A basic closed economy model • More formally, the building blocks of a prototype New Keynesian model are: 1 Households: consume differentiated goods, hold money, own firms, offer labor 2 Firms: produce differentiated intermediate goods (they are price setters) and ‘bundle’ final products in perfectly competetive markets; all firms demand labor, pay dividends 3 Government: make money transfer payments financed by the creation of money Centre for Central Banking Studies Modelling and Forecasting 6
A basic closed economy model • More formally, the building blocks of a prototype New Keynesian model are: 1 Households: consume differentiated goods, hold money, own firms, offer labor 2 Firms: produce differentiated intermediate goods (they are price setters) and ‘bundle’ final products in perfectly competetive markets; all firms demand labor, pay dividends 3 Government: make money transfer payments financed by the creation of money Centre for Central Banking Studies Modelling and Forecasting 6
Representative household • During each period t = 0 , 1 , 2 . . . , the representative household chooses { C t , L t , M t / P t , B t / P t } , to maximise � ∞ � �� C 1 − σ � M t � − 1 � β t η L η t E 0 1 − σ + a m ln , t P t t = 0 • Subject to the budget constraint M t − 1 + B t − 1 + T t + W t L t + D t = C t + B t / R t + M t , P t P t • Where M , B , R , and W are nominal money, nominal bonds, gross interest rates and the nominal wage respectively; D , L and C are real profits, hours worked and total consumption Centre for Central Banking Studies Modelling and Forecasting 7
Representative household • During each period t = 0 , 1 , 2 . . . , the representative household chooses { C t , L t , M t / P t , B t / P t } , to maximise � ∞ � �� C 1 − σ � M t � − 1 � β t η L η t E 0 1 − σ + a m ln , t P t t = 0 • Subject to the budget constraint M t − 1 + B t − 1 + T t + W t L t + D t = C t + B t / R t + M t , P t P t • Where M , B , R , and W are nominal money, nominal bonds, gross interest rates and the nominal wage respectively; D , L and C are real profits, hours worked and total consumption Centre for Central Banking Studies Modelling and Forecasting 7
Representative household • During each period t = 0 , 1 , 2 . . . , the representative household chooses { C t , L t , M t / P t , B t / P t } , to maximise � ∞ � �� C 1 − σ � M t � − 1 � β t η L η t E 0 1 − σ + a m ln , t P t t = 0 • Subject to the budget constraint M t − 1 + B t − 1 + T t + W t L t + D t = C t + B t / R t + M t , P t P t • Where M , B , R , and W are nominal money, nominal bonds, gross interest rates and the nominal wage respectively; D , L and C are real profits, hours worked and total consumption Centre for Central Banking Studies Modelling and Forecasting 7
Representative household • To solve the problem of the representative household we can set up the Lagrangian function as: � � C 1 − σ � � η L η M t − 1 ∞ 1 − σ + a m ln t t P t � β t � � M t − 1 + B t − 1 + T t + W t L t + D t − C t − B t / R t + M t +Λ t t = 0 P t P t • Where Λ t is the Lagrange multiplier on the budget constraint • The representative household chooses { C t , L t , M t / P t , B t / P t } to maximize the Lagrangian Centre for Central Banking Studies Modelling and Forecasting 8
Representative household • To solve the problem of the representative household we can set up the Lagrangian function as: � � C 1 − σ � � η L η M t − 1 ∞ 1 − σ + a m ln t t P t � β t � � M t − 1 + B t − 1 + T t + W t L t + D t − C t − B t / R t + M t +Λ t t = 0 P t P t • Where Λ t is the Lagrange multiplier on the budget constraint • The representative household chooses { C t , L t , M t / P t , B t / P t } to maximize the Lagrangian Centre for Central Banking Studies Modelling and Forecasting 8
Representative household • To solve the problem of the representative household we can set up the Lagrangian function as: � � C 1 − σ � � η L η M t − 1 ∞ 1 − σ + a m ln t t P t � β t � � M t − 1 + B t − 1 + T t + W t L t + D t − C t − B t / R t + M t +Λ t t = 0 P t P t • Where Λ t is the Lagrange multiplier on the budget constraint • The representative household chooses { C t , L t , M t / P t , B t / P t } to maximize the Lagrangian Centre for Central Banking Studies Modelling and Forecasting 8
First order conditions • The first order conditions for this problem are: C − σ C t : = Λ t (1) t W t = L η − 1 L t : Λ t (2) t P t Λ t Λ t + 1 B t / P t : = β E t (3) R t P t P t + 1 � − 1 � M t = Λ t Λ t + 1 M t / P t : a m − β E t (4) P t P t P t + 1 Centre for Central Banking Studies Modelling and Forecasting 9
First order conditions • We can substitute equation (1) into (2) and (3) to write: L η − 1 W t t = C − σ P t t C − σ C − σ t + 1 t = β E t R t P t P t + 1 • These two equations represent the labor supply and the Euler equation for consumption respectively • We can substitute equations (1) and (3) into (4) to write: M t R t a m = C − σ P t R t − 1 t • This equation is the LM equation Centre for Central Banking Studies Modelling and Forecasting 10
