Summary

 In this chapter, we augmented the two-period economy to include money, assuming that prices are flexible. This assumption is a reasonable one when the focus is on the long run.

 Money is demanded because it provides liquidity services. The demand for money depends positively on the volume of transactions and negatively on the nominal interest rate, reflecting the opportunity cost of holding money.

 The assumption of price flexibility ensures the insulation of the real side of the economy from monetary developments. Yet the price level depends both on monetary and real factors.

 A permanent monetary expansion produces a proportional increase in the price level, today and in the future. When the monetary expansion is temporary, the impact on current prices is half-way. An anticipated monetary expansion causes the price level

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to jump today, ahead of the monetary expansion, because the opportunity cost of holding money increases.

 Real shocks impact differently on the price level depending on whether the economy is closed or open to capital flows.

 Fiscal and monetary policy are not independent in the long run. Money creation is a source of government revenues alternative to formal taxation. Since changes in money supply have fiscal implications, the Treasury must stand ready to adjust taxes and transfers whenever the central bank’ changes the path of money supply. In other words, consistency requires fiscal policy to provide an appropriate backing to monetary policy.

 The fact that money is a source of government finance implies that inflation is ultimately a fiscal phenomenon.

Appendix 1: Money demands and parity conditions in a context of uncertainty

In this appendix, we extend the basic model outlined in sections 11.2 and 11.3 to a context of uncertainty, to see how the parity conditions and the money demand might change.

Assume that output in period 1, Q₁, is known, but output in period 2 is stochastic. As for a reference, let’s consider the following process:





 

p Q

p Q Q_L

H

2 1

2

2 (a1)

In a context of uncertainty, it worth considering different types of securities, according to their exposure to inflation and exchange rate risks. In what follows, we consider a portfolio with the following assets: a domestic (government) nominal bond (D_t ) paying a nominal return i₁ , a real asset (b_t ) paying a certain real return, and foreign bonds paying a nominal return i₁^*. You can think the real bond as property, for instance, that is purchased a price P₁ and sold in period 2 at price P₂ , and paid a certain real return r₁ between period 1 and period 2 (say a rent).

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When buying foreign bonds, the household may opt to cover the exchange rate premium contracting a forward rate (F₁), or simply assume the exchange rate risk and convert the foreign bond into domestic currency at the spot exchange rate in period 2 (e₂).

Let B_t^* be the amount of foreign bonds covered by a forward exchange rate contract, and B_t^* the amount invested in foreign bonds that is uncovered. For simplicity, we assume no bond holdings in period 0, and that the household pays no taxes.

At the end of periods 1, the household’ asset position will be:

 

* *

1 1 1 1 1 1 1 1 1 1 0

DP e B B Pb M P Q C M (a2)

At the end of period 2, the household’ asset position will depend on which state of nature materializes. The flow budget constrains in the good scenario and in the bad scenario will be, respectively:

  

^*



^{* *}

^{ }  

2^H 1 1 1^P 1 1 1 1 2^H 1 2^H 1 1 1 2^H 2^H 2^H 1

M  i D  i F B e B P b r P Q C M (a3)

  

^*



^{* *}

^{ }  

2^L 1 1 1^P 1 1 1 1 2^L 1 2^L 1 1 1 2^L 2^L 2^L 1

M  i D  i F B e B P b r P Q C M (a4) The household maximizes expected utility considering the two states of nature, and the corresponding ex post inter-temporal constraints. The problem is to maximize:

   

1 ¹

 

2 ²

^{ }  

2 ²

1 2 2

1 1

1

H L

H L

H L

M M M

U u C v p u C v p u C v

P  P P

    

     

               

subject to (a3) and (a4). The easiest way to solve this problem is to solve (a2), (a3), and (a4) for C₁ , C₂^H, and C₂^L, and replace the resulting expressions in the expected utility function.

Expected utility is then maximized in respect to b₁, D₁ , B₁^*, B₁^*, M₁, M₂^H , and M₂^L. The first order conditions in respect to b₁ and D₁ deliver:

 

1 ¹

 

2

' 1 '

1

u C r E u C



  

    (a5)

 

1

 

2

1 1

2

1 '

' 1

i u C

u C P E

 P

 

  

   

(a6)

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Where we used ^{E u C} ^'

 

^{2 }^{ pu C'}

 

^{2H  }

^

^{1 p u C}

^

^'

 

^2L and

 

^{2 2}

 

^{2 2}

^{ }  

^{2 2}

' ' ^{H H} 1 ' ^{L L}

E u C P pu C P   p u C P . Solving together (a5) and (a6), we

obtain

    ^{ }

 

2 1 2

1 1

2

1 1 '

'

E u C P P

r i

E u C

 

 

  

 

 

, implying:

      ^{ }

 

2 1 2

1 1 1 2

2

' ,

1 1

'

Cov u C P P

r i E P P

E u C

  

 

        

. (a7)

Comparing to (12), this condition states that the nominal interest rate and the Fisher parity condition holds in expected terms in case the covariance between the marginal utility of consumption and the inflation rate is zero. Otherwise, there will be a risk premium on the nominal interest rate, to compensate for the inflation risk. In case the household is risk neutral, the marginal utility of consumption in period 2 will be a constant and the risk premium disappears. In that case, 1  r1



1 i E P P1

 

 1 2



.

