8 Consumption, uncertainty, and the current account

8 Consumption, uncertainty, and the current account

Index:

8 Consumption, uncertainty, and the current account ... 1

8.1 Introduction ... 2

8.2 The benchmark (deterministic) open economy case ... 3

8.3 The stochastic Euler equation ... 4

8.4 The random walk theory ... 6

8.4.1 Smoothing ex ante... 6

8.4.2 No smoothing “ex post” ... 7

8.4.3 An illustrative example ... 8

8.4.4 Does openness cause a smoother consumption? ... 9

8.5 The case with precautionary savings ... 10

8.5.1 Open economy ... 10

8.5.2 The logarithmic utility function ... 11

8.5.3 An illustrative example ... 11

8.5.4 The closed economy case ... 13

8.5.5 The Great Moderation, the crisis, and the Current Account ... 15

8.6 Portfolio decisions ... 15

8.6.1 The Capital Asset Pricing Model (CAPM) ... 15

8.6.2 The optimal demand for the risky asset (Merton problem) ... 17

8.7 Main ideas ... 17

Further reading ... 18

Appendix: Risk-aversion versus prudence ... 18

8.8 Review questions and exercises ... 21

Review questions ... 21

Problems ... 21

8.1 Introduction

When people become more uncertain about the future, a possible reaction is to save more, as a matter of prudence. Some economist have argued that one of the reasons why the world interest rates have declined towards zero in recent years is that precautionary savings have increased. Since the world as a whole cannot be a net saver, this exerts a downward pressure on real interest rates.

The idea of precautionary savings fits well in the Keynesian doctrine of “animal spirits”. But is this reasoning supported at the theoretical level? Why should increased uncertainty come along with a higher saving rate? Is this a general case? Or it depends?

In this handout we reassess the 2-period model of consumption, extending it to the case of uncertainty. So far, we have assumed that consumers know their future incomes before decisions regarding consumption in period 1 are taken. Now, we assume that consumption decisions regarding the current period are taken before future income is known.

Our question is whether consumption smoothing will still hold as a general rule.

The note is organized as follows. In Section 2 we consider a benchmark deterministic case where the interest rate is equal to the rate of time preference and consequently optimal consumption is constant over time. In Section 3 we solve the consumer maximization problem in a context of uncertainty, obtaining a stochastic version of the Euler equation.

Then, we consider two particular cases for consumer preferences: in Section 4 we analyse the case in which the marginal utility of consumption is linear. In this case, consumption smoothing holds ex ante: increased uncertainty reduces the household welfare, but it does not impact on savings. In Section 5 we turn to the more interesting case, featured by the most popular utility functions: the case in which the marginal utility of consumption declines at decreasing rates with consumption. In this case, a precautionary saving arises under uncertainty, and the household is said to be “prudent”. Finally, in Section 6 we turn to the case where uncertainty regards the real return of a financial asset. In that case, the consumer must decide the optimal composition of its portfolio between the safe assets and the risky asset. Section 7 summarizes the main ideas.

8.2 The benchmark (deterministic) open economy case

Consider a two-period endowment open economy with no government. As a benchmark, consider the deterministic case, in which future income of the representative is revealed before current consumption is decided.

The household life-time utility function is assumed additively separable:

   

_

 

 1

1 2

C C u

u

U with u'0, u''0. (1) Where  is the subjective rate of discount.

Assuming for simplicity that there are no initial assets

 

^b0 , the consumer’ life-time budget constraint is equal to:

2 2

1 1

1 1

1 1

C Q

C Q

r r

  

  (2)

Maximization of (1) subject to (2) delivers the Euler equation

 

1 ¹

 

2

' 1 '

1

u C r u C



 

 (3)

The Euler equation, together with (2) determines the household’ optimal consumption plan.

For comparative purposes, in what follows, we will refer to the particular case where

*

r1 . In that case, the Euler equation simplifies to

 

₁ ^'

 

₂

' C u C

u  (4) The solution to (4) is

2

1 C

C  (5)

Substituting this in the budget constraint (2), one obtains the optimal demand for current consumption:

1 1 2

1

2 1

C  Q Q

 

 

      (6)

The saving function is:

 

1 1 1 1 2

1

S Q C 2 Q Q

    

 (7)

This equation tells us that savings will be positive when current output is higher than future output; negative when current output is less than future output; nil when the output pattern is flat. Since we are considering an open economy with no investment and no government, private savings are equal to the current account:

 

1 1 2

1

CA 2 Q Q

  

 (8)

The case with Q₁Q₂ will be a benchmark throughout the discussion. In that case, the current account is zero.

