Birkbeck MSc/Phd Economics Advanced Macro Spring 2006 Consumption - - PowerPoint PPT Presentation

Birkbeck MSc/Phd Economics Advanced Macro Spring 2006 Consumption and Asset Prices Lecture 1: Optimal Consumption under Uncertainty: (Mainly) Partial Equilibrium Models

1 Introductory

1.1 Overall Course Structure

1. Optimal consumption under uncertainty: partial equilibrium models 2. Consumption and asset prices in an endowment economy 3. Optimal consumption and asset prices in the stochastic growth model

1.2 Exercises

There are no classes scheduled for this lecture course, but I believe it is essential that you do some worked exercises. These will be included as we proceed; you’ll be expected to have worked through them by the start of the next lecture, when we’ll go over solutions.

1.3 Reading for Lecture 1

• Deaton’s Understanding Consumption is excellent for anyone who wants

to understand the consumption problem in any depth. He does not how- ever deal in any depth with the maths of dynamic programming, dynamic consistency, etc. For this see references given by Yunus Aksoy.

• If you have never come across the introductory material I shall cover in early
• n in this lecture, you could also look at Chapter 7 of Romer, or, for cover-

age of the two-period intertemporal problem, any micro text (Williamson’s macro text also covers well.

1.4 Notation

• Capital letters, eg, C, Y will always refer to underlying levels before any

transformations

• Everything is measured in real terms, including all return series, so, 1 +

Rt = (I + St) / (1 + Πt) where S is the nominal return, and Π is the inflation rate.

• Stock variables, eg Kt, At are values at the beginning of period t (contrast

with most published data - often a source of confusion)

• Link with convention that, under uncertainty, any t−dated variable is in

the information set It used to make decisions.

• Lower case letters will usually refer to logs, ie c = log C (where log x =

ln x)

• Except in case of interest rates, growth rates, etc where r = log(1+R) ≈

R ; g = log(1 + G) ≈ G by the log approximation....

• .... and later on in the course lower case letters may sometimes be used

to represent log deviations from steady state values, eg I may define ct = log

³

Ct/ b Ct

´

≈ Ct− b

Ct

b

Ct

2 The Consumption Decision with Perfect Fore- sight and a Finite Horizon

2.1 The Objective

The representative consumer at time t is assumed to maximise U = U(Ct, Ct+1, .....CT) (1) For now we shall ignore the posssibility that the utility function might contain

• ther arguments, in particular leisure.

It is very common to restrict this very general formulation to be “additive separable” in per period consumption, so that the maximand becomes the discounted sum of per period utility (or “felicity”) functions: U =

T−t

X

i=0

1 (1 + Θ)iu(Ct+i) (2)

• Θ is normally referred to as the “subjective discount rate”, or the “rate of

pure time preference”. Rationales?

• (nb lower case convention will apply as for interest rates so θ = log(1+Θ))
• Highly restrictive, but restrictions are useful in modelling terms and (I shall

argue) insightful

• Deaton Ch1 also discusses aggregation across goods.

2.2 The Constraint

We can represent the constraint in two equivalent ways. First we can look at the evolution of assets: At+1 = (At + Yt − Ct) (1 + Rt+1) , (3) where Yt is labour income; At is net assets, and then impose a condition on the consumer’s terminal wealth: AT + YT − CT ≥ 0

i.e. terminal net assets cannot be negative. If no bequest motive is included in the model, and the constraint is assumed to bind, this means that CT = YT + AT. Just for now assume, for simplicity, that R is constant in all periods. Then we can use the asset evolution identity and solve backwards recursively from this terminal condition, and thus derive the constraint in present value terms as:

T−t

X

i=0

Ct+i (1 + R)i = At +

T−t

X

i=0

Yt+i (1 + R)i (4) = At + Wt It is this form that we shall use initially here; in more complex problems, how- ever, (most notably when uncertainty enters the picture) we shall not be able to do so.

2.3 First Order Conditions

If the consumer maximises (2), by choice of Ct, Ct+1, ...CT subject to (4), the first order conditions for any two adjacent periods are: Ct+i :

µ1 + R

1 + Θ

¶i

u

0(Ct+i) = λ

(5) Ct+i+1 :

µ1 + R

1 + Θ

¶i+1

u

0(Ct+i+1) = λ

(6)

where λ is the Lagrange multiplier on the constraint (4).∗ Hence the optimal consumption path between any two periods will satisfy the “Euler Equation”: u

0(Ct+i) =

µ1 + R

1 + Θ

¶

u

0(Ct+i+1)

(7) Note what is in, and is not in this expression. Bear in mind the inverse relationship between marginal utility and consumption. Implications of R = Θ; R > Θ; R < Θ? The assumption that R = Θ is particularly popular because it gives a straight- forward result whatever the form of the felicity function.