First order conditions • We can substitute equation (1) into (2) and (3) to write: L η − 1 W t t = C − σ P t t C − σ C − σ t + 1 t = β E t R t P t P t + 1 • These two equations represent the labor supply and the Euler equation for consumption respectively • We can substitute equations (1) and (3) into (4) to write: M t R t a m = C − σ P t R t − 1 t • This equation is the LM equation Centre for Central Banking Studies Modelling and Forecasting 10
First order conditions • We can substitute equation (1) into (2) and (3) to write: L η − 1 W t t = C − σ P t t C − σ C − σ t + 1 t = β E t R t P t P t + 1 • These two equations represent the labor supply and the Euler equation for consumption respectively • We can substitute equations (1) and (3) into (4) to write: M t R t a m = C − σ P t R t − 1 t • This equation is the LM equation Centre for Central Banking Studies Modelling and Forecasting 10
First order conditions • We can substitute equation (1) into (2) and (3) to write: L η − 1 W t t = C − σ P t t C − σ C − σ t + 1 t = β E t R t P t P t + 1 • These two equations represent the labor supply and the Euler equation for consumption respectively • We can substitute equations (1) and (3) into (4) to write: M t R t a m = C − σ P t R t − 1 t • This equation is the LM equation Centre for Central Banking Studies Modelling and Forecasting 10
Basic intuition • FOC on L t : Captures the consumption/leisure “trade-off” • The marginal rate of substitution between consumption and leisure must be equal to the real wage • FOC on B t / P t : Euler equation • The marginal utility of today’s consumption must be equal to the marginal utility of tomorrow’s consumption in present discounted terms • FOC on M t / P t : Money demand • At the margin, the cost (measured in terms of utility) of not consuming today must be equal to the benefit of holding another unit of money. Another unit of money yields utility and also allows future consumption Centre for Central Banking Studies Modelling and Forecasting 11
Basic intuition • FOC on L t : Captures the consumption/leisure “trade-off” • The marginal rate of substitution between consumption and leisure must be equal to the real wage • FOC on B t / P t : Euler equation • The marginal utility of today’s consumption must be equal to the marginal utility of tomorrow’s consumption in present discounted terms • FOC on M t / P t : Money demand • At the margin, the cost (measured in terms of utility) of not consuming today must be equal to the benefit of holding another unit of money. Another unit of money yields utility and also allows future consumption Centre for Central Banking Studies Modelling and Forecasting 11
Basic intuition • FOC on L t : Captures the consumption/leisure “trade-off” • The marginal rate of substitution between consumption and leisure must be equal to the real wage • FOC on B t / P t : Euler equation • The marginal utility of today’s consumption must be equal to the marginal utility of tomorrow’s consumption in present discounted terms • FOC on M t / P t : Money demand • At the margin, the cost (measured in terms of utility) of not consuming today must be equal to the benefit of holding another unit of money. Another unit of money yields utility and also allows future consumption Centre for Central Banking Studies Modelling and Forecasting 11
Basic intuition • FOC on L t : Captures the consumption/leisure “trade-off” • The marginal rate of substitution between consumption and leisure must be equal to the real wage • FOC on B t / P t : Euler equation • The marginal utility of today’s consumption must be equal to the marginal utility of tomorrow’s consumption in present discounted terms • FOC on M t / P t : Money demand • At the margin, the cost (measured in terms of utility) of not consuming today must be equal to the benefit of holding another unit of money. Another unit of money yields utility and also allows future consumption Centre for Central Banking Studies Modelling and Forecasting 11
Basic intuition • FOC on L t : Captures the consumption/leisure “trade-off” • The marginal rate of substitution between consumption and leisure must be equal to the real wage • FOC on B t / P t : Euler equation • The marginal utility of today’s consumption must be equal to the marginal utility of tomorrow’s consumption in present discounted terms • FOC on M t / P t : Money demand • At the margin, the cost (measured in terms of utility) of not consuming today must be equal to the benefit of holding another unit of money. Another unit of money yields utility and also allows future consumption Centre for Central Banking Studies Modelling and Forecasting 11
Basic intuition • FOC on L t : Captures the consumption/leisure “trade-off” • The marginal rate of substitution between consumption and leisure must be equal to the real wage • FOC on B t / P t : Euler equation • The marginal utility of today’s consumption must be equal to the marginal utility of tomorrow’s consumption in present discounted terms • FOC on M t / P t : Money demand • At the margin, the cost (measured in terms of utility) of not consuming today must be equal to the benefit of holding another unit of money. Another unit of money yields utility and also allows future consumption Centre for Central Banking Studies Modelling and Forecasting 11