The first order conditions in respect to B₁^* delivers:

 

1 1 1^*

 

2 1

1 2

1 '

' 1

PF i u C

u C E

e  P

 

  

   

(a8)

Combining with (a5), we obtain:



^*



¹

1 1

1

1 1 F

i i

   e (a9)

This condition is known as the Covered Interest Rate Parity condition. It states that the forward exchange rate shall obey to an arbitrage condition implying that investments in domestic and in foreign currency must pay the same. Since in this case there is no exchange rate uncertainty, equation (a9) just replicates our result in (5).

The first order conditions in respect to B₁^* is:

 

1 1^*

 

2 2 1

1 2

1 '

' 1

u C e P i

u C E

e  P

 

  

   

(a10)

Combining with (a9) and proceeding as before, we obtain:

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   

 

^{2 2 2}

1 2

2 2

' ,

'

Cov u C P e F E e

E u C P

  

 

 

   

 

(a11)

This equation reveals that the forward exchange rate will differ from the expected spot exchange rate by a risk premium, which in turn is a function of the covariance between the marginal utility of income in period 2 and the exchange rate in period 2. In case the household is risk neutral, the marginal utility of consumption in period 2 will be a constant and the risk premium disappears. In that case, F1E e

 

2 .

We can now use the first order conditions in respect to M₁ to see how the demand for money looks like in the stochastic model:

 

1 1

 

2

1

1 2

' ' '

1 M P u C

u C v E

P  P

 

 

 

     

(a12)

Combining with (a5), the money demand will be such that:

 

1 1

1 1

' '

1 _i

M i

v u C

P i

 

 

    

    (28a)

In case of a log utility, this delivers exactly equation (28). This is a very important result: although money is a nominal asset and holding money exposes the household to the inflation risk, this risk is entirely reflected (mediated by) the nominal interest rate. The nominal interest rate adjusts to the inflation rate in light with equation (a7), and by then the demand for money is impacted indirectly, but there is no direct effect of inflation or of uncertainty regarding inflation in the demand for money. By the same token, no other asset return is influencing the demand for money: changes in the foreign interest rate or in the real

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interest rate induce portfolio reallocations across bonds, but money is impacted only in case the nominal interest rate changes. Money demand depends only on transaction on an opportunity cost, and is independent from portfolio considerations⁷.

Finally, it is easy to chow that the first order conditions in respect to M₂^H , and M₂^L

imply ²

 

2

2

' '

H

H H

v M u C

P

 

 

  and ²

 

2

2

' '

L

L L

v M u C

P

 

 

  . With a log utility, the money demands functions become the same as (29), with the difference that there will be two different possible materialization depending on the state of nature.

7 A formal discussion in Lebre de Freitas, M., 2022. International Currency Substitution and the Demand for Money in the Euro Area. Economic Modelling, DOI: 10.1016/j.econmod.2022.106064].

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Review questions and exercises

Review questions

11.1. Many economists have argued that the level of prices is determined by believes. Explain this in the context of our two-period model. What is the role of the central bank in respect to these believes?

11.2. Does the classical dichotomy imply that nominal variables are unaffected by the real side of the economy?

11.3. Explain what the zero lower bound is. Does it have consequences under flexible prices?

Exercises

11.4. (Changes in money supply) Consider a two-period closed economy where output in periods 1 and 2 are given: Q₁ Q₂ 50. In this economy, the preferences of the representative agent are U lnC1ln



M P1 1



lnC2 ln



M P2 2



. The government does not consume, so the lifetime seigniorage revenues are transferred back to households in period 1. The government budget constraint is given by

   

1 1 1 2 2 1 1

T M P M P r

      .

a) From the consumer optimization problem, find out: (a1) the optimal consumption in period 1; (a2) the optimal demand for money in periods 1 and 2.

b) Find out the expression of aggregate supply and national expenditure in this economy and find out the autarky real interest rate. Represent in a graph. [A:

1r1^a 1].

c) Assume that M0 M1 M2 100.Find out: (c1) the nominal interest rate in period 1; (c2) the price levels in period 2 and in period 1; (c3) the real money supply in period 1. (c4) The lump sum transfer in period 1. Describe the money market equilibrium for period 1 in a graph. [A: P1 1; P2 2].

d) (Temporary monetary expansion) Departing from (c), assume that the Treasury decides to expand the money supply in the first period to M1 400.

Find out the implications of this policy on (d1) the nominal interest rate in period 1; (d2) the price levels in period 2 and in period 1; (d3) the real money supply in period 1. (d4) Describe the money market equilibrium for period 1.

[A: P1 1.6]

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e) (Helicopter money) Departing from (c), assume that the Treasury decides a lump sum transfer today, fully financed by a permanent monetary expansion to

1 2 400.

M M  . Find out (e1) the nominal interest rate in period 1; (e2) the price levels in period 2 and in period 1; (e3) the real money supply in period 1.