8.3 The stochastic Euler equation

Assume now that output in period 1, Q₁, is known with certainty, but output in period 2 is stochastic. In particular, let’s consider the following process:





 

p Q

p Q Q_L

H 2 1

2

2 (9)

Once C₁ is decided, the net asset position of the representative consumer at the end of period 1 will be:

1 1 1

b Q C (10)

Since Q₂ is uncertain, future consumption will also be a stochastic variable. In case the good scenario materializes, future consumption will be:

 

2^H 1 1 1 2^H

C b r Q (11)

If instead the bad scenario materializes, future consumption will be:

 

2^L 1 1 1 2^L

C b r Q (12)

In this problem, we have two “ex post” inter-temporal budget constraints, which one materializing in a given scenario. Using (10) in (11), the inter-temporal budget constraint in the good scenario is:

2 2

1 1

1 1

1 1

H H

C Q

C Q

r r

  

  (13)

The inter-temporal budget constraint in the bad scenario case is:

2 2

1 1

1 1

1 1

L L

C Q

C Q

r r

  

  (14)

The expected consumption in period 2 will be:

 

C₂



Q₁ C₁



1 r



E

 

Q₂

E     (15)

Note however that (15) only holds in expected terms. After the future endowment is revealed, the relevant budget constraints to determine future consumption will be (13) or (14). The household maximizes expected utility considering the two states of nature, and the corresponding ex post inter-temporal constraints. The problem is:

      

^H

   

^L



C C

CMax U u C puC p uC

L

H 1 2 2

,

, 1

1 1

2 2 1



 





 

subject to (13) and (14). The first order conditions of this problem deliver the stochastic version of the Euler equation:

 

^{1 1 1}

 

²

^{ }  

²

' ' 1 '

1

H L

u C r pu C p u C

  

      (3a)

Equation (3a) is the key equation of all our discussion. It states that the marginal utility of consumption in period 1 must equal the expected marginal utility of consumption in period 2 multiplied by the factor relating the interest rate to the rate of time preference:

 

1 ¹

 

2

' 1 '

1

u C r E u C



    (3b)

To solve (3b) we need to specify an utility function, and eventually some simplifying assumptions regarding the interest rate. In case the economy is open, we conveniently assume that r₁   r₁^*. In that case, equation (3b) becomes:

 

₁



'

 

₂



' C Eu C

u  (4a)

This condition states that the optimal plan will be such that the marginal utility of current consumption is equal to the expected value of the marginal utility of future consumption. Mathematically, equation (4a) cannot be solved, unless we know the utility

function. In the following, three classes of utility functions are distinguished, according to the following properties:



u'

 

C₂



u'



E

 

C₂



E  (16)



u'

 

C₂



u'



E

 

C₂



E  (17)



u'

 

C₂



u'



E

 

C₂



E  (18)

As shown in the appendix, these three classes of utility function differ regarding the sign of the third derivative relative to consumption, u' ''(C).

8.4 The random walk theory

8.4.1 Smoothing ex ante

Let’s maintain the assumption that the economy is open, with r₁^*  . In this section, we consider the case where consumer’ preferences are such that (16) holds. Replacing this in the Euler equation (4a), one obtains:

 

₁ '

  

₂



' C u EC

u  (4b)

The solution of (4b) is:

 

2

1 E C

C  (5b)

Equation (5b) states that the household will set current consumption so as to achieve a horizontal consumption path, in expected terms. In other words, the household smoothes consumption “ex ante”.

Replacing (5b) in (15), the current optimal consumption solves for:

 

2

1 1

1

2 1

C  Q E Q

 

 

      (6b)

The household saving in period 1 will be:

1 1

 

2

1

S 2 Q E Q

^{ }

     (7b)

Savings will be positive, negative or nil, depending on whether the household expects future output to decline, to increase or to remain equal to current output. Whenever

1

 

2

Q E Q , savings will be zero. Note that savings are influenced by expected values only.

Uncertainty (the variance) does not impact on savings.

