Intermediate Macroeconomics II
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ECON – 2221
Cash in Advanced Model
Sergio Ocampo Daz
Department of Economics - Western University
Outline
Context
- Dynamics and monetary economies: the role of inflation
- Where does money demand come from?
Cash in Advanced Model
- Overview of the model
- Firms
- Government
- Consumers
- Nominal to real variables
Equilibrium Analysis
- First order conditions
- Market clearing
- Steady state
Optimal level of inflation
Outline
Context
- Dynamics and monetary economies: the role of inflation
- Where does money demand come from?
Cash in Advanced Model
- Overview of the model
- Firms
- Government
- Consumers
- Nominal to real variables
Equilibrium Analysis
- First order conditions
- Market clearing
- Steady state
Optimal level of inflation
Outline
Context
- Dynamics and monetary economies: the role of inflation
- Where does money demand come from?
Cash in Advanced Model
- Overview of the model
- Firms
- Government
- Consumers
- Nominal to real variables
Equilibrium Analysis
- First order conditions
- Market clearing
- Steady state
Optimal level of inflation
Outline
Context
- Dynamics and monetary economies: the role of inflation
- Where does money demand come from?
Cash in Advanced Model
- Overview of the model
- Firms
- Government
- Consumers
- Nominal to real variables
Equilibrium Analysis
- First order conditions
- Market clearing
- Steady state
Optimal level of inflation
Context
The role of inflation
IPrevious module: What is money and how can we add it to the model?
I Moneys key feature: facilitate transactions
I Classical dichotomy: solve separately for real and nominal variables
I Money neutrality: Level of money does not matter!
I Level of money changes prices... but nothing else
IWe do not think money is neutral in the real world
I Prices are not entirely flexible
IEven if money is neutral it can still affect the dynamics of the model
I Inflation affects the value of money over time
I The level of money might not matter... but its growth rate might!
The role of inflation
IPrevious module: What is money and how can we add it to the model?
I Moneys key feature: facilitate transactions
I Classical dichotomy: solve separately for real and nominal variables
I Money neutrality: Level of money does not matter!
I Level of money changes prices... but nothing else
IWe do not think money is neutral in the real world
I Prices are not entirely flexible
IEven if money is neutral it can still affect the dynamics of the model
I Inflation affects the value of money over time
I The level of money might not matter... but its growth rate might!
The role of inflation
IPrevious module: What is money and how can we add it to the model?
I Moneys key feature: facilitate transactions
I Classical dichotomy: solve separately for real and nominal variables
I Money neutrality: Level of money does not matter!
I Level of money changes prices... but nothing else
IWe do not think money is neutral in the real world
I Prices are not entirely flexible
IEven if money is neutral it can still affect the dynamics of the model
I Inflation affects the value of money over time
I The level of money might not matter... but its growth rate might!
Back to money demand
IKey to determine effects of money:
I What do we use money for?
IPrevious module: A simple model of money demand
I Md=PL(Y,R)Demand depends on demand forrealliquidity
IThis module: Model transaction motive
I Agents keep money because they need it to make transactions
I This is called acash in advanced constraint
I Moving cash between periods makes agents vulnerable to inflation
Back to money demand
IKey to determine effects of money:
I What do we use money for?
IPrevious module: A simple model of money demand
I Md=PL(Y,R)Demand depends on demand forrealliquidity
IThis module: Model transaction motive
I Agents keep money because they need it to make transactions
I This is called acash in advanced constraint
I Moving cash between periods makes agents vulnerable to inflation
Back to money demand
IKey to determine effects of money:
I What do we use money for?
IPrevious module: A simple model of money demand
I Md=PL(Y,R)Demand depends on demand forrealliquidity
IThis module: Model transaction motive
I Agents keep money because they need it to make transactions
I This is called acash in advanced constraint
I Moving cash between periods makes agents vulnerable to inflation
Cash in Advanced (CIA) Model
Overview of the model
IThe economy works for infinitely many periods
I No longer a two period economy, instead time goes on forever
I This requires being careful with the Lagrangian
IProduction uses only labor (no capital)
I This will simplify the problem, no need to solve for investment
IGovernment prints money and makes transfers to consumers
I We simplify the model by settingG=0, no government spending
IKey:Consumers have a CIA constraint
I They need liquid assets to pay for consumption (like cash!)
Overview of the model
IThe economy works for infinitely many periods
I No longer a two period economy, instead time goes on forever
I This requires being careful with the Lagrangian
IProduction uses only labor (no capital)
I This will simplify the problem, no need to solve for investment
IGovernment prints money and makes transfers to consumers
I We simplify the model by settingG=0, no government spending
IKey:Consumers have a CIA constraint
I They need liquid assets to pay for consumption (like cash!)
Overview of the model
IThe economy works for infinitely many periods
I No longer a two period economy, instead time goes on forever
I This requires being careful with the Lagrangian
IProduction uses only labor (no capital)
I This will simplify the problem, no need to solve for investment
IGovernment prints money and makes transfers to consumers
I We simplify the model by settingG=0, no government spending
IKey:Consumers have a CIA constraint
I They need liquid assets to pay for consumption (like cash!)
Overview of the model
IThe economy works for infinitely many periods
I No longer a two period economy, instead time goes on forever
I This requires being careful with the Lagrangian
IProduction uses only labor (no capital)
I This will simplify the problem, no need to solve for investment
IGovernment prints money and makes transfers to consumers
I We simplify the model by settingG=0, no government spending
IKey:Consumers have a CIA constraint
I They need liquid assets to pay for consumption (like cash!)