Representative final goods producer • The representative final goods producing firm uses Y t ( i ) units of each intermediate good i to produce Y t according to the CES technology ε �� 1 � ε − 1 ε − 1 ε di Y t ( i ) = Y t 0 • It sells at the nominal price P t in order to maximize its profits subject to the CES technology: hence, the problem of the representative goods producing firm is to choose Y t ( i ) to maximize ε �� 1 � 1 � ε − 1 ε − 1 ε di max Y t ( i ) P t Y t ( i ) − P t ( i ) Y t ( i ) di 0 0 Centre for Central Banking Studies Modelling and Forecasting 12
Representative final goods producer • The representative final goods producing firm uses Y t ( i ) units of each intermediate good i to produce Y t according to the CES technology ε �� 1 � ε − 1 ε − 1 ε di Y t ( i ) = Y t 0 • It sells at the nominal price P t in order to maximize its profits subject to the CES technology: hence, the problem of the representative goods producing firm is to choose Y t ( i ) to maximize ε �� 1 � 1 � ε − 1 ε − 1 ε di max Y t ( i ) P t Y t ( i ) − P t ( i ) Y t ( i ) di 0 0 Centre for Central Banking Studies Modelling and Forecasting 12
Representative final goods producer • The FOC is Y t ( i ) = ( P t ( i ) / P t ) − ε Y t • This represents the demand for intermediate goods that the representative final goods producing firm has: The parameter ε is the elasticity of demand • If ↑ P t ( i ) then ↓ Y t ( i ) : Also, as ε → ∞ , individual goods become closer substitutes and individual firms have less power • If we substitute this last expression into the CES production technology, after a bit of algebra, we obtain the price level 1 �� 1 � ε − 1 P t ( i ) ε − 1 di P t = 0 Centre for Central Banking Studies Modelling and Forecasting 13
Representative final goods producer • The FOC is Y t ( i ) = ( P t ( i ) / P t ) − ε Y t • This represents the demand for intermediate goods that the representative final goods producing firm has: The parameter ε is the elasticity of demand • If ↑ P t ( i ) then ↓ Y t ( i ) : Also, as ε → ∞ , individual goods become closer substitutes and individual firms have less power • If we substitute this last expression into the CES production technology, after a bit of algebra, we obtain the price level 1 �� 1 � ε − 1 P t ( i ) ε − 1 di P t = 0 Centre for Central Banking Studies Modelling and Forecasting 13
Representative final goods producer • The FOC is Y t ( i ) = ( P t ( i ) / P t ) − ε Y t • This represents the demand for intermediate goods that the representative final goods producing firm has: The parameter ε is the elasticity of demand • If ↑ P t ( i ) then ↓ Y t ( i ) : Also, as ε → ∞ , individual goods become closer substitutes and individual firms have less power • If we substitute this last expression into the CES production technology, after a bit of algebra, we obtain the price level 1 �� 1 � ε − 1 P t ( i ) ε − 1 di P t = 0 Centre for Central Banking Studies Modelling and Forecasting 13
Representative final goods producer • The FOC is Y t ( i ) = ( P t ( i ) / P t ) − ε Y t • This represents the demand for intermediate goods that the representative final goods producing firm has: The parameter ε is the elasticity of demand • If ↑ P t ( i ) then ↓ Y t ( i ) : Also, as ε → ∞ , individual goods become closer substitutes and individual firms have less power • If we substitute this last expression into the CES production technology, after a bit of algebra, we obtain the price level 1 �� 1 � ε − 1 P t ( i ) ε − 1 di P t = 0 Centre for Central Banking Studies Modelling and Forecasting 13
Representative intermediate goods producer • Demand for each intermediate firm’s good is given by (solution to the previous slide’s maximisation problem): Y t ( i ) = ( P t ( i ) / P t ) − ε Y t • Note that this is a downward-sloping demand curve, ε is the elasticity of demand • They face this given demand function for their goods and must either choose the price or the quantity produced • Since intermediate good producing firms are monopolistic competitors, they choose P t ( i ) to maximise profits subject to their demand function and a given production technology (to be defined) Centre for Central Banking Studies Modelling and Forecasting 14
Representative intermediate goods producer • Demand for each intermediate firm’s good is given by (solution to the previous slide’s maximisation problem): Y t ( i ) = ( P t ( i ) / P t ) − ε Y t • Note that this is a downward-sloping demand curve, ε is the elasticity of demand • They face this given demand function for their goods and must either choose the price or the quantity produced • Since intermediate good producing firms are monopolistic competitors, they choose P t ( i ) to maximise profits subject to their demand function and a given production technology (to be defined) Centre for Central Banking Studies Modelling and Forecasting 14
Representative intermediate goods producer • Demand for each intermediate firm’s good is given by (solution to the previous slide’s maximisation problem): Y t ( i ) = ( P t ( i ) / P t ) − ε Y t • Note that this is a downward-sloping demand curve, ε is the elasticity of demand • They face this given demand function for their goods and must either choose the price or the quantity produced • Since intermediate good producing firms are monopolistic competitors, they choose P t ( i ) to maximise profits subject to their demand function and a given production technology (to be defined) Centre for Central Banking Studies Modelling and Forecasting 14
