A Monetary Intertemporal Model - University of Missourifaculty.missouri.edu/~hedlunda/teaching/2013b/11 - money_banking.pdf · A Monetary Intertemporal Model Economics 3307 - Intermediate

A Monetary Intertemporal Model

Economics 3307 - Intermediate Macroeconomics

Aaron Hedlund

Baylor University

Fall 2013

Econ 3307 (Baylor University) A Monetary Intertemporal Model Fall 2013 1 / 18

Introduction

Thus far, agents have exchanged goods directly in all of our models.

What is money, and how does it affect the economy?

Three important functions of money:I Medium of exchange: It is accepted in exchange for goods purely

because it can be traded for other goods. The advantage of money isits liquidity and transparency.

I Store of value: Like other assets, money can be used to tradecurrent goods for future goods.

I Unit of account: Almost all contracts are denominated in terms ofmoney (rather than some other good).


Measuring the Money Supply

Many forms of money: commodity money, private bank notes, etc.

The definition of money depends on drawing a line about whichassets satisfy the medium of exchange property.

Various measures of money supply in the United States:I M0, referred to as the monetary base or outside money, represents

liabilities to the Fed. M0 = currency in circulation + Fed deposits.

I M1 is a measure of the assets most widely used by the private sector inmaking transactions. M1 = currency in circulation + travelers’ checks+ demand deposits + transactions deposits.

I M2 = M1 + savings deposits + retail money market funds.


Monetary Models

Two approaches to modeling money:I “Deep models” explicitly model the frictions that lead people to use

money as a medium of exchange.F Most frictions arise from agents’ double coincidence of wants.

I “Applied models” simply assume that money is used in some or alltransactions.

For now, we assume that goods must be purchased either with cashon hand or costly transaction services (credit) provided by banks.


Nominal and Real Interest Rates

The price level– the amount of money required to purchase 1 unit ofconsumption– is P1 in period 1 and P2 in period 2.

Household portfolios include money, which pays no interest, andnominal bonds, which pay interest 1 + R in money in period 2.

The real interest rate on savings is r and is related to R by theFisher relation,

1 + r =1 + R

1 + i, where 1 + i =

P2

P1⇒ r ≈ R − i

The nominal interest rate on money is Rm = 0 and the real rate isrm = 1

1+i ≈ −i , so why do agents hold money?


Historical Interest Rate Data


A Monetary Intertemporal Model

In the money-in-utility model, utility comes from consumption Ct ,leisure lt , and period 1 real money balances, m1 ≡ M1

P1.

Households receive labor income Ptwt(h − lt), dividends Ptπt , andpay taxes PtTt .

Households save in period 1 using money M1 and bonds S1 which payM1 + (1 + R)S1 in period 2.

The government sets G1, G2, T1, T2, and Ms1 .


Households

Nominal budget constraints:

P1C1 + M1 + S1 = P1 (w1(h − l1) + π1 − T1)

P2C2 = P2 (w2(h − l2) + π2 − T2) + M1 + (1 + R)S1

Substitute out S1 to get the nominal intertemporal constraint:

P1C1 +R

1 + RM1 +

P2C2

1 + R= P1 (w1(h − l1) + π1 − T1)

+P2 (w2(h − l2) + π2 − T2)

1 + R

Divide through by P1 and recall that 1 + r = 1+R1+i = (1 + R)P1

P2to get

C1 +R

1 + Rm1 +

C2

1 + r= w1(h− l1) +π1−T1 +

w2(h − l2) + π2 − T2

1 + r


Households

Households solve

maxC1,l1,m1,C2,l2

u(C1, l1) + φ(m1) + βu(C2, l2)

subject to

C1 +R

1 + Rm1 +

C2

1 + r= w1(h− l1) +π1−T1 +

w2(h − l2) + π2 − T2

1 + r

Optimality conditions (continued on next slide):

Within-period decisions:

ul (C1,l1)uC (C1,l1)

= w1


= w2

φ′(m1)uC (C1,l1)

= R1+R


Households

Optimality conditions continued:

Intertemporal decisions:uC (C1, l1)

βuC (C2, l2)= 1 + r

C1 +R

1 + Rm1 +

C2

1 + r= w1(h− l1) +π1−T1 +

w2(h − l2) + π2 − T2

1 + r

These conditions tell us that m1 ≡ M1P1

depends negatively on R andpositively on C1 (and therefore positively on aggregate income Y1).

Thus, Md1 = P1L(Y1,R), i.e. Md

1 = P1L(Y1, r + i) ≈ P1L(Y1, r).


Firms

The firm solves

maxN1,I1,N2

P1 (z1F (K1,N1)− w1N1 − I1)

+P2 (z2F ((1− d)K1 + I1,N2)− w2N2 + (1− d)[(1− d)K1 + I1])

1 + R

Optimality conditions:

Within-period decisions:

{z1FN(K1,N1) = w1

z2FN(K2,N2) = w2

Investment decision: z2FK (K2,N2)− d =P1

P2(1 + R)− 1 = r

The firm’s decisions are independent of nominal variables!