Firms
IFirms produce output using only labor with technology
Yt=ztNt
the subindextgives us the time (or period) in which output is produced
IFirms take prices as given:PtandWt(both nominal prices!)
I Real wage iswt=WPtt
IFirms want to maximize profits every period
I With no capital or investment problem is static
max
Nt
PtYtWtNt
Firms: Labor demand
max
Nt
PtztNtWtNt
Solution:
IFirms marginal profit per worker is:PtztWt
I Marginal profit does not depend on the number of workers hired!
IIf marginal profit is positive the firm should hire infinite workers
IIf marginal profit is negative the firm should not hire any workers
IIf marginal profit is zero the firm is indifferent between any number of hires
Firm labor demand
Nd=
ifWt<ztPt
R+ ifWt=ztPt
0 ifWt>ztPt
Firms: Labor demand
max
Nt
PtztNtWtNt
Solution:
IFirms marginal profit per worker is:PtztWt
I Marginal profit does not depend on the number of workers hired!
IIf marginal profit is positive the firm should hire infinite workers
IIf marginal profit is negative the firm should not hire any workers
IIf marginal profit is zero the firm is indifferent between any number of hires
Firm labor demand
Nd=
ifWt<ztPt
R+ ifWt=ztPt
0 ifWt>ztPt
Firms: Labor demand
max
Nt
PtztNtWtNt
Solution:
IFirms marginal profit per worker is:PtztWt
I Marginal profit does not depend on the number of workers hired!
IIf marginal profit is positive the firm should hire infinite workers
IIf marginal profit is negative the firm should not hire any workers
IIf marginal profit is zero the firm is indifferent between any number of hires
Firm labor demand
Nd=
ifWt<ztPt
R+ ifWt=ztPt
0 ifWt>ztPt
Government
ITwo functions:
I Print money
I Make transfers/Charge taxes
IFor simplicity we will setG=0 in the slides
Governments Budget
Mts+ 1 Mts=PtTt
IThe change in money supply must be given to consumers as transfers
ITransfers are negative taxes
Government
ITwo functions:
I Print money
I Make transfers/Charge taxes
IFor simplicity we will setG=0 in the slides
Governments Budget
Mts+ 1 Mts=PtTt
IThe change in money supply must be given to consumers as transfers
ITransfers are negative taxes
Government: Monetary rule
IWe assume that the government follows a monetary rule
IMoney grows at a constant rate of
Mts+ 1
Mts
= 1 +
I If >0 the government is printing more money every period
I New money is transferred to households
I If <0 the government is actually reducing the amount of money
I Money is taxed away from households and taken out of circulation
Government: Monetary rule
IWe assume that the government follows a monetary rule
IMoney grows at a constant rate of
Mts+ 1
Mts
= 1 +
I If >0 the government is printing more money every period
I New money is transferred to households
I If <0 the government is actually reducing the amount of money
I Money is taxed away from households and taken out of circulation
Government: Monetary rule
IWe assume that the government follows a monetary rule
IMoney grows at a constant rate of
Mts+ 1
Mts
= 1 +
I If >0 the government is printing more money every period
I New money is transferred to households
I If <0 the government is actually reducing the amount of money
I Money is taxed away from households and taken out of circulation
Consumers: What do they want? What can they choose?
max
t= 0
t[U(Ct)V(Nt)]
IConsumers want to maximize the present (discounted) value of their utility
IUtility of every period is
U(Ct)V(Nt)
Period Utility
and depends on consumption and labor
IThe utility of future periods is discounted at a ratet
I Key:The utility between any two periods is discounted at a rate!
Consumers: What can they choose?
IConsumption and labor for all periods{Ct,Nt}t= 0
IAssets to transfer resources across periods:
I Cash(Mt), nominal bonds(Bt), and real bonds(Xt)
IAll bonds are zero-coupon bonds
I Buy/Sell nominal bonds at a priceqtand get paid/pay their face value next period
I Buy/Sell nominal bonds at a pricestand get paid/pay their face value next period
IConsumers are choosing for a whole series of variables fromt=0 to
I How to solve that problem? Take it one step at at time! (more on this later)
Consumers: What can they choose?
IConsumption and labor for all periods{Ct,Nt}t= 0
IAssets to transfer resources across periods:
I Cash(Mt), nominal bonds(Bt), and real bonds(Xt)
IAll bonds are zero-coupon bonds
I Buy/Sell nominal bonds at a priceqtand get paid/pay their face value next period
I Buy/Sell nominal bonds at a pricestand get paid/pay their face value next period
IConsumers are choosing for a whole series of variables fromt=0 to
I How to solve that problem? Take it one step at at time! (more on this later)
Consumers: What can they choose?
IConsumption and labor for all periods{Ct,Nt}t= 0
IAssets to transfer resources across periods:
I Cash(Mt), nominal bonds(Bt), and real bonds(Xt)
IAll bonds are zero-coupon bonds
I Buy/Sell nominal bonds at a priceqtand get paid/pay their face value next period
I Buy/Sell nominal bonds at a pricestand get paid/pay their face value next period
IConsumers are choosing for a whole series of variables fromt=0 to
I How to solve that problem? Take it one step at at time! (more on this later)
Consumers: What can they choose?
IConsumption and labor for all periods{Ct,Nt}t= 0
IAssets to transfer resources across periods:
I Cash(Mt), nominal bonds(Bt), and real bonds(Xt)
IAll bonds are zero-coupon bonds
I Buy/Sell nominal bonds at a priceqtand get paid/pay their face value next period
I Buy/Sell nominal bonds at a pricestand get paid/pay their face value next period
IConsumers are choosing for a whole series of variables fromt=0 to
I How to solve that problem? Take it one step at at time! (more on this later)
Consumers: What can they do?