Representative intermediate goods producer • Demand for each intermediate firm’s good is given by (solution to the previous slide’s maximisation problem): Y t ( i ) = ( P t ( i ) / P t ) − ε Y t • Note that this is a downward-sloping demand curve, ε is the elasticity of demand • They face this given demand function for their goods and must either choose the price or the quantity produced • Since intermediate good producing firms are monopolistic competitors, they choose P t ( i ) to maximise profits subject to their demand function and a given production technology (to be defined) Centre for Central Banking Studies Modelling and Forecasting 14
Representative intermediate goods producer • During each period t = 0 , 1 , 2 . . . , the representative intermediate goods producer hires L t ( i ) units of labor from the household to manufacture Y t ( i ) units of intermediate good i according to the production technology Y t ( i ) = Z t L t ( i ) • The aggregate technology shock, Z t , follows the law of motion ln ( Z t / Z ) = ρ ln ( Z t − 1 / Z ) + ξ t where 0 < ρ < 1, and ξ t ∼ N ( 0 , µ 2 ) Centre for Central Banking Studies Modelling and Forecasting 15
Representative intermediate goods producer • During each period t = 0 , 1 , 2 . . . , the representative intermediate goods producer hires L t ( i ) units of labor from the household to manufacture Y t ( i ) units of intermediate good i according to the production technology Y t ( i ) = Z t L t ( i ) • The aggregate technology shock, Z t , follows the law of motion ln ( Z t / Z ) = ρ ln ( Z t − 1 / Z ) + ξ t where 0 < ρ < 1, and ξ t ∼ N ( 0 , µ 2 ) Centre for Central Banking Studies Modelling and Forecasting 15
Representative intermediate goods producer • As in Rotemberg (1982), the representative intermediate goods producer faces a quadratic cost of adjusting its nominal price between periods � 2 φ p � P t ( i ) Π P t − 1 ( i ) − 1 Y t 2 • Hence, any change in price is costly for the firm • With this formulation both upwards and downwards changes affect the decision of the firm • This is Stiglitz’s idea of the implicit cost of changing prices Centre for Central Banking Studies Modelling and Forecasting 16
Representative intermediate goods producer • As in Rotemberg (1982), the representative intermediate goods producer faces a quadratic cost of adjusting its nominal price between periods � 2 φ p � P t ( i ) Π P t − 1 ( i ) − 1 Y t 2 • Hence, any change in price is costly for the firm • With this formulation both upwards and downwards changes affect the decision of the firm • This is Stiglitz’s idea of the implicit cost of changing prices Centre for Central Banking Studies Modelling and Forecasting 16
Representative intermediate goods producer • As in Rotemberg (1982), the representative intermediate goods producer faces a quadratic cost of adjusting its nominal price between periods � 2 φ p � P t ( i ) Π P t − 1 ( i ) − 1 Y t 2 • Hence, any change in price is costly for the firm • With this formulation both upwards and downwards changes affect the decision of the firm • This is Stiglitz’s idea of the implicit cost of changing prices Centre for Central Banking Studies Modelling and Forecasting 16
Representative intermediate goods producer • As in Rotemberg (1982), the representative intermediate goods producer faces a quadratic cost of adjusting its nominal price between periods � 2 φ p � P t ( i ) Π P t − 1 ( i ) − 1 Y t 2 • Hence, any change in price is costly for the firm • With this formulation both upwards and downwards changes affect the decision of the firm • This is Stiglitz’s idea of the implicit cost of changing prices Centre for Central Banking Studies Modelling and Forecasting 16
Representative intermediate goods producer • Hence, the problem of the intermediate goods producer is to choose P t ( i ) to maximize its total market value � � 2 � ∞ P t ( i ) Y t ( i ) − W t L t ( i ) − φ p β t Λ t � P t ( i ) � Π P t − 1 ( i ) − 1 P t Y t P t 2 t = 0 subject to the production technology Y t ( i ) = Z t L t ( i ) and the demand for intermediate goods Y t ( i ) = ( P t ( i ) / P t ) − ε Y t Centre for Central Banking Studies Modelling and Forecasting 17
Representative intermediate goods producer • The first order condition for the representative intermediate goods producer is � − ε − 1 W t � − ε � P t ( i ) � P t ( i ) 0 = ( 1 − ε ) + ε P t P t P t Z t � � P t ( i ) P t ( i ) − φ p Π P t − 1 ( i ) − 1 Π P t − 1 ( i ) � P t + 1 ( i ) � Λ t + 1 � P t + 1 ( i ) � P t Y t + 1 + βφ p E t Π P t ( i ) − 1 Λ t P t ( i ) Y t Π P t ( i ) • This is the (non-approximated version of the) New Keynesian Phillips curve Centre for Central Banking Studies Modelling and Forecasting 18