Since there is no investment and government expenditures, savings are equal to the current account¹:

 

1 1 2

1

CA 2 Q E Q

 ^{ }

     (8b).

8.4.2 No smoothing “ex post”

Equation (5b) states that when condition (16) holds, the household smooths consumption in expected terms. Whether consumption smoothing will materialize ex post or not, it depends on the realization of the shock.

Given the first-period choices, C₂ is determined according to (13) or (14), depending on the scenario. Solving (6b) for Q₁ and replacing in (13) and (14), one obtains, respectively:

 

₂

2 1

2 C Q EQ

C^H   ^H  (13b)

 

₂

2 1

2 C Q EQ

C^L   ^L (14b)

This means that consumption in period 2 differs from consumption in period 1 by the forecasting error. As long as the actual Q₂ differs from the expected value, there will be no consumption smoothing ex-post: future consumption will differ from current consumption in the unexpected component of future output. Thus, an output expansion ahead of expectations will impact positively on consumption, implying that consumption will be positively correlated with output. Consumption will be a pro-cyclical variable.

1 Note that one cannot use this expression and set CA=0 to find the autarky interest rate. The reason is that we already assumed that the interest rate is equal to the rate of time preference (eq. 4a). To solve for the closed economy, one must depart from (3a).

Denoting by  Q2E Q

 

2 the forecasting error, equations (13b) and (14b) imply that:





 ₁

2 C

C (19)

This equation is known as the Random Walk Hypothesis²: the household consumption this year is the best predictor of household consumption next year. It doesn’t matter whether output is expected to increase or to decrease over time, as long as these moves are anticipated: anticipated changes in income were already reflected in the household expected life-time wealth and will not change current consumption. Only unexpected changes in income will influence private consumption.

This result also has policy implications: it implies that, as long as households have access to financial markets, consumption will be as stable as possible given the information available. Since the only source of instability is the unpredictable component, policymakers drawing on the same information pool as the private sector have little to do in help private consumption to become more stable.

8.4.3 An illustrative example

As for an illustration, assume that the stochastic process (9) takes the following particular form:





 ₁

2 Q

Q with





2 1

2 1



  (9b)

The stochastic process  has an average equal to zero and standard deviation equal to

 . Hence, the production pattern is expected to be flat: E

 

Q₂ Q₁. In this case, the optimal consumption and savings will be C₁Q₁ and S₁ 0.

2 Hall, Robert. (1978). "Stochastic Implications of the Life Cycle-Permanent Income Hypothesis:

Theory and Evidence". Journal of Political Economy 86 (6): 971–987.

Figure 1 illustrates the optimal consumer choice in a graph. Ex ante, the household expects future output to be equal to current output, so its consumption will be equal to current output and the trade balance will be zero (point 1). After the realization of the shock, the actual inter-temporal budget constraint will materialize above or below the expected budget constraint. However, it will be too late to change C₁: hence, the actual consumption pattern will be 2H or 2L, depending on how the shock materializes. The efficiency condition MRS=1+r that underlies point 1 will not, in general, hold “ex post”.

Figure 1: The random walk case

8.4.4 Does openness cause a smoother consumption?

The fact that the household smooths consumption in expected terms does not imply that consumption will be actually smoothed ex post. Actually, it may happen that the opportunity to borrow or lend ends up causing consumption to be more volatile than if the economy was closed to capital flows.

To see this with a simple numerical example, suppose that r₁^* 0 and that the production pattern was Q₁ 100 and E

 

Q₂ 120 . From (6b), we know that current consumption will be C₁E

 

C₂ 110 implyingTB₁10 . Then, suppose that after the consumption decision for the first period was taken, future output materialized as Q₂ 80.

Since the external debt had been already hired and needed to be reimbursed, consumption in period 2 will be C₂ Q₂TB₂ 70.

Of course, if the consumer anticipated correctly the output pattern Q₁100and

2 80

Q , he would have chosen C₁C₂ 90. But the fact that the decision was based on an expected value resulted in a pattern of consumption more volatile in the open economy (C₁110, C₂ 70) than it would have been if the economy was closed to capital flows (C₁ 100,C₂ 80).

Does this mean that there should be a government intervention to moderate private consumption in the first period? Not necessarily. Unless the government had more information than the private sector regarding the future (better forecasting tool), any intervention could also be destabilizing.