Budget Constraint
PtCt+qtBt+ 1 +PtstXt+ 1 +PtTt+Mt+ 1 =Mt+Bt+PtXt+WtNt+Ptt
Consumers: What can they do?
Budget Constraint
PtCt
Consumption
+ qtBt+ 1 +PtstXt+ 1
Buy Bonds that pay in t+1
+ PtTt
Pay Taxes
+ Mt+ 1
Save Cash
=Mt+Bt+PtXt+WtNt+Ptt
Consumers: What can they do?
Budget Constraint
PtCt+qtBt+ 1 +PtstXt+ 1 +PtTt= Mt
Cash from t-1
+ Bt+PtXt
Bond Payments
+ WtNt
Labor Earnings
+Ptt
Profits
Consumers: What can they do?
Budget Constraint
PtCt+qtBt+ 1 +PtstXt+ 1 +PtTt+Mt+ 1 =Mt+Bt+PtXt+WtNt+Ptt
Cash in Advance Constraint
PtCt+qtBt+ 1 +PtstXt+ 1 +PtTt=Mt+Bt+PtXt
Consumers: What can they do?
Budget Constraint
PtCt+qtBt+ 1 +PtstXt+ 1 +PtTt+Mt+ 1 =Mt+Bt+PtXt+WtNt+Ptt
Cash in Advance Constraint
PtCt+qtBt+ 1 +PtstXt+ 1 +PtTt
Total Expenditure (Net of Taxes)
=Mt+Bt+PtXt
Cash in Hand
Consumers: What can they do?
Budget Constraint
PtCt+qtBt+ 1 +PtstXt+ 1 +PtTt+Mt+ 1 =Mt+Bt+PtXt+WtNt+Ptt
Cash in Advance Constraint
PtCt+qtBt+ 1 +PtstXt+ 1 +PtTt=Mt+Bt+PtXt
INote that bonds are like cash because you can trade them for their cash value
I We are assuming that bonds are (perfectly) liquid assets
Nominal vs Real variables
IWe have written everything in nominal terms
IBut the consumer cares about real variables!
IWe need to make variables real
I Divide all terms byMtsto make them real.
I Here real will mean relative to the amount of money there is
I This is equivalent to dividing all terms byPt(a nice exercise)
mt=Mt
Mts
Relative Money Holdings
pt=Pt
Mts
Real Price
bt=Bt
Mts
Relative Value of N. Bonds
IWhat is next?
I We need to express the constraints in real terms
Nominal vs Real variables
IWe have written everything in nominal terms
IBut the consumer cares about real variables!
IWe need to make variables real
I Divide all terms byMtsto make them real.
I Here real will mean relative to the amount of money there is
I This is equivalent to dividing all terms byPt(a nice exercise)
mt=Mt
Mts
Relative Money Holdings
pt=Pt
Mts
Real Price
bt=Bt
Mts
Relative Value of N. Bonds
IWhat is next?
I We need to express the constraints in real terms
Nominal vs Real variables
IWe have written everything in nominal terms
IBut the consumer cares about real variables!
IWe need to make variables real
I Divide all terms byMtsto make them real.
I Here real will mean relative to the amount of money there is
I This is equivalent to dividing all terms byPt(a nice exercise)
mt=Mt
Mts
Relative Money Holdings
pt=Pt
Mts
Real Price
bt=Bt
Mts
Relative Value of N. Bonds
IWhat is next?
I We need to express the constraints in real terms
Passing from nominal to real: Inflation shows up!
Budget Constraint
PtCt+qtBt+ 1 +PtstXt+ 1 +PtTt+Mt+ 1 =Mt+Bt+PtXt+WtNt+Ptt
Passing from nominal to real: Inflation shows up!
Budget Constraint
Pt
Mts
Ct+qt
Bt+ 1
Mts
+
Pt
Mts
stXt+ 1 +
Pt
Mts
Tt+
Mt+ 1
Mts
=
Mt
Mts
+
Bt
Mts
+
Pt
Mst
Xt+
Wt
Mts
Nt+
Pt
Mts
t
IDivide all terms byMtto make them real.
IHere real will mean relative to the amount of money there is
IThis is equivalent to dividing all terms byPt(a nice exercise)
Passing from nominal to real: Inflation shows up!
Budget Constraint
ptCt+qt
Bt+ 1
Mts
+ptstXt+ 1 +ptTt+
Mt+ 1
Mts
=mt+bt+ptXt+
Wt
Mts
Nt+ptt
IWe now getmt,btandwt... but what do we do withBMt+s^1
t and
Mt+ 1
Mst and
Wt
Mts?
Passing from nominal to real: Inflation shows up!
Budget Constraint
ptCt+qt
Bt+ 1
Mts
+ptstXt+ 1 +ptTt+
Mt+ 1
Mts
=mt+bt+ptXt+
Wt
Mts
Nt+ptt
IWe now getmt,btandwt... but what do we do withBMt+s^1
t and
Mt+ 1
Mst and
Wt
Mts?
Mt+ 1
Mts
=
Mts+ 1
Mts
Mt+ 1
Mts+ 1
= ( 1 +)mt+ 1
Bt+ 1
Mts
=
Mts+ 1
Mts
Bt+ 1
Mts+ 1
= ( 1 +)bt+ 1
Wt
Mst
=
Pt
Mts
Wt
Pt
=ptwt
Passing from nominal to real: Inflation shows up!