Representative intermediate goods producer • The first order condition for the representative intermediate goods producer is � − ε − 1 W t � − ε � P t ( i ) � P t ( i ) 0 = ( 1 − ε ) + ε P t P t P t Z t � � P t ( i ) P t ( i ) − φ p Π P t − 1 ( i ) − 1 Π P t − 1 ( i ) � P t + 1 ( i ) � Λ t + 1 � P t + 1 ( i ) � P t Y t + 1 + βφ p E t Π P t ( i ) − 1 Λ t P t ( i ) Y t Π P t ( i ) • This is the (non-approximated version of the) New Keynesian Phillips curve Centre for Central Banking Studies Modelling and Forecasting 18
Representative intermediate goods producer • To develop the intuition, suppose φ p → 0, so that there are no nominal rigidities in the economy • Then the Phillips curve collapses to � − 1 W t � P t ( i ) 0 = ( 1 − ε ) + ε P t P t Z t which can be re-written as P t ( i ) ε W t = P t ε − 1 P t Z t that is the familiar condition that states that the price is set as a markup over marginal costs Centre for Central Banking Studies Modelling and Forecasting 19
Representative intermediate goods producer • To develop the intuition, suppose φ p → 0, so that there are no nominal rigidities in the economy • Then the Phillips curve collapses to � − 1 W t � P t ( i ) 0 = ( 1 − ε ) + ε P t P t Z t which can be re-written as P t ( i ) ε W t = P t ε − 1 P t Z t that is the familiar condition that states that the price is set as a markup over marginal costs Centre for Central Banking Studies Modelling and Forecasting 19
Government, and aggregate constraint • Government budget constraint: ( M t − M t − 1 ) = T t • The economy resource constraint can be derived from the representative household, intermediate goods producer budget, and government constraints: � 2 Y t = C t + φ p � P t ( i ) Π P t − 1 ( i ) − 1 Y t 2 Centre for Central Banking Studies Modelling and Forecasting 20
Government, and aggregate constraint • Government budget constraint: ( M t − M t − 1 ) = T t • The economy resource constraint can be derived from the representative household, intermediate goods producer budget, and government constraints: � 2 Y t = C t + φ p � P t ( i ) Π P t − 1 ( i ) − 1 Y t 2 Centre for Central Banking Studies Modelling and Forecasting 20
Symmetric equilibrium • In a symmetric equilibrium, all intermediate goods producers make identical decisions, so that Y t ( i ) = Y t , L t ( i ) = L t , P t ( i ) = P t , and D t ( i ) = D t , for all t = 0 , 1 , 2 . . . . • In addition, market-clearing conditions B t = B t − 1 = 0, and M t = M t − 1 + T t must hold for all t = 0 , 1 , 2 . . . . • After imposing these conditions we can derive the equilibrium of the model Centre for Central Banking Studies Modelling and Forecasting 21
Symmetric equilibrium • In a symmetric equilibrium, all intermediate goods producers make identical decisions, so that Y t ( i ) = Y t , L t ( i ) = L t , P t ( i ) = P t , and D t ( i ) = D t , for all t = 0 , 1 , 2 . . . . • In addition, market-clearing conditions B t = B t − 1 = 0, and M t = M t − 1 + T t must hold for all t = 0 , 1 , 2 . . . . • After imposing these conditions we can derive the equilibrium of the model Centre for Central Banking Studies Modelling and Forecasting 21
Symmetric equilibrium • In a symmetric equilibrium, all intermediate goods producers make identical decisions, so that Y t ( i ) = Y t , L t ( i ) = L t , P t ( i ) = P t , and D t ( i ) = D t , for all t = 0 , 1 , 2 . . . . • In addition, market-clearing conditions B t = B t − 1 = 0, and M t = M t − 1 + T t must hold for all t = 0 , 1 , 2 . . . . • After imposing these conditions we can derive the equilibrium of the model Centre for Central Banking Studies Modelling and Forecasting 21
Symmetric equilibrium • The model describes the behaviour of 8 variables { C t , W t , M t / P t , Z t , L t , Π t , R t , Y t } by C − σ C − σ 1. IS curve R t P t = β E t t t + 1 P t + 1 L η − 1 W t 2. Labor supply P t = t C − σ t M t R t a m 3. LM curve P t = R t − 1 C − σ t 4. Taylor rule ln ( R t / R ) = θ y ln ( Y t / Y ) + θ π ln (Π t / Π) + ǫ t Centre for Central Banking Studies Modelling and Forecasting 22
Symmetric equilibrium • The model describes the behaviour of 8 variables { C t , W t , M t / P t , Z t , L t , Π t , R t , Y t } by C − σ C − σ 1. IS curve R t P t = β E t t t + 1 P t + 1 L η − 1 W t 2. Labor supply P t = t C − σ t M t R t a m 3. LM curve P t = R t − 1 C − σ t 4. Taylor rule ln ( R t / R ) = θ y ln ( Y t / Y ) + θ π ln (Π t / Π) + ǫ t Centre for Central Banking Studies Modelling and Forecasting 22
Symmetric equilibrium • The model describes the behaviour of 8 variables { C t , W t , M t / P t , Z t , L t , Π t , R t , Y t } by C − σ C − σ 1. IS curve R t P t = β E t t t + 1 P t + 1 L η − 1 W t 2. Labor supply P t = t C − σ t M t R t a m 3. LM curve P t = R t − 1 C − σ t 4. Taylor rule ln ( R t / R ) = θ y ln ( Y t / Y ) + θ π ln (Π t / Π) + ǫ t Centre for Central Banking Studies Modelling and Forecasting 22
Symmetric equilibrium • The model describes the behaviour of 8 variables { C t , W t , M t / P t , Z t , L t , Π t , R t , Y t } by C − σ C − σ 1. IS curve R t P t = β E t t t + 1 P t + 1 L η − 1 W t 2. Labor supply P t = t C − σ t M t R t a m 3. LM curve P t = R t − 1 C − σ t 4. Taylor rule ln ( R t / R ) = θ y ln ( Y t / Y ) + θ π ln (Π t / Π) + ǫ t Centre for Central Banking Studies Modelling and Forecasting 22