∗Alternative is to have T − t Lagrange multipliers, one for each period’s constraint but this

alternative also disappears under uncertainty so won’t be pursued here (cf Yunus’s lectures)

2.4 A Popular Special Case: Quadratic Utility

If per period utility is quadratic, eg, of form ubliss − β 2

³

Ct − Cbliss

t

´2

marginal utility is linear. Exercise 1: Why should we worry about assuming quadratic utility? Linearity of marginal utlility is a very useful property once uncertainty is allowed into the problem, since (as future lectures will show) quadratic utility implies “certainty equivalence”. If in addition we assume that R = Θ, we get a special case which Deaton refers to as the “Permanent Income Consumption” representation, whereby optimal consumption is constant, and equal to the perfect foresight solution, even under uncertainty. More on this later

2.5 Another Popular Special Case: Power Utility

If per period utility function is of the “constant relative risk aversion” (or “power utility”) form: u(C) = C1−γ 1 − γ (8) = ln(C) for γ = 1 u0(C) = C−γ (9) the Euler equation can be manipulated to yield: Ct+i+1 Ct+i =

µ1 + R

1 + Θ

¶1

γ

(10)

Hence this specification implies that the elasticity of intertemporal substitution is equal to the reciprocal of the coefficient of relative risk aversion. Some economists have objected to this linkage, and have produced formulations which separate the two; my personal view is that the link is insightful. Exercise 2: The first line of (8) is undefined for γ = 1. Show however, using L’Hopital’s Rule, that lim

γ→1

C1−γ 1 − γ = ln C Exercise 3: Show that with power utility the coefficient of relative risk aversion c.u00/u0 is constant and equal to γ

2.6 The “Closed Form Solution” when R = Θ.

The Euler Equation only tells us about the time profile of consumption, not its absolute level. A simple closed form solution can however be derived if R = Θ,.(A possible, but quite restrictive general equilibrium rationale for this assumption is given below). If this is the case optimal consumption is constant in all period, ie Ct = C∀t and we can use the budget constraint to derive the level of C, thus: C

T−t

X

i=0

1 (1 + R)i = At + Wt (11)

which can be manipulated to yield C =

µ

R 1 + R

¶

κ (At + Wt) (12) where κ =

Ã

1 1 − (1 + R)−(T−t+1)

!

> 1 (13) Exercise 4: Show how! Note that as T goes to infinity, κ goes to unity, and the consumer simply consumes the annuity value of their total wealth (financial and human capital)

• hence the link with Friedman’s “Permanent Income” model.

How much does it matter (in modelling terms) that people do not live for ever?

Do not forget the dependence of this solution on the assumption that R = Θ. For more general cases, the solution is nastier (and can only be arrived at by making explicit assumptions on the nature of the felicity function). It is though relatively easy to understand conceptually.

2.7 Comparitive Statics for An Atomistic Consumer

Why do we need to assume the consumer is atomistic? 2.7.1 Changes in Labour Income In the special case above, with R = Θ, it is easy to see that if current labour income changes with income in all future periods constant:

∂C/∂Yt =

µ

R 1 + R

¶

κ ≈ R (14) What however if the change in Yt persists for all periods? ie, is “permanent”: think of a shock dYt = dYt+i... = dYT. In this case: dC/dYt =

µ

R 1 + R

¶

κ

T−t

X

i=0

∂Wt/∂Yt+i (15) =

µ

R 1 + R

¶

κ

T−t

X

i=0

1 (1 + R)i = 1

2.7.2 Changes in the Interest Rate The Euler Equation tells us that the growth of consumption is positively related to the interest rate.But this does not tell us what the closed form response will be except under certain more restrictive assumptions. Exercise 5 Show geometrically in a 2 period model that for a consumer with initial assets the impact of a permanent change in the interest rate on the level of consumption is ambiguous. Exercise 6 In a multi-period model, if the interest rate changes only temporarily, describe a restriction on the nature of the path of interest rates that will ensure that consumption unambiguously falls when the interest rate rises.