Budget Constraint
ptCt+( 1 +)qtbt+ 1 +ptstXt+ 1 +ptTt+( 1 +)mt+ 1 =mt+bt+ptXt+ptwtNt+ptt
Passing from nominal to real: Inflation shows up!
Budget Constraint
ptCt+( 1 +)qtbt+ 1 +ptstXt+ 1 +ptTt+( 1 +)mt+ 1 =mt+bt+ptXt+ptwtNt+ptt
Cash in Advance Constraint
Passing from nominal to real: Inflation shows up!
Budget Constraint
ptCt+( 1 +)qtbt+ 1 +ptstXt+ 1 +ptTt+( 1 +)mt+ 1 =mt+bt+ptXt+ptwtNt+ptt
Cash in Advance Constraint
PtCt+qtBt+ 1 +PtstXt+ 1 +PtTt=Mt+Bt+PtXt
Passing from nominal to real: Inflation shows up!
Budget Constraint
ptCt+( 1 +)qtbt+ 1 +ptstXt+ 1 +ptTt+( 1 +)mt+ 1 =mt+bt+ptXt+ptwtNt+ptt
Cash in Advance Constraint
ptCt+ ( 1 +)qtbt+ 1 +ptstXt+ 1 +ptTt=mt+bt+ptXt
Passing from nominal to real: Inflation shows up!
Budget Constraint
ptCt+( 1 +)qtbt+ 1 +ptstXt+ 1 +ptTt+( 1 +)mt+ 1 =mt+bt+ptXt+ptwtNt+ptt
Cash in Advance Constraint
ptCt+ ( 1 +)qtbt+ 1 +ptstXt+ 1 +ptTt=mt+bt+ptXt
Governments Budget Constraint
Passing from nominal to real: Inflation shows up!
Budget Constraint
ptCt+( 1 +)qtbt+ 1 +ptstXt+ 1 +ptTt+( 1 +)mt+ 1 =mt+bt+ptXt+ptwtNt+ptt
Cash in Advance Constraint
ptCt+ ( 1 +)qtbt+ 1 +ptstXt+ 1 +ptTt=mt+bt+ptXt
Governments Budget Constraint
Mts+ 1 Mts=PtTt
Passing from nominal to real: Inflation shows up!
Budget Constraint
ptCt+( 1 +)qtbt+ 1 +ptstXt+ 1 +ptTt+( 1 +)mt+ 1 =mt+bt+ptXt+ptwtNt+ptt
Cash in Advance Constraint
ptCt+ ( 1 +)qtbt+ 1 +ptstXt+ 1 +ptTt=mt+bt+ptXt
Governments Budget Constraint
=ptTt
Cash in Advanced (CIA) Model
Equilibrium
Equilibrium definition
An equilibrium is a sequence of (real) quantities
{
Ct,Nts,Ntd,Xt,Tt,mt,bt,t
}
t= 0 and
prices{pt,qt,st,wt}t= 0 such that, taking as given the governments monetary rule and
the productivity for every period{zt}t= 0 , the following conditions are satisfied:
Equilibrium definition
An equilibrium is a sequence of (real) quantities
{
Ct,Nts,Ntd,Xt,Tt,mt,bt,t
}
t= 0 and
prices{pt,qt,st,wt}t= 0 such that, taking as given the governments monetary rule and
the productivity for every period{zt}t= 0 , the following conditions are satisfied:
- Firms max{t}t= 0 choosing
{
Ntd
}
t= 0 taking{pt,wt}
t= 0 and{zt}
t= 0 as given
t=max
Nd
ptztNtdptwtNtdNd=
ifwt<zt
R+ ifwt=zt
0 ifwt>zt
2.
3.
4.
Equilibrium definition
An equilibrium is a sequence of (real) quantities
{
Ct,Nts,Ntd,Xt,Tt,mt,bt,t
}
t= 0 and
prices{pt,qt,st,wt}t= 0 such that, taking as given the governments monetary rule and
the productivity for every period{zt}t= 0 , the following conditions are satisfied:
- Firms max{t}t= 0 choosing
{
Ntd
}
t= 0 taking{pt,wt}
t= 0 and{zt}
t= 0 as given
- Consumer maximizes P.V. of utility choosing{Ct,Nts,Xt,mt,bt}t= 0 subject to the series of budget and CIA constraints taking as given prices and transfers
max
{Ct,Nts,Xt,mt,bt}t= 0
t= 0
t[U(Ct)V(Nt)]
ptCt+( 1 +)qtbt+ 1 +ptstXt+ 1 +ptTt+( 1 +)mt+ 1 =mt+bt+ptXt+ptwtNt+ptt
ptCt+( 1 +)qtbt+ 1 +ptstXt+ 1 +ptTt=mt+bt+ptXt
3.
4.