Symmetric equilibrium • The model describes the behaviour of 8 variables { C t , W t , M t / P t , Z t , L t , Π t , R t , Y t } by C − σ C − σ 1. IS curve R t P t = β E t t t + 1 P t + 1 L η − 1 W t 2. Labor supply P t = t C − σ t M t R t a m 3. LM curve P t = R t − 1 C − σ t 4. Taylor rule ln ( R t / R ) = θ y ln ( Y t / Y ) + θ π ln (Π t / Π) + ǫ t Centre for Central Banking Studies Modelling and Forecasting 22
Symmetric equilibrium (ctd.) • And � Z t − 1 � � � Z t 5. Stochastic technology shock ln = ρ ln + ξ t Z Z 6. Production technology Y t = Z t L t 7. Phillips curve � Π t � Π t ( 1 − ε ) + ε W t 0 = − φ p − 1 P t Z t Π Π � Λ t + 1 � Π t + 1 � Π t + 1 Y t + 1 � + βφ p E t − 1 Λ t Y t Π Π 8. Aggregate resource constraint � 2 � φ p P t ( i ) Y t = C t + − 1 Y t 2 Π P t − 1 ( i ) Centre for Central Banking Studies Modelling and Forecasting 23
Symmetric equilibrium (ctd.) • And � Z t − 1 � � � Z t 5. Stochastic technology shock ln = ρ ln + ξ t Z Z 6. Production technology Y t = Z t L t 7. Phillips curve � Π t � Π t ( 1 − ε ) + ε W t 0 = − φ p − 1 P t Z t Π Π � Λ t + 1 � Π t + 1 � Π t + 1 Y t + 1 � + βφ p E t − 1 Λ t Y t Π Π 8. Aggregate resource constraint � 2 � φ p P t ( i ) Y t = C t + − 1 Y t 2 Π P t − 1 ( i ) Centre for Central Banking Studies Modelling and Forecasting 23
Symmetric equilibrium (ctd.) • And � Z t − 1 � � � Z t 5. Stochastic technology shock ln = ρ ln + ξ t Z Z 6. Production technology Y t = Z t L t 7. Phillips curve � Π t � Π t ( 1 − ε ) + ε W t 0 = − φ p − 1 P t Z t Π Π � Λ t + 1 � Π t + 1 � Π t + 1 Y t + 1 � + βφ p E t − 1 Λ t Y t Π Π 8. Aggregate resource constraint � 2 � φ p P t ( i ) Y t = C t + − 1 Y t 2 Π P t − 1 ( i ) Centre for Central Banking Studies Modelling and Forecasting 23
Symmetric equilibrium (ctd.) • And � Z t − 1 � � � Z t 5. Stochastic technology shock ln = ρ ln + ξ t Z Z 6. Production technology Y t = Z t L t 7. Phillips curve � Π t � Π t ( 1 − ε ) + ε W t 0 = − φ p − 1 P t Z t Π Π � Λ t + 1 � Π t + 1 � Π t + 1 Y t + 1 � + βφ p E t − 1 Λ t Y t Π Π 8. Aggregate resource constraint � 2 � φ p P t ( i ) Y t = C t + − 1 Y t 2 Π P t − 1 ( i ) Centre for Central Banking Studies Modelling and Forecasting 23
Symmetric equilibrium (ctd.) • And � Z t − 1 � � � Z t 5. Stochastic technology shock ln = ρ ln + ξ t Z Z 6. Production technology Y t = Z t L t 7. Phillips curve � Π t � Π t ( 1 − ε ) + ε W t 0 = − φ p − 1 P t Z t Π Π � Λ t + 1 � Π t + 1 � Π t + 1 Y t + 1 � + βφ p E t − 1 Λ t Y t Π Π 8. Aggregate resource constraint � 2 � φ p P t ( i ) Y t = C t + − 1 Y t 2 Π P t − 1 ( i ) Centre for Central Banking Studies Modelling and Forecasting 23
A ‘workhorse’ monetary policy rule • Many models closed using a Taylor rule � � r t = γ r t − 1 + ( 1 − γ ) θ π π t + i + θ y y t (5) • The original Taylor (1993) empirical proposal was for θ π = 1 . 5 , θ y = 0 . 5, γ = 0 (no smoothing), i = 0 (contemporaneous feedback) • Is the model microfounded if the policy rule is not? • Should it be? Centre for Central Banking Studies Modelling and Forecasting 24
A ‘workhorse’ monetary policy rule • Many models closed using a Taylor rule � � r t = γ r t − 1 + ( 1 − γ ) θ π π t + i + θ y y t (5) • The original Taylor (1993) empirical proposal was for θ π = 1 . 5 , θ y = 0 . 5, γ = 0 (no smoothing), i = 0 (contemporaneous feedback) • Is the model microfounded if the policy rule is not? • Should it be? Centre for Central Banking Studies Modelling and Forecasting 24
A ‘workhorse’ monetary policy rule • Many models closed using a Taylor rule � � r t = γ r t − 1 + ( 1 − γ ) θ π π t + i + θ y y t (5) • The original Taylor (1993) empirical proposal was for θ π = 1 . 5 , θ y = 0 . 5, γ = 0 (no smoothing), i = 0 (contemporaneous feedback) • Is the model microfounded if the policy rule is not? • Should it be? Centre for Central Banking Studies Modelling and Forecasting 24
A ‘workhorse’ monetary policy rule • Many models closed using a Taylor rule � � r t = γ r t − 1 + ( 1 − γ ) θ π π t + i + θ y y t (5) • The original Taylor (1993) empirical proposal was for θ π = 1 . 5 , θ y = 0 . 5, γ = 0 (no smoothing), i = 0 (contemporaneous feedback) • Is the model microfounded if the policy rule is not? • Should it be? Centre for Central Banking Studies Modelling and Forecasting 24
Why Taylor rules? • Become a cornerstone of modern monetary policy analysis • Simple to understand • Reflects a move to ‘models without money’: Interest rate becomes the policy instrument • The restriction that θ π > 1 should hold has become enshrined as the ‘Taylor Principle’ • Variations used to explain monetary policy across regimes • Smoothing introduced to make dynamics more realistic • Forecast inflation targeting ( i > 0) intended to reflect policy aims: To bring inflation back to target in the future • Influential empirical papers: Clarida et al. (1998); Theory in Taylor (ed.) (1999) Monetary Policy Rules Centre for Central Banking Studies Modelling and Forecasting 25
Why Taylor rules? • Become a cornerstone of modern monetary policy analysis • Simple to understand • Reflects a move to ‘models without money’: Interest rate becomes the policy instrument • The restriction that θ π > 1 should hold has become enshrined as the ‘Taylor Principle’ • Variations used to explain monetary policy across regimes • Smoothing introduced to make dynamics more realistic • Forecast inflation targeting ( i > 0) intended to reflect policy aims: To bring inflation back to target in the future • Influential empirical papers: Clarida et al. (1998); Theory in Taylor (ed.) (1999) Monetary Policy Rules Centre for Central Banking Studies Modelling and Forecasting 25