3 General Equilibrium in an Endowment Econ-

• my With Perfect Foresight

What if the representative consumer is not atomistic, but, in contrast, repre- sents all the consumers in the economy? If we think in these terms, we can get some insights into why we might, or might not, want to make our assumption that R = Θ. Let aggregate income be simply an endowment, which is con- stant, and equal to aggregate consumption. Look again at the version of the Euler Equation in (10). In general equilibrium, this reduces to: 1 =

µ1 + R

1 + Θ

¶1

γ

(16)

so the combination of optimal consumption and the supply side constraint must imply R = Θ. However, suppose instead that aggregate output (and hence consumption) is growing at a constant proportional rate. By the same reasoning, we will get 1 + G =

µ1 + R

1 + Θ

¶1

γ

(17) which will only be satisfied if R > Θ. Exercise 7: Show that for power utility, this implies

r = θ + γg (18) and hence (19) R ≈ Θ + γG (20) where r = log(1 + R); g = log(1 + G)

4 The Stochastic Consumption Problem

4.1 The Stochastic Objective

With expected utility, we now assume that the consumer’s problem now be- comes max Et

T−t

X

i=0

1 (1 + Θ)i(u(Ct+i)) where Etxt+i = E (xt+i) |It where It is the information set at time t. We shall sometimes assume that uncertainty is of the discrete state variety, ie, in any time period there are s = 1.. S states of nature, with probability πs, ie

the objective is of the form max

S

X

s=1 T−t

X

i=0

πs (1 + Θ)i(u(Cs,t+i)) This objective is, as Deaton (Ch1) puts it “doubly additive separable”, and puts quite a lot of structure on preferences, but also makes the solution of the problem very much easier.

4.2 From the Value Function to the Stochastic Euler Equa- tion

The objective looks like, and indeed is, a multi-period one, but we cannot simply write max

Ct,Ct+1,...CT

Et

T−t

X

i=0

1 (1 + Θ)i(u(Ct+i)) and solve using standard Lagrangian techniques, because we can’t choose C in periods other than period t The dynamic budget constraint is still (3), ie, At+1 = (At + Yt − Ct) (1 + Rt+1) ,

but both Y and R may in principle be stochastic. At is the “state variable” - it is exogenously given in the current period (but not in the future). We shall in due course assume that there are many assets, but for now (and in many problems) we’ll assume a single one Given the intertemporal budget constraint, there again is tradeoff between At+1 and Ct . But this tradeoff is itself stochastic. Hence budget constraint cannot simply be iterated forward to give a present value constraint (as in the perfect foresight case). Nor can we use multiple Lagrange Multipliers Exercise 8 : Why can we not simply take expectations of the budget constraint in future periods? To solve the problem, we apply a sequence of tricks. The most crucial (and

• ften forgotten), is the first:

• 1. Assume the problem has been solved!

We then analyse what the consumer’s behaviour will look like given this solution. But in doing so, we shall find out at least something (though by no means everything) we need to know about how the consumer would solve the problem.

• 2. Turn the problem into a sequence of two period problems: Define

the “Value Function” recursively as the maximised objective, in terms of its next period value Vt (At) = max

Ct

u(Ct) + 1 1 + ΘEtVt+1 note that there is only a single choice variable, Ct, all other choices will have to be made in the future and cannot be committed to in period t.At, the state variable, is not of course the single argument of the function, but we suppress dependence on others.

• 3. Exploit dependence of Vt+1 on At+1, and hence on Ct

Vt = max

Ct

u(Ct) + 1 1 + ΘEt{Vt+1(At+1(Ct), ...)}

• 4. Hence multiperiod problem becomes 2 period problem. With first
• rder condition:

u0(Ct) + 1 1 + ΘEt

(

∂Vt+1 ∂At+1 ∂At+1 ∂Ct

)

= 0 Exercise 9 Why is it OK to maximise Etf() by setting Etf0() = 0?

• 5. Use budget constraint to derive....

∂At+1 ∂Ct = − (1 + Rt+1)

(nb: this derivative may be stochastic when evaluated at period t.)

• 6. And the Envelope Theorem to derive....

∂Vt+1 ∂At+1 = u0(Ct+1) This last stage may not be obvious, but bear in mind: a) In period t + 1, At+1 will be exogenous, just like At in period t. So we can derive this by thinking about ∂Vt

∂At.

b) Given that we have defined Vt as the maximised objective function, if At rises today, and induces a rise in consumption, we do not need to worry about the second-round impacts on future values of the value function, since, all such

terms are multiplied by ∂Vt

∂Ct = 0 at the optimum (see Yunus’s notes). Intu-

itively,at the optimum, the consumer will be indifferent, in present value terms, as to period in which any marginal increase in current wealth is consumed: “any period will do” (cf standard envelope theorem results for, eg choice between different goods, or different factors of production).