Equilibrium definition
An equilibrium is a sequence of (real) quantities
{
Ct,Nts,Ntd,Xt,Tt,mt,bt,t
}
t= 0 and
prices{pt,qt,st,wt}t= 0 such that, taking as given the governments monetary rule and
the productivity for every period{zt}t= 0 , the following conditions are satisfied:
- Firms max{t}t= 0 choosing
{
Ntd
}
t= 0 taking{pt,wt}
t= 0 and{zt}
t= 0 as given
- Consumer maximizes P.V. of utility choosing{Ct,Nts,Xt,mt,bt}t= 0 subject to the series of budget and CIA constraints taking as given prices and transfers
- Government chooses{Tt}t= 0 to balance its budget taking the rate of money growth()and prices as given =ptTt
Equilibrium definition
An equilibrium is a sequence of (real) quantities
{
Ct,Nts,Ntd,Xt,Tt,mt,bt,t
}
t= 0 and
prices{pt,qt,st,wt}t= 0 such that, taking as given the governments monetary rule and
the productivity for every period{zt}t= 0 , the following conditions are satisfied:
- Firms max{t}t= 0 choosing
{
Ntd
}
t= 0 taking{pt,wt}
t= 0 and{zt}
t= 0 as given
- Consumer maximizes P.V. of utility choosing{Ct,Nts,Xt,mt,bt}t= 0 subject to the series of budget and CIA constraints taking as given prices and transfers
- Government chooses{Tt}t= 0 to balance its budget taking the rate of money growth()and prices as given
- Markets Clear Nst=Ntd Labor Market
mt= (^1) Money Market bt=Xt= (^0) Bond Markets I Remember thatmt=MMtst so for markets to clearmt= 1 I The conditionbt=Xt=0 is a no-trade equilibrium. Why? I There is a single representative agent. Who will the agent trade with?
Labor market
IThe labor demand function is perfectly elastic atwt=zt
ISo the only possible equilibrium requireswt=zt
I Or else labor demand would be larger()or lower( 0 )than labor supply
IThis solves for the equilibrium in the labor market
IAs a consequencet=0 in equilibrium
Consumers problem
max
{Ct,Nts,Xt,mt,bt}t= 0
t= 0
t[U(Ct)V(Nt)]
ptCt+ ( 1 +)qtbt+ 1 +ptstXt+ 1 +ptTt+ ( 1 +)mt+ 1 =mt+bt+ptXt+ptztNt
ptCt+ ( 1 +)qtbt+ 1 +ptstXt+ 1 +ptTt=mt+bt+ptXt
Key:This is just like any other utility maximization problem, but with more constraints
IWe can use a Lagrangian with a multiplier for each constraint
IThere is a budget constraint in each period and a CIA constraint in each period
I twill be the multiplier of the periodtbudget constraint
I twill be the multiplier of the periodtCIA constraint
IMultipliers measure the (utility) value of resources in each period
I Because there are multiple periods we will discount the multipliers with
Consumers problem
max
{Ct,Nts,Xt,mt,bt}t= 0
t= 0
t[U(Ct)V(Nt)]
ptCt+ ( 1 +)qtbt+ 1 +ptstXt+ 1 +ptTt+ ( 1 +)mt+ 1 =mt+bt+ptXt+ptztNt
ptCt+ ( 1 +)qtbt+ 1 +ptstXt+ 1 +ptTt=mt+bt+ptXt
Key:This is just like any other utility maximization problem, but with more constraints
IWe can use a Lagrangian with a multiplier for each constraint
IThere is a budget constraint in each period and a CIA constraint in each period
I twill be the multiplier of the periodtbudget constraint
I twill be the multiplier of the periodtCIA constraint
IMultipliers measure the (utility) value of resources in each period
I Because there are multiple periods we will discount the multipliers with
Consumers problem
max
{Ct,Nts,Xt,mt,bt}t= 0
t= 0
t[U(Ct)V(Nt)]
ptCt+ ( 1 +)qtbt+ 1 +ptstXt+ 1 +ptTt+ ( 1 +)mt+ 1 =mt+bt+ptXt+ptztNt
ptCt+ ( 1 +)qtbt+ 1 +ptstXt+ 1 +ptTt=mt+bt+ptXt
Key:This is just like any other utility maximization problem, but with more constraints
IWe can use a Lagrangian with a multiplier for each constraint
IThere is a budget constraint in each period and a CIA constraint in each period
I twill be the multiplier of the periodtbudget constraint
I twill be the multiplier of the periodtCIA constraint
IMultipliers measure the (utility) value of resources in each period
I Because there are multiple periods we will discount the multipliers with
Lagrangian
L=
t= 0
t(U(Ct)V(Nt))
+
t= 0
tt
mtd+bt+ptXt
Cash on Hand
ptCtqt( 1 +)bt+ 1 ptstXt+ 1 ptTt
Expenditures
+
t= 0
tt
mdt+bt+ptXt+ptwtNts+ptt
Total Income
ptCtqt( 1 +)bt+ 1 ptstXt+ 1 ptTt( 1 +)mtd+ 1
Expenditures Plus Savings
Before taking derivatives
What is the consumer choosing in periodt?
- Ct: How much consumption to have
- Nts: How much to work (at the current wage)
- How much to accumulate of each type of asset! I bt+ 1 ,Xt+ 1 andmt+ 1
What about the value of current assetsbt,Xtandmt?
IThis value is pre-determined by the consumers past actions!
IThe decision in timetis about how much to accumulate for the future
Before taking derivatives
What is the consumer choosing in periodt?
- Ct: How much consumption to have
- Nts: How much to work (at the current wage)
- How much to accumulate of each type of asset! I bt+ 1 ,Xt+ 1 andmt+ 1
What about the value of current assetsbt,Xtandmt?
IThis value is pre-determined by the consumers past actions!