Why Taylor rules? • Become a cornerstone of modern monetary policy analysis • Simple to understand • Reflects a move to ‘models without money’: Interest rate becomes the policy instrument • The restriction that θ π > 1 should hold has become enshrined as the ‘Taylor Principle’ • Variations used to explain monetary policy across regimes • Smoothing introduced to make dynamics more realistic • Forecast inflation targeting ( i > 0) intended to reflect policy aims: To bring inflation back to target in the future • Influential empirical papers: Clarida et al. (1998); Theory in Taylor (ed.) (1999) Monetary Policy Rules Centre for Central Banking Studies Modelling and Forecasting 25
Why Taylor rules? • Become a cornerstone of modern monetary policy analysis • Simple to understand • Reflects a move to ‘models without money’: Interest rate becomes the policy instrument • The restriction that θ π > 1 should hold has become enshrined as the ‘Taylor Principle’ • Variations used to explain monetary policy across regimes • Smoothing introduced to make dynamics more realistic • Forecast inflation targeting ( i > 0) intended to reflect policy aims: To bring inflation back to target in the future • Influential empirical papers: Clarida et al. (1998); Theory in Taylor (ed.) (1999) Monetary Policy Rules Centre for Central Banking Studies Modelling and Forecasting 25
Why Taylor rules? • Become a cornerstone of modern monetary policy analysis • Simple to understand • Reflects a move to ‘models without money’: Interest rate becomes the policy instrument • The restriction that θ π > 1 should hold has become enshrined as the ‘Taylor Principle’ • Variations used to explain monetary policy across regimes • Smoothing introduced to make dynamics more realistic • Forecast inflation targeting ( i > 0) intended to reflect policy aims: To bring inflation back to target in the future • Influential empirical papers: Clarida et al. (1998); Theory in Taylor (ed.) (1999) Monetary Policy Rules Centre for Central Banking Studies Modelling and Forecasting 25
Why Taylor rules? • Become a cornerstone of modern monetary policy analysis • Simple to understand • Reflects a move to ‘models without money’: Interest rate becomes the policy instrument • The restriction that θ π > 1 should hold has become enshrined as the ‘Taylor Principle’ • Variations used to explain monetary policy across regimes • Smoothing introduced to make dynamics more realistic • Forecast inflation targeting ( i > 0) intended to reflect policy aims: To bring inflation back to target in the future • Influential empirical papers: Clarida et al. (1998); Theory in Taylor (ed.) (1999) Monetary Policy Rules Centre for Central Banking Studies Modelling and Forecasting 25
Why Taylor rules? • Become a cornerstone of modern monetary policy analysis • Simple to understand • Reflects a move to ‘models without money’: Interest rate becomes the policy instrument • The restriction that θ π > 1 should hold has become enshrined as the ‘Taylor Principle’ • Variations used to explain monetary policy across regimes • Smoothing introduced to make dynamics more realistic • Forecast inflation targeting ( i > 0) intended to reflect policy aims: To bring inflation back to target in the future • Influential empirical papers: Clarida et al. (1998); Theory in Taylor (ed.) (1999) Monetary Policy Rules Centre for Central Banking Studies Modelling and Forecasting 25
Why Taylor rules? • Become a cornerstone of modern monetary policy analysis • Simple to understand • Reflects a move to ‘models without money’: Interest rate becomes the policy instrument • The restriction that θ π > 1 should hold has become enshrined as the ‘Taylor Principle’ • Variations used to explain monetary policy across regimes • Smoothing introduced to make dynamics more realistic • Forecast inflation targeting ( i > 0) intended to reflect policy aims: To bring inflation back to target in the future • Influential empirical papers: Clarida et al. (1998); Theory in Taylor (ed.) (1999) Monetary Policy Rules Centre for Central Banking Studies Modelling and Forecasting 25
Why Taylor rules? • Taylor suggests the following policy rule table y t -2 0 2 π t 0 .5 1 2 2 3 4 5 4 6 7 8 6 9 10 11 8 12 13 14 • Does this imply we can all go home? Centre for Central Banking Studies Modelling and Forecasting 26
Why Taylor rules? • Taylor suggests the following policy rule table y t -2 0 2 π t 0 .5 1 2 2 3 4 5 4 6 7 8 6 9 10 11 8 12 13 14 • Does this imply we can all go home? Centre for Central Banking Studies Modelling and Forecasting 26
Why not Taylor rules? • The universality of Taylor rules doesn’t mean that they have desirable properties • Svensson (2003) (amongst many others) argues that a lot of effort has gone into microfounding models and none at all into microfounding Taylor rules • It may be that some variant of a Taylor rule is (nearly) optimal • It may be that there are many better simple alternatives • Long literature on simplicity and robustness • These use rules that don’t rely on the model but instead on ‘control principles’ Centre for Central Banking Studies Modelling and Forecasting 27