• 1. Substitute into first order condition to give stochastic Euler Equa-

tion... u0(Ct) = 1 1 + ΘEt

n

u0(Ct+1) (1 + Rt+1)

• (21)

4.3 Is it so simple?

The stochastic Euler Equation for optimal consumption in period t is so ubiq- uitous that it is worth trying to understand it properly. Obvious question: How can there be just a single first order condition for

• ptimal consumption, when in the perfect foresight case there were T − t first
• rder conditions?

Answer: The Euler Equation in period t is a necessary, but not sufficient condition for optimal consumption. To see why think about what the Euler equation for optimal consumption will look like in period t + i... u0(Ct+i) = 1 1 + ΘEt+i

n

u0(Ct+i+1) (1 + Rt+i+1)

The definition of the value function implies that the consumer must expect this to hold for all i; ie, forming expectations in period t Etu0(Ct+i) = 1 1 + ΘEtEt+i

n

u0(Ct+i+1) (1 + Rt+i+1)

• ∀i

(22) = 1 1 + ΘEt

n

u0(Ct+i+1) (1 + Rt+i+1)

• ∀i

where EtEt+i = Et by the law of iterated expectations. They will also expect the terminal condition on assets to hold in period T.Thus there are effectively T − t conditions determining optimal consumption, but they are not strictly first-order conditions. They are the requirements that the optimal consump- tion decision be dynamically consistent. So optimal consumption is not determined solely by t− and .expected t + 1−dated variables, as (21) would appear to suggest. Exercise 10 Show that if (21) is satisfied in period t, but (22) is not satisfied

for some i, then the definition of the value function will not be satisfied, and hence the optimality conditions for consumption will not be satisfied.

4.4 The Impact of Uncertainty on the Euler Equation†

To simplify, assume, as before, no “relative impatience” in terms of the expected real interest rate: ie, assume: Et(Rt+i) = Θ ∀i Using this assumption, and the property that the expectation of a product of two variables equals the product of their expectations plus their covariance, means that the Euler Equation can be expressed as: u0(Ct) = Et[u0(Ct+1)] + 1 1 + ΘCov

h

u0(Ct+1), Rt+1

i

. (23)

†This section is a simplified version of the introduction to Mason and Wright (2000), “The

Impact of Uncertainty on Optimal Consumption” Journal of Economic Dynamics and Control. I would not recommend the rest of the paper itself except to those of an especially masochistic

• r sycophantic disposition.

With convex marginal utility, Jensen’s inequality implies that Et(u0(Ct+1)] ≥ u0(Et(Ct+1)), implying that the Euler Equation can be further decomposed as follows: u0(Ct) = u0(Et(Ct+1)) + J + 1 1 + ΘCov

h

u0(Ct+1), Rt+1

i

. (24) where J = Et(u0(Ct+1)] − u0(Et(Ct+1)) ≥ 0. Exercise 11 Check that you can derive (23) from (21). What initial assump- tion is crucial to getting this result? What assumption on preferences implies (usually strictly) convex marginal utility? We can now look at various special cases that allow us to restrict the Euler Equation further.

4.4.1 No Asset Return Uncertainty, and Quadratic Utility The first assumption eliminates the covariance term, and since marginal utility is linear, J is also zero. ie, with u = ubliss − β 2

³

Ct − Cbliss

t

´2

→ u0 = −β

³

Ct − Cbliss

t

´

= α − βCt > 0 for Ct < Cbliss

t

so u0(Et(Ct+1)) = α − βEtCt+1 = Et(u0(Ct+1)) By implication, therefore the euler equation reduces to α − βCt = α − βEtCt+1 ⇒ Ct = Et(Ct+1)

which leads, as we shall see, to the stochastic version of the permanent income

• model. It also leads to the feature that consumption is a “Martingale”, ie by

applying the law of iterated expectations, EtCt+i = Ct∀i But note how strong the restrictions are that are required to get this result. 4.4.2 No Asset Return Uncertainty and Strictly Convex Marginal Utility u0(Ct) = u0(Et(Ct+1)) + J hence u0(Ct) > u0(Et(Ct+1)) hence Ct < Et(Ct+1)

This is commonly referred to as implying “precautionary saving”. Since con- sumption is expected to rise over time, due solely (in this restrictive case) to the existence of labour income uncertainty, forward-looking consumers post- pone consumption until later in life. NB: this result is crucially a result for the atomistic consumer, ie, it treats the return process as exogenous and the consumption response as endogenous. Once we allow for general equilibrium, at least in an endowment economy, we shall see that this feature is effectively turned on its head. 4.4.3 Both Asset Return Uncertainty and Strictly Convex Marginal Utility When both J and the covariance term are present, Mason and Wright (2000) show that, for the general case, there can be ambiguity, since the covariance

term will be negative for net savers. There is however one special case (log utility, for a consumer with no labour income), which I deal with below.