The Government

The government’s period budget constraints are

P1G1 = P1T1 + B1 + Ms1

P2G2 + Ms1 + (1 + R)B1 = P2T2

Substituting out B1 gives the nominal intertemporal constraint

P1G1 +P2G2

1 + R= P1T1 +

R

1 + RMs

1 +P2T2

1 + R

Dividing by P1 gives the real intertemporal constraint

G1 +G2

1 + r= T1 +

R

1 + R

Ms1

P1+

T2

1 + r


Monetary Equilibrium

A competitive equilibrium is prices P1, P2, r , w1, w2; household allocationsC1, Ns

1 , Md1 , C2, Ns

2 ; firm allocations K1, Nd1 , I1, Nd

2 ; and allocations for thegovernment G1, G2, T1, T2, Ms

1 such that:

1 C1, Ns1 , Md

1 , C2, and Ns2 solve the household’s problem.

2 Nd1 , I1, and Nd

2 maximize profits V = P1π1 + P2π2

1+R , given K1.

3 The government’s budget constraint is satisfied:

G1 + G2

1+r = T1 + R1+R

Ms1

P1+ T2

1+r .

4 Labor market clearing: Nd1 = Ns

1 and Nd2 = Ns

2 .

5 Credit market clearing: Sp1 + Sg

1 = 0⇔ Sp1 = B1.

6 Goods market clearing: C1 + I1 + G1 = z1F (K1,Nd1 ) and

C2 + G2 = z2F (K2,Nd2 ) + (1− d)K2 where K2 = (1− d)K1 + I1.

7 Money market clearing: Md1 = Ms

1 .


The Classical Dichotomy

Rearrange the household constraint and substitute Md1 = Ms

1 to get

C1+C2

1 + r= w1(h− l1)+π1+

w2(h − l2) + π21 + r

−(T1 +

R

1 + R

Ms1

P1+

T2

1 + r

)

Substitute in the government’s budget constraint to get

C1 +C2

1 + r= w1(h − l1) + π1 +

w2(h − l2) + π21 + r

−(G1 +

G2

1 + r

)

Thus, the household’s budget constraint in equilibrium looks the sameas in the real intertemporal model.


The Classical Dichotomy

We can divide the remaining optimality conditions into two groups:

Real variables:


= w1 and ul (C2,l2)uC (C2,l2)

= w2

uC (C1,l1)βuC (C2,l2)

= 1 + r

z1FN(K1,N1) = w1 and z2FN(K2,N2) = w2

z2FK (K2,N2)− d = r

Nominal variables:φ′(m1)

uC (C1, l1)=

R

1 + R

Households/firms make consumption, labor, and investment decisionsindependent of nominal variables: the classical dichotomy.


Monetary Equilibrium

w1 =

Pe

rio

d 1

Re

al W

age

N1 = Period 1 Employment

N1(w1 ;r) s

N1

w1

N1(w1) d

r

r =

Re

al In

tere

st R

ate

Y1 = Period 1 Output

Y1 (r) d

Y1

Y1(r) s

P1

P1 =

Pri

ce L

eve

l

M1 = Period 1 Money

M1 s

M1 = P1L(Y1 ,r) d

To find the monetary equilibrium, find r and Y1 from the realequilibrium as before and then solve Md

1 = Ms1 ⇔ P1L(Y1, r) = Ms

1 .

Suppose Ms1 jumps to M̃s

1 . Then P̃1L(Y1, r) = M̃s1 ⇒

P̃1P1

=M̃s

1Ms

1.

Changes in Ms1 cause changes in P1 but not in any real variables:

money is neutral.


Effects of Higher Present TFP z1

Output Supply

Higher z1 increases MPN this period.

Labor demand Nd1 increases, causing Y s

1 to shift to the right.

Output Demand

No changes in Y d1 .

Equilibrium and the Money Market

Equilibrium r must decrease and Y1 must increase.

Decreasing r and increasing Y1 cause Md1 = P1L(Y1, r) to increase,

resulting in lower equilibrium P1.


Effects of Higher Present TFP z1

w1 =

Pe

rio

d 1

Re

al W

age

r =

Re

al In

tere

st R

ate

N1 = Period 1 Employment Y1 = Period 1 Output

r

Y1 (r) d

N1(w1 ;r) s

N1 Y1

w1

N1(w1) d Y1 (r)

s

Y1(r) s ~

r ~

Y1 ~

N1(w1 ;r ) s ~

N1(w1) d ~

N1 ~

w1 ~

P1

P1 =

Pri

ce L

eve

l

M1 = Period 1 Money

M1 s

M1 = P1L(Y1 ,r) d

P1 ~

M1 = P1L(Y1 ,r ) ~ ~ ~ d

Equilibrium r decreases and Y1 increases, causing an increase inmoney demand Md

1 = P1L(Y1, r).

The price level P1 decreases to clear the money market.


Documents

A Monetary Intertemporal Model - University of Missourifaculty.missouri.edu/~hedlunda/teaching/2013b/11 - money_banking.pdf · A Monetary Intertemporal Model Economics 3307 - Intermediate