(e4) Describe the money market equilibrium for period 1 in a graph. (e5) how much will be the amount in the transfer? [A: P2 8; T1  75].

f) (Anticipated monetary expansion) Departing from (c), assume that the government transfers  T1 37.5 to the household, financing this with a monetary expansion in the future, M2. Find out: (f1) the required level of M2

; the impacts on (f2) the price levels in period 2 and in period 1; (f3) the nominal interest rate in period 1; (f4) the real money supply in period 1. (f5) Describe the money market equilibrium for period 1 in a graph. [A: P1 1.6] 11.5. (Unpleasant monetarist arithmetic) Consider a two-period closed economy

where the preferences of the representative agent are given by

   

1 1 1 2 2 2

ln ln ln ln

U  C  M P  C  M P . There is no government consumption.

Taxes are collected only in period 1. The government budget constraint is given by

   

1 1 1 2 2 1

0  T M P  M P 1r . Further assume that M₀ M₂ 450 ,

1 2 50

Q Q  . Initially, M₁ 450.

a) Find out: (a1) the expressions of aggregate supply and expenditure; (a2) the equilibrium real interest rate; (a3) the nominal interest rate in period 1; (a4) the price levels in period 2 and in period 1; (a5) the real money supply in period 1; (a6) The government budget constraint; (a7) Describe the initial equilibrium in a graph. [A: 1r1^a 1;T1 0].

b) Departing from (a), assume that the central bank wanted to decrease the current price level, and accordingly it decreased the current money supply to

1 300

M  , maintaining M2 450 Find out: (b1) the nominal interest rate in period 1; (b2) the real money supply in period 1; (b3) The required increase in taxes (b4) Describe the initial equilibrium in a graph. [A:

1 1 83.33

M P  ;T1 25].

c) Departing from (b) suppose instead that government taxes didn’t increase in period 1 . In that case, (c1) how much should M₂increase to offset the fall in the current money supply to M₁ 300? What would be the implications for:

(c2) the current and future price levels? (c3) nominal interest rate; (c4) real money supply; (c5) Describe the equilibrium in a graph. [A:

1 1 66.6

M P  ;P₂ 18].

11.6. (Temporary output expansion, closed economy) Consider a two-period closed economy where initially Q₁ 50 and Q₂ 100 . In this economy, the preferences of the representative agent are given by

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   

1 1 1 2 2 2

ln ln ln ln

U  C  M P  C  M P . The government does not consume, so any seigniorage revenue will be transferred back to the household in period 2. The government budget constraint is given by   T2 M P2 2 



1 r1



M P1 1



. Further assume that initially M₀ M₁ M₂ 100.

a) Find out: (a1) the expressions for aggregate supply and expenditure; (a2) the equilibrium real interest rate; (a3) the nominal interest rate in period 1; (a4) the price levels in period 2 and in period 1; (a5) the real money supply in period 1; (a6) The lump sum transfer in period 1. (a7) Describe the initial equilibrium in the 3-panels graph. [A: 1r1^a 2; T2 0]

b) Departing from (a), examine the implications of a temporary output expansion to Q1 80, presuming that the money supplies remain unchanged. Find out:

(b1) the new expressions for aggregate supply and expenditure; (b2) the equilibrium real interest rate; (b3) the nominal interest rate in period 1; (b4) the price levels in period 2 and in period 1; (b5) the real money supply in period 1; (b6) The lump sum transfer in period 1. (b7) Describe the change in the 3-panels graph. [A: 1r1^a 1.25; M P1 1 160]

c) Departing from (b), supposes that the central bank was instead concerned with price stability. In that case, how much will be: (c1) the nominal interest rate?

(c2) M1and M2? (c3) the real money supply? (c4) the transfer to households in period 1? (c5) Describe the adjustment in light of the 3-panel graph.

[A:M P1 1 400; T2  75]

11.7. (Temporary output expansion, open economy) Consider a two-period open economy where the preferences of the representative agent are given by

   

1 1 1 2 2 2

ln ln ln ln

U  C  M P  C  M P . Government expenditure and taxes (or transfers if negative) only take place in period 1. The government budget constraint is given by G1   T1 M P1 1 



M P2 2

 

1r1



. Further assume that initially

0 1 2 100

M M M  . Y₁ Q₁ 100, G₁ T₁ 50 and Q₂ 100. The international interest rate and prices are 1r₁^* 1 and P₁^* P₂^* 1.

a) Find out: (a1) the expressions for aggregate supply and expenditure; (a2) the current account; (a3) the nominal interest rate in period 1; (a4) the price levels in period 2 and in period 1; (a5) the real money supply in period 1; (a6) Describe the initial equilibrium in the 3-panels graph. [A:

1 25

CA   ;M P1 1 150].

b) Departing from (a), examine the implications of a temporary output expansion to Q1 200, presuming that the money supplies remain unchanged. Find out:

(b1) the new expressions for aggregate supply and expenditure; (b2) the current account; (b3) consumption in periods 1 and 2; (b4) the nominal interest rate in period 1; (b5) the price levels in period 2 and in period 1; (b5) the real

No documento 11. The two-period monetary economy under flexible prices (páginas 32-41)