𝑟)

𝑟 = (𝜌 + 𝛾𝜍) + (𝜇 − 𝛾)

𝛿 ≔ 𝑟 − 𝜇

𝑣 = 𝛿 − 𝑟 𝑚(𝑟)



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!



Fiscal Constraint:

• The Policy 𝑟 = 0 is not feasible.

• The Friedman Rule requires qualification.




• Gains utility in each period, from leisure share of its time : 𝑥

• Supplies 1 − 𝑥 unit of Labor in each period with productivity 𝑦𝑡 = 𝑦𝑡−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




• 𝑈 𝑐𝑡, 𝑧𝑡 =1

1−𝜎𝑐𝜑(𝑧

𝑐)∅(𝑥)

1−𝜎

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.




• Maximize the total utility over his lifetime: 𝑉 = 1/ 1 + 𝜌 −𝑡𝑈(𝑐𝑡, 𝑧𝑡, 𝑥𝑡)

∞𝑡=0

Subjected to his constraint by choosing 𝑐𝑡, 𝑧𝑡, 𝑥𝑡




• 𝑀𝑡 = 1 + 𝜇 𝑀𝑡−1

• Government purchase in each time: 𝐺𝑡 = 𝑔𝑡𝑦𝑡

• Government collect tax from household income with rate of 𝜏.



Household Constraint:

∀𝑡 ∶ 𝑀𝑡+1 = 𝑀𝑡 + 𝑃𝑡(1 − 𝜏)(1 − 𝑥𝑡)𝑦𝑡−𝑃𝑡𝑐𝑡


𝑦𝑡=𝑀𝑡


𝑐𝑡

𝑦𝑡 , 1 + 𝜋𝑡 =

𝑃𝑡

𝑃𝑡−1

1 + 𝛾 1 + 𝜋𝑡+1 𝑚𝑡+1 = 𝑚𝑡 + (1 − 𝜏)(1 − 𝑥𝑡) − 𝜔𝑡



Household Behavior:

• Household begins in period 1 with balance 𝑀 and productivity 𝑦.

• 𝑉 = 𝑉 𝑦, 𝑧 = 𝑚𝑎𝑥 1/ 1 + 𝜌 −𝑡𝑈(𝑐𝑡, 𝑧𝑡, 𝑥𝑡)∞𝑡=0

𝑐1, 𝑐2, … 𝑥1, 𝑥2, …



Household Behavior:


𝑣(𝑚) = max *1


𝜔)∅(𝑥)

1−𝜎

+1 + 𝛾 1−𝜎

1 + 𝜌𝑣(𝑚′)+

𝑚′ =𝑚 + (1 − 𝜏)(1 − 𝑥) − 𝜔

(1 + 𝜋)(1 + 𝛾)

𝜔, 𝑥



Balanced Growth Path: 1 + 𝛾 1 + 𝜋𝑡+1 𝑚𝑡+1 = 𝑚𝑡 + (1 − 𝑥𝑡)(1 − 𝜏) − 𝜔𝑡

• 𝜔𝑡 =𝑐𝑡

𝑦𝑡= 𝜔

• 𝑥𝑡 = 𝑥



• 1 + 𝜋𝑡 = 1 + 𝜋 = (1 + 𝜇)/(1 + 𝛾)

• 𝜇𝑚 = (1 − 𝑥)(1 − 𝜏) − 𝜔



Household Behavior:

• There are 2 First Order and 1 Envelope Conditions

𝑟 𝜑𝑚

𝜔∗−𝑚

𝜔∗ 𝜑′𝑚

𝜔∗= 𝜑′

𝑚

𝜔∗

𝜑𝑚

𝜔∗−𝑚

𝜔∗ 𝜑′𝑚

𝜔∗𝜙 𝑥∗ 1 − 𝜏 = 𝜔∗𝜑

𝑚

𝜔∗𝜙′(𝑥∗)



Market Clearing & Budget Constraint:

• There are 2 relations because of Market

Clearing and Budget Constraint:

𝑐𝑡 + 𝐺𝑡 = 𝑦𝑡 1 − 𝑥𝑡 ∶ 𝜔

∗ + 𝑔 + 𝑥∗ = 1 𝜇𝑚 = 1 − 𝜏 1 − 𝑥∗ −𝜔∗



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.



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 = 𝑥𝛽 , 𝛽 is a constant.



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)-




𝛿 = 0.02 1 − 𝑔 = 0.35

𝛽1 = 0.0001 𝛽2 = 0.3 𝛽3 = 0.6 𝛽4 = 0.9

Welfare Cost


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.



