Macro assignment

Macroeconomic Theory

1 Chapter 15: Monetary Policy in the Intertemporal Framework

Our dynamic framework has had no role for money, which acts as a medium of exchange, unit

of account, and store of value. We introduce money into the in�nite-period framework as a store

of value. This feature allows for us to use the model to study how the monetary policy of central

banks may in�uence real economic activity and con�ict with the classical dichotomy.

→ Introducing money requires us to consider an additional market, The Money Market. • Money Demand: The nominal quantity demanded of money demanded by households

(as well as �rms and the government).

• Money Supply: The nominal quantity of money in circulation (e�ectively determined by the banking system)

⇒ Equilibrium in the Money Market occurs when the nominal quantity of money supplied equals nominal quantity of money demanded.

→ To generate money demand for households, we will use the Money-in-the-Utility (MIU) approach, where demand for real money holdings is an explicit argument in the households'

utility function. For any given period t:

u

( ct, lt,

MDt Pt

) (1)

where MDt is the nominal money demand (so that M D t /Pt is real money demand).

1.1 MIU Model

1.1.1 Households

max {ct+s,lt+s,MDt+s}

∞ s=0

V = ∞∑ s=0

βsu

( ct+s, lt+s,

MDt+s Pt+s

) (2)

subject to the nominal period-t budget constraint:1

Ptct + At + M D t = (1 + it)At−1 + M

D t−1 + Wt(1− lt) (3)

→ The intratemporal and intertemporal conditions, along with a new `consumption-money' optimality condition, can be derived by setting up the sequential Lagrangian, and taking the

�rst-order conditions with respect to ct,ct+1, lt,At, and M D t :

L = ∞∑ s=0

{βsu ( ct+s, lt+s,

MDt+s Pt+s

) +λt+s(Pt+sct+s + At+s + M

D t+s − (1 + it+s)At+s−1

−MDt+s−1 −Wt+s(1− lt+s))}

1 recall that for any nominal variable X, the real quantity is de�ned as x = X/P

1

Writing this out for s = 0,1:

L =u ( ct, lt,

MDt Pt

) + λt

( Ptct + At + M

D t − (1 + it)At−1 −M

D t−1 −Wt(1− lt)

) + βu

( ct+1, lt+1,

MDt+1 Pt+1

) + λt+1

( Pt+1ct+1 + At+1 + M

D t+1 − (1 + it+1)At −M

D t −Wt+1(1− lt+1)

) + ...

Taking FOCs:

∂L ∂ct

= 0 −→ ∂u

∂ct + λtPt = 0 −→

∂u

∂ct

1

Pt = −λt (4)

∂L ∂ct+1

= 0 −→ β ∂u

∂ct+1 + λt+1Pt+1 = 0 −→ β

∂u

∂ct+1

1

Pt+1 = −λt+1 (5)

∂L ∂lt

= 0 −→ ∂u

∂lt + λtWt = 0 −→

∂u

∂lt

1

Wt = −λt (6)

∂L ∂At

= 0 −→ λt − (1 + it+1)λt+1 = 0 −→−λt = −(1 + it+1)λt+1 (7)

∂L ∂MDt

= 0 −→ ∂u

∂MDt

1

Pt + λt −λt+1 = 0 −→

∂u

∂MDt

1

Pt + λt = λt+1 (8)

→ Intratemporal Optimality Condition (Nominal terms): Using Equations (4) and (6):

∂u

∂ct

1

Pt = ∂u

∂lt

1

Wt

⇒ ∂u/∂lt ∂u/∂ct

= Wt Pt

(9)

→ Intertemporal Optimality Condition (Nominal terms): Using Equations (4)-(5) into (7):

−λt = −(1 + it+1)λt+1

∂u

∂ct

1

Pt = (1 + it+1)β

∂u

∂ct+1

1

Pt+1

⇒ ∂u/∂ct

β∂u/∂ct+1 =

Pt Pt+1

(1 + it+1) (10)

2

→ Consumption-Money Optimality Condition: Using Equation (7) into (8):

∂u

∂MDt

1

Pt + λt = λt+1

∂u

∂MDt

1

Pt + λt =

λt (1 + it+1)

∂u

∂MDt

1

λtPt + 1 =

1

(1 + it+1)

∂u

∂MDt

1

λtPt =

1

(1 + it+1) −1

∂u

∂MDt

1

λtPt =

1−1− it+1 (1 + it+1)

∂u

∂MDt

1

λtPt =

−it+1 (1 + it+1)

Using Equation (4) into the above:

( ∂u

∂MDt

) −Pt(∂u ∂ct

) Pt

  = −it+1(1 + it+1)

⇒ ∂u/∂MDt ∂u/∂ct

= it+1

(1 + it+1) (11)

⇒ Equation (11) is the new consumption-money optimality condition. → Intuition: The household desires to have real money holdings such that the MRS of money

for consumption is equal to the relative price of holding money to consumption, each in terms of

their opportunity cost (i.e. savings in interest bearing asset).

⇒ This optimality condition characterizes the households' demand for money: → An increase in the nominal interest increases the right-hand side of Equation (11), which implies that the rate of exchange of money for consumption increases.

→ Assuming a dominant substitution e�ect, the household responds by reducing their money holdings relative to their consumption.

→ Graphically, the household's money demand function is expressed in (Mt, it+1) space:

3