IThe decision in timetis about how much to accumulate for the future
First order conditions
tU
(Ct)t(t+t)pt= 0 [Ct]
t+^1 (t+ 1 +t+ 1 )t(t+t)qt( 1 +) = 0 [bt+ 1 ]
t+^1 (t+ 1 +t+ 1 )pt+ 1 t(t+t)ptst= 0 [Xt+ 1 ]
t+^1 (t+ 1 +t+ 1 )tt( 1 +) = 0 [mt+ 1 ]
tV
(Nts) +ttptwt= 0 [Nt]
First order conditions: Mrg Benefit = Mrg Cost
U
(Ct) = (t+t)pt [Ct]
(t+ 1 +t+ 1 ) = (t+t)qt( 1 +) [bt+ 1 ]
(t+ 1 +t+ 1 )pt+ 1 = (t+t)ptst [Xt+ 1 ]
(t+ 1 +t+ 1 ) =t( 1 +) [mt+ 1 ]
tptwt=V
(Nts) [Nt]
Interpretation of first order conditions
IWhat ist? The value of more resources in your budget constraint at timet
IWhat ist? The value of more cash in hand at timet
IThe termt+tshows up a lot!
I When you accumulate a liquid asset (money or bonds) you get more resources
tomorrow (t+ 1 ) but those resources are liquid, so you also get more cash in hand
which gives your asset an additional value oft+ 1
I So the total value of having the asset is(t+ 1 +t+ 1 )
I Theis discounting because you pay for the asset intbut get the payment int+ 1
I When you accumulate the asset you have to pay for it. Because of the CIA you have
to pay it with your current liquid assets. So the cost reflects the fact that you spend
resources, but also that those resources reduce your liquid assets.
IWorking gives you resources, but not liquid assets so onlytshows up in FOC
Interpretation of first order conditions
IWhat ist? The value of more resources in your budget constraint at timet
IWhat ist? The value of more cash in hand at timet
IThe termt+tshows up a lot!
I When you accumulate a liquid asset (money or bonds) you get more resources
tomorrow (t+ 1 ) but those resources are liquid, so you also get more cash in hand
which gives your asset an additional value oft+ 1
I So the total value of having the asset is(t+ 1 +t+ 1 )
I Theis discounting because you pay for the asset intbut get the payment int+ 1
I When you accumulate the asset you have to pay for it. Because of the CIA you have
to pay it with your current liquid assets. So the cost reflects the fact that you spend
resources, but also that those resources reduce your liquid assets.
IWorking gives you resources, but not liquid assets so onlytshows up in FOC
Interpretation of first order conditions
IWhat ist? The value of more resources in your budget constraint at timet
IWhat ist? The value of more cash in hand at timet
IThe termt+tshows up a lot!
I When you accumulate a liquid asset (money or bonds) you get more resources
tomorrow (t+ 1 ) but those resources are liquid, so you also get more cash in hand
which gives your asset an additional value oft+ 1
I So the total value of having the asset is(t+ 1 +t+ 1 )
I Theis discounting because you pay for the asset intbut get the payment int+ 1
I When you accumulate the asset you have to pay for it. Because of the CIA you have
to pay it with your current liquid assets. So the cost reflects the fact that you spend
resources, but also that those resources reduce your liquid assets.
IWorking gives you resources, but not liquid assets so onlytshows up in FOC
Interpretation of first order conditions
What about the growth of money()?
Ishows up in the FOC ofmt+ 1 andbt+ 1
I It shows up on the cost side
IWe still dont know why for sure
I But higher growth rate of money will affect the inflation rate
I If there is inflation our nominal assets will be worth less tomorrow
I That is a cost when thinking about buying them
Interpretation of first order conditions
What about the growth of money()?
Ishows up in the FOC ofmt+ 1 andbt+ 1
I It shows up on the cost side
IWe still dont know why for sure
I But higher growth rate of money will affect the inflation rate
I If there is inflation our nominal assets will be worth less tomorrow
I That is a cost when thinking about buying them
Interpretation of first order conditions
What about the growth of money()?
Ishows up in the FOC ofmt+ 1 andbt+ 1
I It shows up on the cost side
IWe still dont know why for sure
I But higher growth rate of money will affect the inflation rate
I If there is inflation our nominal assets will be worth less tomorrow
I That is a cost when thinking about buying them
Solving for equilibrium
IWe can interpret the first order conditions, but we cannot solve them
I Them solution is an infinite sequence of values!
I We only know the sequence of some variables:
mt= 1 bt= 0 Xt= 0
wt=zt = 0
IWe can go further by exa mining the constraints and FOC
Solving for equilibrium: The CIA constraint
ptCt+ ( 1 +)qtbt+ 1 +ptstXt+ 1 +ptTt=mt+bt+ptXt Original Eq.
ptCt+ptTt= 1 Eq. Values
ptCt= 1 Gov. Budget
pt=
1 +
Ct
Price Level
So now we can express the price level in terms of aggregate consumption!
IThis also gives us taxes in terms of consumption becauseptTt=
Solving for equilibrium: The budget constraint
ptCt+ptTt+ ( 1 +) = 1 +ptztNt Original Eq.+Eq.Values
ptCt+ ( 1 +) = 1 +ptztNt Gov. Budget
Ct=ztNt Consumption
Now we have consumption in terms of labor!
IWe can use the FOC to get an expression for labor
Solving for equilibrium: The FOC
U
(Ct) = (t+t)pt [Ct]
(t+ 1 +t+ 1 ) =t( 1 +) [mt+ 1 ]
tptzt=V
(Nt) [Nt]
How to link labor and consumption?
IWorking gives resources valued att(third FOC)
I But those resources cannot be consumed right away
INeed to save them as money, to spend them in the future (second FOC)
IThen that money (liquid asset) can be used to buy consumption in the future!