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!



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!



Results:



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.


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.




• 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 𝑦𝑡 = 𝑦𝑡−1(1 + 𝛾)

• Gains utility in each period, from the consumption of one nondurable good: 𝑐




• 𝑈 𝑐𝑡 =1

1−𝜎𝑐𝑡1−𝜎

• 𝑐𝑡 = 𝑧𝑡𝑓(𝑠𝑡)

f is the transaction technology : 𝑓′ > 0 and

𝑓 0 = 0




• Maximize the total utility over his lifetime: 𝑉 = 1/ 1 + 𝜌 −𝑡𝑈(𝑐𝑡)

∞𝑡=0

Subjected to his constraint by choosing 𝑐𝑡, 𝑠𝑡.




• 𝑀𝑡 = 1 + 𝜇 𝑀𝑡−1

• Government take the lump sum tax 𝐻𝑡 in each

period



Household Budget Constraint :

∀𝑡 ∶ 𝑀𝑡+1 = 𝑀𝑡 −𝐻𝑡 + 𝑃𝑡(1 − 𝑠𝑡)𝑦𝑡−𝑃𝑡𝑐𝑡


𝑦𝑡=𝑀𝑡


𝑐𝑡

𝑦𝑡 , 1 + 𝜋𝑡 =

𝑃𝑡

𝑃𝑡−1 , 𝑣𝑡 = 𝐻𝑡/𝑦𝑡

1 + 𝛾 1 + 𝜋𝑡+1 𝑚𝑡+1 = 𝑚𝑡 − 𝑣𝑡 + (1 − 𝑠𝑡) − 𝜔𝑡



Household Transaction Constraint:

𝑐𝑡 = 𝑧𝑡𝑓(𝑠𝑡)

𝜔𝑡 = 𝑚𝑡𝑓(𝑠𝑡)



Household Behavior:

• Household begins in period 1 with balance 𝑀 and productivity 𝑦.

• 𝑉 = 𝑉 𝑦, 𝑧 = 𝑚𝑎𝑥 1/ 1 + 𝜌 −𝑡𝑈(𝑐𝑡)∞𝑡=0

𝑐1, 𝑐2, … 𝑠1, 𝑠2, …



Household Behavior:


𝑣(𝑚) = max *1

1 − 𝜍𝜔1−𝜎 +

1 + 𝛾 1−𝜎

1 + 𝜌𝑣(𝑚′)+

𝑚′ =𝑚−𝑣+1−𝑠−𝜔

(1+𝜋)(1+𝛾) , 𝜔 = 𝑚𝑓(𝑠)

𝜔, 𝑠



Balanced Growth Path:

1 + 𝛾 1 + 𝜋𝑡+1 𝑚𝑡+1 = 𝑚𝑡 − 𝑣𝑡 + 1 − 𝑠𝑡 −𝜔𝑡

• 𝜔𝑡 =𝑐𝑡

𝑦𝑡= 𝜔

• 𝑣𝑡 = 𝑣



• 1 + 𝜋𝑡 = 1 + 𝜋 = (1 + 𝜇)/(1 + 𝛾)

• 𝜇𝑚 = −𝑣 + 1 − 𝜔 − 𝑠



Household Behavior:

• From The F.O.C and envelope condition:

𝑓 𝑠∗ = 𝑟𝑚𝑓′(𝑠∗)



Market Clearing:

𝑐𝑡 = 𝑦𝑡 1 − 𝑠𝑡 ∶ 1 − 𝑠

∗ = 𝑚𝑓(𝑠∗)



Solving the Model:

• Having the functional form of 𝑓(𝑠), 𝑚(𝑟) and

𝑠(𝑟) could be solved as a function of 𝑟.



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



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)




Welfare Cost


Welfare Cost :

• For small 𝑟, 𝑠 𝑟 ≪ 1, so :

𝑠′ 𝑟 = −𝑟𝑚′ 𝑟 , 𝑠 𝑟 = − 𝜓 𝑚 𝑑𝑚𝑟

This is the same formula based on consumer surplus of money demand!



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



Results:

• In U.S. Economy with 𝐴 = 0.05, 𝑠 ≅ 1 % for

𝑟 = 4% and 𝑠 ≅ 2 % for 𝑟 = 16%

• 𝑠′ 𝑟 > 0, so the optimal nominal interest rate

is 0.



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.


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.



M1 as a Measure of Money Holding for Transactions:

• In this paper, M1 is taken to be a measure of non-

interest 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.



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)



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.




Questions ?


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