8.5 The case with precautionary savings

8.5.1 Open economy

Sticking to the case of an open economy, and assuming r₁^* , we now consider the case where condition (17) holds. Replacing this in the Euler equation (4a), one obtains:

 

₁ '

  

₂



' C u EC

u  (4c)

Since the marginal utility is decreasing, u''0, this implies

 

^L

H p C

pC

C₁  ₂  1 ₂ , that is

 

₂

1 EC

C  (5c)

Equation (5c) reveals a clear distinction relative to the earlier model: in this case, the consumer does not smooth consumption, even “ex ante”. Because consumers are uncertain regarding the future, they prefer to consume less today, giving rise to precautionary savings.

Using (5c) in (15), one obtains:

 

2

1 1

1

2 1

C  Q E Q

 

 

      (6c)

And savings will be:

1 1

 

2

1

S 2 Q E Q

^{ }

     (7c)

Thus, even if the consumer expects the output path to be steady, Q₁ E

 

Q₂ , he will save, just because he is uncertain about the future.

In sum, when condition (17) holds, on the top of a positive correlation between consumption and output ex post, there will be precautionary savings. At the country level, all else equal, higher uncertainty comes along with a higher current account surplus.

8.5.2 The logarithmic utility function

We now consider a specific functional form for the utility function. In particular, we turn to the logarithmic case, that we have been using along the course:

1 2

ln ln 1 U C C

  

 (1b)

The logarithmic utility function satisfies condition (17) and delivers a precautionary saving. In what follows, we use this production function to describe both the cases of an open economy and of a closed economy. Hence, we must depart from the general formulation of the stochastic Euler equation (3a). Using (1b), its equivalent will be:

¹

1 2 2

1

1 1

1 ^{H L}

r p p

C  C C

 

 

     (3c)

Equation (3c) will be solved for two particular setups: open economy, and closed economy.

8.5.3 An illustrative example

Considering again the stochastic process (9b), and using (13) and (14), the Euler equation (3c) becomes:

       

1

2

1 1 1 1 1 1 1 1 1

1

1 0.5 0.5

1 1 1 '

r E u C

C  Q C r Q  Q C r Q 

 

               (3d)

In what follows, we assume that t r₁^*  0. In that case, (3d) simplifies to

  

₂



1 1 1

1 1

2 ' 1 2

1 2

1

1 Eu C

C Q C

Q

C 



 







 



 



 (4d)

Since this equation is quadratic, the solution has two roots, but one of them leads to utility equal to minus infinity. The valid solution is:

2 2

1

3

2 2 2

C  Q   Q 

  (6d) Implying:

2 2

1 1 1

1

2 2 2

S Q C   Q    Q  (7d)

These equations show that the optimal savings depend on uncertainty. In case of no uncertainty,  0 , implying C₁ Q and S₁ 0, just like in our benchmark example. When uncertainty increases, current consumption declines and savings increase.

For a graphical illustration of the impact of uncertainty on consumption, we turn to Figures 2 and 3. Figure 2 displays the marginal utility of C₂ as a function of C₂, when preferences are logarithmic. The expected value of the marginal utility of C₂is identified by point 1.

Figure 2: The case with prudence

1 C₂^H

C₂^L2Q₁C₁ 1

C₂^L u'

 

C₂ ^_C¹

2

E[u'

 

C₂ ^]_C¹

1

C₂^H 2Q₁C₁

1

E[C₂] C₁

Now, remember from (5.4a) that consumer optimization implies the expected value of the marginal utility of C₂ to be equal to the marginal utility of C₁. This condition is re-stated in the vertical axes in Figure 2, to determines the level of current consumption, C₁. In the figure, you see that C₁E

 

C₂ , revealing the prudent behaviour.

With this model in mind, you can now visualise what happens to current consumption when uncertainty increases. This is illustrated in figure 3. When uncertainty, captured by the parameter  , increases, the variance of future output increases, but its expected level remains unchanged. In the figure, there will be a larger distance between C₂^H and C₂^L, but E

 

C₂ remains the same. The dashed line in the figure shifts upwards, determining a higher expected value of the marginal utility of C₂ , and by then a higher marginal utility of current consumption. Thus, C₁ must decrease.