Solving for equilibrium: The FOC
tptzt=V
(Nt) Labor FOC
(
t+ 1 +t+ 1
1 +
)
ptzt=V
(Nt) Use resources for money
(
U
(Ct+ 1 )
1 +
)
pt
pt+ 1
zt=V
(Nt) Use money for consumption
1 +
U
(Ct+ 1 )
Ct+ 1
Ct
zt=V
(Nt) Replace prices
To finalize you would replaceCt=ztNtandCt+ 1 =zt+ 1 Nt+ 1
Solving for equilibrium
IWe were able to get an expression that determines the equilibrium
1 +
U
(Ct+ 1 )
Ct+ 1
Ct
zt=V
(Nt)
But we cannot solve it...
ITwo problems
- The equation involves variables in different periods
- We need a functional form for utility(U()andV()) IWe also dont have values for the prices{qt,st}t= 0 Solution: Focus on the steady state
Solving for equilibrium
IWe were able to get an expression that determines the equilibrium
1 +
U
(Ct+ 1 )
Ct+ 1
Ct
zt=V
(Nt)
But we cannot solve it...
ITwo problems
- The equation involves variables in different periods
- We need a functional form for utility(U()andV()) IWe also dont have values for the prices{qt,st}t= 0
Solution: Focus on the steady state
Solving for equilibrium
IWe were able to get an expression that determines the equilibrium
1 +
U
(Ct+ 1 )
Ct+ 1
Ct
zt=V
(Nt)
But we cannot solve it...
ITwo problems
- The equation involves variables in different periods
- We need a functional form for utility(U()andV()) IWe also dont have values for the prices{qt,st}t= 0 Solution: Focus on the steady state
A special type of equilibrium: Steady State
In steady state we are solving for values that do not change over time
Ct=Ct+ 1 =C Nt=Nt+ 1 =N
pt=pt+ 1 =p qt=qt+ 1 =q st=st+ 1 =s
t=t+ 1 = t=t+ 1 =
Prices
These are our original FOC:
(t+ 1 +t+ 1 ) = (t+t)qt( 1 +) [bt+ 1 ]
(t+ 1 +t+ 1 )pt+ 1 = (t+t)ptst [Xt+ 1 ]
Prices
These are the FOC in steady state:
(+) = (+)q( 1 +) [bt+ 1 ]
(+)p= (+)ps [Xt+ 1 ]
Prices
We can solve for prices:
1 +
=q Price of nominal bonds
=s Price of real bonds
IAt these prices the consumer is indifferent between accumulating any type of assets
IThese prices are in fact no-arbitrage conditions!
IWe indifference between types of assets to guarantee thatXt=bt= 0
I So that the consumer chooses to only accumulate cash
Prices
We can solve for prices:
1 +
=q Price of nominal bonds
=s Price of real bonds
What about the interest rate?
IFrom non-arbitrage between normal bonds and zero-coupon bonds (module 5):
1 +R=
1
q
1 +r=
1
s
1 +R=
1 +
1 +r=
1
Quantities
Quantities are obtained implicitly, depending only on the functional form of utility
1 +
U
(Ct+ 1 )
Ct+ 1
Ct
zt=V
(Nt) Eq. Condition
1 +
U
(C)z=V
(N) Steady State
1 +
U
(zN)z=V
(N) Market Clearing
IOnce we knowNwe also know the remaining variables:
C=zN p=
1 +
C
T=
1 +
C
Quantities
Quantities are obtained implicitly, depending only on the functional form of utility
1 +
U
(Ct+ 1 )
Ct+ 1
Ct
zt=V
(Nt) Eq. Condition
1 +
U
(C)z=V
(N) Steady State
1 +
U
(zN)z=V
(N) Market Clearing
IOnce we knowNwe also know the remaining variables:
C=zN p=
1 +
C
T=
1 +
C
Quantities – A graphical analysis
Labor (N)
Utils
Steady State Condition for Labor - CIA Model
Quantities – A graphical analysis
Labor (N)
Utils
Nss
Steady State Condition for Labor - CIA Model
What about inflation?
IWe managed to solve the whole model without talking about inflation!
IIn reality inflation is hiding in plain sight!
1 +it+ 1 =
Pt+ 1
Pt
=
Pt+ 1
Mt+ 1
Mt+ 1
Mt
Mt
Pt
=pt+ 1 ( 1 +)
1
pt
= ( 1 +)
pt+ 1
pt
= ( 1 +)
Ct
Ct+ 1
=
1 +
1 +gtC+ 1
What about inflation?
IWe managed to solve the whole model without talking about inflation!
IIn reality inflation is hiding in plain sight!
1 +it+ 1 =
Pt+ 1
Pt
=
Pt+ 1
Mt+ 1
Mt+ 1
Mt
Mt
Pt
=pt+ 1 ( 1 +)
1
pt
= ( 1 +)
pt+ 1
pt
= ( 1 +)
Ct
Ct+ 1
=
1 +
1 +gtC+ 1
Inflation, money growth and aggregate demand
1 +it+ 1 =
1 +
1 +gtC+ 1
Inflation depends on two forces
- How fast is the money supply growing I More cash in circulation (given some demand) increases prices I Less cash (negative) requires prices to go down
- How fast is the demand for liquidity growing I Consumers need cash to conduct transaction (purchase consumption) I A higher growth rate of consumption(gC), given some money supply, decreases prices because more transaction have to be made with the same money I The opposite happens if the growth of consumption is low
Inflation, money growth and aggregate demand
1 +it+ 1 =
1 +
1 +gtC+ 1
Inflation depends on two forces
- How fast is the money supply growing I More cash in circulation (given some demand) increases prices I Less cash (negative) requires prices to go down 2. How fast is the demand for liquidity growing I Consumers need cash to conduct transaction (purchase consumption) I A higher growth rate of consumption(gC), given some money supply, decreases prices because more transaction have to be made with the same money I The opposite happens if the growth of consumption is low
Inflation, money growth and aggregate demand
1 +it+ 1 =
1 +
1 +gtC+ 1
Inflation depends on two forces
- How fast is the money supply growing I More cash in circulation (given some demand) increases prices I Less cash (negative) requires prices to go down
- How fast is the demand for liquidity growing I Consumers need cash to conduct transaction (purchase consumption) I A higher growth rate of consumption(gC), given some money supply, decreases prices because more transaction have to be made with the same money I The opposite happens if the growth of consumption is low
Inflation, money growth and aggregate demand
1 +it+ 1 =
1 +
1 +gtC+ 1
What about steady state?