Why not Taylor rules? • The universality of Taylor rules doesn’t mean that they have desirable properties • Svensson (2003) (amongst many others) argues that a lot of effort has gone into microfounding models and none at all into microfounding Taylor rules • It may be that some variant of a Taylor rule is (nearly) optimal • It may be that there are many better simple alternatives • Long literature on simplicity and robustness • These use rules that don’t rely on the model but instead on ‘control principles’ Centre for Central Banking Studies Modelling and Forecasting 27
Why not Taylor rules? • The universality of Taylor rules doesn’t mean that they have desirable properties • Svensson (2003) (amongst many others) argues that a lot of effort has gone into microfounding models and none at all into microfounding Taylor rules • It may be that some variant of a Taylor rule is (nearly) optimal • It may be that there are many better simple alternatives • Long literature on simplicity and robustness • These use rules that don’t rely on the model but instead on ‘control principles’ Centre for Central Banking Studies Modelling and Forecasting 27
Why not Taylor rules? • The universality of Taylor rules doesn’t mean that they have desirable properties • Svensson (2003) (amongst many others) argues that a lot of effort has gone into microfounding models and none at all into microfounding Taylor rules • It may be that some variant of a Taylor rule is (nearly) optimal • It may be that there are many better simple alternatives • Long literature on simplicity and robustness • These use rules that don’t rely on the model but instead on ‘control principles’ Centre for Central Banking Studies Modelling and Forecasting 27
Why not Taylor rules? • The universality of Taylor rules doesn’t mean that they have desirable properties • Svensson (2003) (amongst many others) argues that a lot of effort has gone into microfounding models and none at all into microfounding Taylor rules • It may be that some variant of a Taylor rule is (nearly) optimal • It may be that there are many better simple alternatives • Long literature on simplicity and robustness • These use rules that don’t rely on the model but instead on ‘control principles’ Centre for Central Banking Studies Modelling and Forecasting 27
Why not Taylor rules? • The universality of Taylor rules doesn’t mean that they have desirable properties • Svensson (2003) (amongst many others) argues that a lot of effort has gone into microfounding models and none at all into microfounding Taylor rules • It may be that some variant of a Taylor rule is (nearly) optimal • It may be that there are many better simple alternatives • Long literature on simplicity and robustness • These use rules that don’t rely on the model but instead on ‘control principles’ Centre for Central Banking Studies Modelling and Forecasting 27
Welfare • What microfoundations do we have for optimal policy? • Woodford (2003) Interest and Prices: Write aggregate one-period utility � 1 v ( Y j U ( C t , L t ) = u ( Y t ; ξ t ) − t ; ξ t ) dj 0 where u ( Y t ; ξ t ) is the utility of aggregate output θ � θ − 1 �� 1 � θ − 1 θ dj � Y j and v ( Y j Y t ≡ t ; ξ t ) the disutility of 0 t supplying Y j t • Uses solution for output Y t = Y 0 + B ǫ t • Y 0 , B are coefficients potentially affected by policy Centre for Central Banking Studies Modelling and Forecasting 28
Welfare • What microfoundations do we have for optimal policy? • Woodford (2003) Interest and Prices: Write aggregate one-period utility � 1 v ( Y j U ( C t , L t ) = u ( Y t ; ξ t ) − t ; ξ t ) dj 0 where u ( Y t ; ξ t ) is the utility of aggregate output θ � θ − 1 �� 1 � θ − 1 θ dj � Y j and v ( Y j Y t ≡ t ; ξ t ) the disutility of 0 t supplying Y j t • Uses solution for output Y t = Y 0 + B ǫ t • Y 0 , B are coefficients potentially affected by policy Centre for Central Banking Studies Modelling and Forecasting 28
Welfare • What microfoundations do we have for optimal policy? • Woodford (2003) Interest and Prices: Write aggregate one-period utility � 1 v ( Y j U ( C t , L t ) = u ( Y t ; ξ t ) − t ; ξ t ) dj 0 where u ( Y t ; ξ t ) is the utility of aggregate output θ � θ − 1 �� 1 � θ − 1 θ dj � Y j and v ( Y j Y t ≡ t ; ξ t ) the disutility of 0 t supplying Y j t • Uses solution for output Y t = Y 0 + B ǫ t • Y 0 , B are coefficients potentially affected by policy Centre for Central Banking Studies Modelling and Forecasting 28
Welfare • What microfoundations do we have for optimal policy? • Woodford (2003) Interest and Prices: Write aggregate one-period utility � 1 v ( Y j U ( C t , L t ) = u ( Y t ; ξ t ) − t ; ξ t ) dj 0 where u ( Y t ; ξ t ) is the utility of aggregate output θ � θ − 1 �� 1 � θ − 1 θ dj � Y j and v ( Y j Y t ≡ t ; ξ t ) the disutility of 0 t supplying Y j t • Uses solution for output Y t = Y 0 + B ǫ t • Y 0 , B are coefficients potentially affected by policy Centre for Central Banking Studies Modelling and Forecasting 28
Recommend
More recommend