Figure 3: increased uncertainty

C₂^L u'

 

C₂ ^_C¹

2

1 C₁

C₂^H

1’

E[C₂]

1

1 C₁'

C₂^H' C₂^L' C₁

C^'₁

In conclusion, when the consumer is prudent, higher uncertainty comes along with less current consumption today, or, in other words, with higher desired savings. In an open economy, the trade balance will improve. In a closed economy, the interest rate will decline.

8.5.4 The closed economy case

So far, we have considered that the economy is open and the interest rate is exogenous. In this section, we briefly discuss the case of a closed economy. As before, we assume that the consumer must decide current consumption before output in period 2 is revealed. In this mode, the interest rate is endogenous, but not random: the individual observes the interest rate at the time of the consumption decision. The interest rate is endogenous in the sense that it adjusts to clear the market for goods and services in period 1.

In a closed endowment economy with no government, the equilibrium in the market for goods and services implies:

Q C 1 (20)

In that case, the Stochastic Euler equation (3c) simplifies to³:

¹

1 2 2

1

1 1

1 ^{H L}

r p p

Q  Q Q

 

 

   

   (3e)

To make the example easy, we consider again the utility function (1b) and the stochastic process (9b). In that case, we obtain

1

1 1 1

1

1 0.5 0.5

1 r

Q  Q  Q 

 

      

Using (20), and solving for the interest rate, we obtain

 

²

1 2

1 r^a 1 1

Q

 ^  ^

     

  (21)

To interpret condition (20), consider first the case with no uncertainty ( 0). In that case, the interest rate is equal to the rate of time preference, reflecting the fact that consumption and output are constant over time. Under uncertainty ( 0), the interest rate

3 Allowing for government consumption is straightforward. In that case, we would have Q_t C_tG_t and the equivalent to (3e) would be ¹

1 1 2 2 2 2

1 1 1

1 ^{H L}

r p p

Q G  Q G Q G

 

 

   

     ^.

falls below the rate of time preference. The reason is that uncertainty induces the prudent consumer to save, even if future output is expected to be equal to current output. Since in a closed economy there are no exports, the domestic interest rate must decline to induce the household to consume all the available output in period 1.

8.5.5 The Great Moderation, the crisis, and the Current Account

In the two decades that preceded the financial crisis, the volatility of output in the US economy declined substantially relatively to the decades before. This period has been labelled

“The Great moderation”. If the precautionary theory of the current account was correct, one would expect the trade deficits in the US to increase during the Great Moderation, and to decrease during the crises period that followed, because of the increased uncertainty.

Coincidently, that was the case⁴.

Arguably, a similar phenomenon happened in Portugal. During the run up to the EMU, the perception of risk in Portugal declined substantially relative to the decades before, creating the conditions for more confident households to reduce their precautionary savings.

In fact, the households saving rate declined to historically levels in Portugal, at the same time the current account was reaching deficits around 10% of GDP.

8.6 Portfolio decisions

8.6.1 The Capital Asset Pricing Model (CAPM)

In Section 8.3, we assumed that the household was endowed with an uncertain amount of output in period 2, and we investigated the consequences for optimal consumption.

A different question regards the pricing of this second period endowment: that is, if the household had the opportunity to sell that lottery in the market, what would be its price?

4 See Schmitt-Grohé, Uribe, M, Woodford, M., 2015. International Macroeconomics, Chapter 4.

To answer this question, assume that the consumer starts out with no initial assets and that it may decide in period 1 to accumulate wealth in the form of a real bond paying a certain return, or claims (shares) in future output, that is uncertain. Let ₁ refers to the number of shares, and a₁ to the price of these shares. In period 2, each share delivers Q₂^H or Q₂^L depending on the scenario that materializes.