1 +i= 1 +
IIn steady state consumption does not grow
(
gC= 0
)
ISo the demand for liquidity is always the same
IPrices move 1-1 with the money supply!
Is money still neutral?
IIt is... We managed to solve the equilibrium level real variables(C,N)without the
level of money
IBut money does have real effects in the model... Through inflation!
Inflation, money growth and aggregate demand
1 +it+ 1 =
1 +
1 +gtC+ 1
What about steady state?
1 +i= 1 +
IIn steady state consumption does not grow
(
gC= 0
)
ISo the demand for liquidity is always the same
IPrices move 1-1 with the money supply!
Is money still neutral?
IIt is... We managed to solve the equilibrium level real variables(C,N)without the
level of money
IBut money does have real effects in the model... Through inflation!
Inflation, money growth and aggregate demand
1 +it+ 1 =
1 +
1 +gtC+ 1
What about steady state?
1 +i= 1 +
IIn steady state consumption does not grow
(
gC= 0
)
ISo the demand for liquidity is always the same
IPrices move 1-1 with the money supply!
Is money still neutral?
IIt is... We managed to solve the equilibrium level real variables(C,N)without the
level of money
IBut money does have real effects in the model... Through inflation!
The effect of inflation
Labor (N)
Utils
Steady State Condition for Labor - CIA Model
The effect of inflation
Labor (N)
Utils
Steady State Condition for Labor - CIA Model
The effect of inflation
Labor (N)
Utils
Nss Nss
Steady State Condition for Labor - CIA Model
Why is inflation bad?
IFirst recall that=iin steady state
IInflation distorts the actions of the agent, but why?
I Agents use money to transfer resources across periods
I Money loses value between periods because of inflation
I This affects the value of labor income!
I Labor income is saved as cash to be spent in the next period
I More inflation reduces the value of labor income when used in the future
I This is what distorts the agents incentives to work
Why is inflation bad?
IFirst recall that=iin steady state
IInflation distorts the actions of the agent, but why?
I Agents use money to transfer resources across periods
I Money loses value between periods because of inflation
I This affects the value of labor income!
I Labor income is saved as cash to be spent in the next period
I More inflation reduces the value of labor income when used in the future
I This is what distorts the agents incentives to work
Why is inflation bad?
IFirst recall that=iin steady state
IInflation distorts the actions of the agent, but why?
I Agents use money to transfer resources across periods
I Money loses value between periods because of inflation
I This affects the value of labor income!
I Labor income is saved as cash to be spent in the next period
I More inflation reduces the value of labor income when used in the future
I This is what distorts the agents incentives to work
Inflation as a tax: Seignorage
IThere are no labor income taxes in the model
IBut inflation works like a tax!
IThe government takes away part of your savings with inflation
I Savings come from your income, so it works like an income tax!
IMore inflation erodes more the value of the cash you hold between periods
IThis is called seignorage
Inflation as a tax: Seignorage
IThere are no labor income taxes in the model
IBut inflation works like a tax!
IThe government takes away part of your savings with inflation
I Savings come from your income, so it works like an income tax!
IMore inflation erodes more the value of the cash you hold between periods
IThis is called seignorage
Cash in Advanced (CIA) Model
Optimal Inflation Rate
Planners vs Consumers Problems
IHow to figure out the optimal monetary rule (hence the optimal inflation rate)?
IWe can look at the planners problem
I The planners problem tell us what the optimal allocation is
IThe planner is a fictional character that gets to choose all variables in the economy
I Key:The planner only constraint is feasibility! No prices or assets!
IWhile the consumer is subject to the CIA constraint the planner is only worried
about real variables!
Planners Problem
IWhat is the highest (feasible) value of utility?
max
{Ct,Nt}t= 0
t= 0
t[U(Ct)V(Nt)] s.t. Ct=ztNt
Feasability Constraint
IWe can solve this with a Lagrangian (just as before)
L=
t= 0
t[U(Ct)V(Nt)] +
t= 0
tt[ztNtCt]
or we can simply replace the constraint in the Objective function
max
{Nt}t= 0
t= 0
t[U(ztNt)V(Nt)]
Optimal inflation rate
IThe optimal allocation of consumption and labor is characterized by the FOC
ztU
(ztNt) =V
(Nt)
ICompare this to the condition from the CIA model
1 +
ztU
(ztNt) =V
(Nt)
IIf 1 +=1 the two conditions are the same!
I If the conditions are the same then{Ct,Nt}in the model are the same as for the
planner
Optimal inflation rate
1 += < 1 < 0
IOptimal to have money decrease at a rate=1. Why?
Iif=1 then the rate of return on money is
1
1 +i
=
1
1 +
=
1
= 1 +r
The rate of return of money is the same as the real interest rate!
IWhen this happens the CIA constraint does not bind (exercise: show thatt=0)
IThis result was first derived by Milton Friedman, so we call it the Friedman Rule