At the end of period 1, the household asset position will be;

1 1 1 1 1

b  a Q C (10a)

Since Q₂ is uncertain, future consumption will be a stochastic variable. In case the good scenario materializes, future consumption will be:

 

2^H 1 1 1 1 2^H

C b r Q (11a)

If instead the bad scenario materializes, future consumption will be:

 

2^L 1 1 1 1 2^L

C b r Q (12a)

As before, the consumer maximizes an expected utility of the form

      ^{ }  

1,1 1 2 2

1 1

1

H L

Max UC u C pu C p u C

 ^{  } ^{   }

subject to (11a) and (12a). The first order condition in respect to current consumption delivers (3a) , or, in a simple form

 

1 ¹

 

2

' 1 '

1

u C r E u C



    (3b)

The novelty here is that the consumer optimally decides its exposure to the uncertain output in period 2, buying more or less shares. The first order condition in respect to ₁ , given (11a) and (12a), will be:

 

²



²

^

¹

^

¹

 ^{ }  

²



²

^

¹

^

¹



' ^{H H} 1 1 ' ^{L L} 1 0

pu C Q  r a   p u C Q  r a 

 

^{2 2}

^

¹

^

1

' Q 1 0

E u C r

a

  

  

  

 

 

The term Q a_{2 1}measures the rate of return of the risky asset. Using the additive properties of the expected value, we get

 

^{2 2}

 

²

^

¹

^

1

' Q ' 1

E u C E u C r

a

     

    

  . Solving for

the interest rate on safe assets, we obtain:

 

 

   

 

2 2 2

2 2 2

1 1 1

1

2 2

' ' cov ' ,

1 ' '

Q Q Q

E u C E u C E u C

a a a

r E u C E u C

         

         

   

  

   

   

That is

   

 

2 2 2 1

1

1 2

cov ' ,

1 '

u C Q Q a

E r

a E u C

  

  

      

   

   

(22)

This equation states that expected return on the risky asset Q a_{2 1} must equal the return on the safe asset plus a term capturing the risk premium. Note that, since the marginal utility is in general decreasing (risk aversion), the term in square brackets is negative implying a positive risk premium. In case the household is risk neutral, the marginal utility will be constant and the covariance term is zero. In that case, the expected return on the risky asset is only required to be equal to the return of the safe asset.

8.6.2 The optimal demand for the risky asset (Merton problem)

8.7 Main ideas

The main conclusions of this handout are as follows:

 Under uncertainty, consumption smoothing does not hold “ex post”, i.e, after the realization of future endowment. Thus, there is scope for consumption to be positively correlated with output.

 Under uncertainty, consumption smoothing may or may not hold “ex ante” (i.e., in expected terms), depending on consumer preferences. In case the utility function is

quadratic, a higher uncertainty, even if affecting negatively the consumer’ welfare, does not translate into higher savings.

 If instead the marginal utility of consumption is convex, as it happens with most popular utility functions, the consumer will save more in response to higher uncertainty, revealing a “prudent” behaviour. This, in turn, materializes into improvements in the current account - in the case of open economies - or into downward pressures in the interest rate, in the case of closed economies.

Further reading

Carlos Vegh, 2013, Open economy macroeconomics in developing countries, Ch 2.3.

Schmitt-Grohé, Uribe, M, Woodford, M., 2015. International Macroeconomics, Chapter 4.

Appendix: Risk-aversion versus prudence

We just saw that the effect of uncertainty on consumption depends critically on the convexity of the marginal utility function. This cannot be confounded with risk aversion, which depends on the concavity of the utility function. This appendix clarifies the distinction.

Risk aversion refers to the fact that consumers dislike uncertainty. This property is illustrated in Figure A1. The fact that the utility function is increasing in C means that the larger the consumption level, the better. The marginal utility is decreasing with the level of consumption, meaning that each new unit of consumption delivers less utility than the unit before. This is reflected in the second derivative u''0 (decreasing marginal utility).

The implication of a decreasing marginal utility is that the utility function is strictly concave in C: thus, given a bad scenario C₂^L and a good scenarioC₂^H, the consumer would be better off if he enjoyed the expected consumption for sure, instead of a lottery (that is,

 



^EC



^E



^U

 

^C



U  .

Risk aversion is a property of any utility function with decreasing marginal utilities. If the utility function was linear, u''0, then the utility of the expected value would be equal to the expected value of utility, meaning that the consumer would be risk neutral.

Prudence refers to consumers preferring to save more in the anticipation of uncertain income. Analytically, this is implied by the convexity of marginal utility: u' ''0.

This is illustrated in Figure A2. In the figure, we display a convex marginal utility function, implying that, in face of two possible outcomes, the marginal utility of the expected value is less than the expected value of the marginal utility. This corresponds to condition (17) and describes a situation in which the consumer is prudent. The most popular utility functions imply convex marginal utilities and, therefore, prudent households.

Figure A1: Risk aversion versus risk neutral: concavity of u

 

C

Figure A2: Prudence versus random walk: convexity of ^u

 

^C

The linear case in figure A2 corresponds to the random walk hypothesis: in that case, the marginal utility is linear, implying that the marginal utility of the expected value is equal to the expected value of the marginal utility (condition 16). In the random walk case, the third derivative of the utility function is equal to zero.

Although not represented, one could also describe a situation in which the consumer is “imprudent”: a concave marginal utility, u'''0 (condition 18). In that case, an increase in

uncertainty regarding the future would cause the consumer to increase consumption. This is however a less realistic case.

8.8 Review questions and exercises

Review questions

8.1. (Consumption under uncertainty). Comment: “The emergence of large current account deficits in peripheral EU countries after the launch of the EMU reveals that consumers were imprudent”.

Problems

8.2. (Trade balance, uncertainty) Consider a small open economy able to borrow or lend in the international markets at the interest rate r^*0. The lifetime utility function of the representative consumer is given by U u C

 

1 u C

 

2 , with

 

' . 0

u  , and ^{u'' .}

 

^0. It is also assumed that the current and future GDP are known with certainty, being Q₁ Q and Q₂  Q  , where of is a deterministic parameter.

a) (Optimal Consumption, deterministic case) Find out the optimal consumption path as a function of . Illustrate the optimal choices with the help of a graph.

b) (Trade Balance, deterministic case) Find out the expression for the TB1 as a function of In which case will the country run a trade deficit? Explain the intuition?

c) (Trade balance, Uncertainty) Now assume that, instead of deterministic, at the time the current consumption decision is taken, future output is expected to be Q₂  Q  with probability ½ and Q₂  Q  with probability ½, where  is some positive parameter. Without making any calculation, explain which further assumptions are needed to ensure that TB in period 1 is positive or negative. In which case would consumption be smoothed ex ante?

8.3. (Prudence): Consider a small economy with no capital controls. The lifetime utility function of the representative consumer is given by U lnC₁ lnC₂ and the international interest rate is constant at r^*  0 . It is also known that the current GDP is equal to Q₁ 8.

a) Assume first that future GDP is known for sure: Q₂ 8. In this case, how much will be optimal consumption in each period? What will be the CA balance in this case? [A:

8,8,0].

b) Now suppose that future output is Q₂ 83 2 with probability ½ and 2

3

2 8

Q . Compute the expected value of output in period 2 as well as its variance. [A: 8, 18]

c) Find out the optimal consumption in period 1 in the case with uncertainty. What can you conclude regarding the impact of uncertainty in the current account? [A: 7].

d) What would be the equilibrium in a closed economy?

8.4. (Fiscal uncertainty): Consider an endowment economy open to capital flows.

The lifetime utility function of the representative consumer is given by

2

1 ln

lnC C

U   In this economy, GDP is constant at Q₁ Q₂ 100. The foreign interest rate is r^* 0.

a) Assume that initially G₁ T₁20 and G₂ T₂ 10 . Find out the optimal consumption pattern, as well as the trade balance. Represent in a graph. Which entity will be hiring liabilities, and to whom?

b) Assume that current taxes declined to T₁15, holding goverment purchases constant.

Would this alter private consumption? What about private savings? Which entity will be hiring liabilities, and to whom?

c) Now assume that government expenditures in period 2 were insteadG₂ 30, while current expenditure was G₁20 . Would this change have any effect on private consumption and on the trade balance? Why?

d) Finally, assume that government consumption could be G₂ 30 with probability ½ and G₂ 10 with probability ½, and that this information would be only revealed after households decided their current consumption. Would private consumption be smoothed in this case? Ex ante? Ex post? How would the trade balance look like?

Explain.

8.5. (Random Walk) Consider an endowment economy open to capital flows. The lifetime utility function of the representative consumer is given by ^{U u}

   

^C₁ ^{u C}₂ ^,

where u

 

C aC2bC². In this economy, Q₁100 future output is stochastic. In particular, assume that Q₂ 125 with probability p0.8, and Q₂ 80 otherwise.

The foreign interest rate is r^* 0.

a) Find out the expected value of future output.

b) Find out the optimal choice for current consumption. What will be the implied TB? Is consumption being smoothed ex ante?

c) Now suppose that the bad scenario materialized. In that case, what will be consumption in period 2? Is the household better of or worse of than in the case in which the economy was closed to capital flows? Explain, with the help of a graph.

8.6. Consider a small open economy. The utility function of the representative household is given by U u

   

C₁ u C₂ , with u

 

C_t C_t 0.5C_t². In this economy,

output is constant and equal to Q₁ Q₂ 140. The international interest rate is

* 0

r .

a) Assume for a moment that G₁ T₁40 and G₂ T₂ 20 . Determine: (a1) the optimal consumption pattern; (a2) the TB in periods 1 and 2; (a3) represent in a graph.

(a4) Who is holding whose liabilities? Alternatively, suppose that T₁ 30. In this case: (a5) what would be the value of the TB? (a6) Which entity would hold whose debt now?

b) Assume now that future government consumption could be G₂ 0 with probability of

½ or G₂ 40 with probability of ½, and that the actual value was only revealed after agents decided their current consumption. (b1) What would the value of TB₁ be? (b2) Would there be consumption smoothing ex ante? (b3) And ex post? (b4) If the negative state materializes, will consumers be better or worse off than if the economy was closed?

8.7. Consider a two-period endowment economy, where the preferences of the representative consumer are U lnC₁lnC₂. Also assume that b₀^*  4, Q₁ 18 , and Q₂ 22. In the basic formulation, there is no government and the economy is open to capital flows, with 1r₁^*1.

a) (Optimal consumption) Find out: (a1) the consumer’s intertemporal budget constraint; (a2) the optimal consumption in periods 1 and 2; (a3) the trade balance in periods 1 and 2. (a4) Represent in a graph. (a5) Explain the intuition for the consumption pattern.

b) (Fiscal policy) Departing from (a), suppose that the government announced

2 8

G  , with G10. Examine the implications, considering two alternative scenarios for taxation: T10 and T14 . (b1) Describe the impact of the policy on household wealth. Find out: (b2) the implied consumption levels in periods 1 and 2; (b3) the trade balance in periods 1 and 2. (b4) By the end of period 1, which agent will hold the debt of which another agent? (b5) Would similar results hold if taxes were proportional to consumption? Discuss.

c) (Sudden stop) Departing from (a), suppose that this economy could not borrow in foreign markets, and that the initial debt matured in period 1. Find out: (c1) the trade balance in period 1; (c2) the implied consumption levels in periods 1 and 2; (c3) the domestic interest rate. (c4) If the sudden stop was accompanied by a debt relief leading to b0^* 0, would the economy still be constrained? Illustrate graphically, comparing to the case of no debt relief.

d) (Uncertainty) Returning to (a), assume that future output was not known with certainty, being instead Q2 14 or Q2 14 , with probability ½ each and  ⁸. (d1) Find out: the expected value of future output; (d2) the optimal consumption in period 1; (d3) the TB in period 1. (d4) Is consumption being smoothed in this case? Discuss.

8.8. (Uncertainty shock, closed economy) Consider a two-period endowment economy, closed to capital flows, where the preferences of the representative

consumer are U lnC₁lnC₂ . Also assume that Q₁ 100 , Q₂ 80 , and that initially there is no government.

a) (Optimal consumption) Find out: (a1) the consumer’s lifetime wealth as a function of the interest rate; (a2) the optimal consumption in period 1 as a function of the interest rate; (a3) national savings as a function of the interest rate; (a4) the autarky interest rate. (a5) Explain the intuition for the autarky interest rate.

b) (Fiscal Policy) Departing from (a), suppose that the government announced

2 30

G  , with G₁ 0. Examine the implications, considering that all taxes are paid in period 1, that is T₂ 0. Quantify: (b1) Consumption levels in periods 1 and 2; (b2) Interest rate; (b3) Private savings and government savings. (b4) Would a postponement of taxation to period 2 impact on the interest rate? What if 20% of the households in this economy were constrained on borrowing?

c) (Uncertainty) Returning to (a), assume that future output was not known with certainty, being instead Q₂ 80 or Q₂ 80 , with probability ½ each and

  20. Find out: (d1) future consumption in the good scenario and in the bad scenario; (d2) the modified Euler equation; (d3) the autarky interest rate; (d4) What do you conclude regarding the effect of uncertainty on savings’ behaviour?