5 “Closed Form” Solutions to the Stochastic Con- sumption Problem

5.1 The General Case

• Using Euler Equation to derivea a closed form solution for consumption is

not generally possible, even if there is only one form of uncertainty. Conse- quently, much of the existing literature has relied on restrictive assumptions to derive analytical results.

• In particular, most papers assume that the representative asset has a return

which is non-stochastic (e.g. is a ‘safe’ short-term government bond).

• In addition either the utility function needs to be of a restrictive form

(normally either quadratic or constant absolute risk aversion, eg Caballero, 1991; Weil (1993)); or some approximation needs to be applied (Mason & Wright (2000)).

• The method of solution is usually “backward induction”:

evaluate the (expected) Euler Equation in the penultimate period of life, exploiting the terminal condition that CT = YT + AT, solve for consumption in period T − 1, then evaluate the expected Euler Equation in period T − 2, etc.

5.2 The “Stochastic Permanent Income Consumption Model” (Deaton’s term): Quadratic Utility with only Labour Income Uncertainty

By far the dominant approach, because it yields such convenient results, is to assume quadratic utility, and assume no asset return uncertainty (normally also making our assumption above of no “relative impatience” (ie, Rt = Θ∀t, ie is non-stochastic). This is convenient, because it implies that consumption behaviour is “certainty equivalent”, ie, we end up with an expression that looks almost exactly the same as that we derived when we looked at the perfect foresight case. Important caveat: this assumption is common only in partial equilibrium models of consumption. When the consumption model is turned into an asset

pricing model in a general equilibrium framework the stochastic properties of R become crucial. With all these assumptions, we get Ct =

µ

Θ 1 + Θ

¶

κ (At + Wt) (25) where κ =

Ã

1 1 − (1 + Θ)−(T−t+1)

!

> 1 (26) which is identical to the solution under certainty (given EtRt+1 = Θ) but Wt is now defined in terms of expected labour income, thus: Wt =

T−t

X

i=0

Et(Yt+i) (1 + Θ)i Thus consumers behave as if they were certain about future labour income: uncertainty does not affect behaviour. This specification has been used very extensively in empirical work (on which more below).

Exercise 12: Derive (25) for the special case T = ∞. (hint: use (4) which is valid because R = Θ is nonstochastic)

5.3 The Pure Rentier Case with Uncertain Asset Returns and Log Utility

There is one other special case which give an almost identical solution. Suppose that u(C) = ln C and that Wt = 0 . Either the consumer is a rentier or labour income risk can be eliminated by choosing a portfolio of assets with offsetting risks (viewing human capital as a particular form of asset with specific (hence diversifiable) risk). Then even with asset return risk, the closed form solution is again certainty-equivalent (as in (25), with Wt = 0). To verify this, note that, with log utility, u0(Ct) = 1

• Ct. Suppose, for simplicity,

that the consumer is infinitely lived‡. Then we plug the conjectured solution

‡The result holds for finite lives too, it is just more complicated to show this.

Ct+1 =

Θ 1+ΘAt+1 into the Euler Equation:

1 Ct = 1 1 + ΘEt{ 1 Ct+1 (1 + Rt+1)} = 1 1 + ΘEt

(

(1 + Θ) ΘAt+1 (1 + Rt+1)

)

but, if we also plug it into the asset evolution equation, At+1 = (At − Ct) (1 + Rt+1) =

½

1 − Θ 1 + Θ

¾

At (1 + Rt+1) = At (1 + Rt+1) 1 + Θ

hence: 1 Ct = 1 1 + ΘEt

(

(1 + Θ)2 ΘAt Rt+1 Rt+1

)

hence Ct = Θ 1 + ΘAt Hence the conjectured solution satisfies the Euler Equation. Intuition? Note that certainty equivalence in the face of the two different types of risk requires two different utility functions.

6 Theory vs Evidence

The vast majority of empirical investigations have been directed at what Deaton terms the “Stochastic Permanent Income Model”: ie, assuming quadratic util- ity, and non-stochastic asset returns, with R = Θ. The most straightforward implication of the Euler Equation Ct = Et(Ct+1) can be seen by subracting both sides from Ct+1: Ct+1 − Ct = Ct+1 − Et(Ct+1) Hall (1978) noted that if expectations are rational, then the right-hand side of this expression should be a white noise error, εt+1 uncorrelated with information available at time t, thus: Ct+1 = Ct + εt+1 (27)

so consumption should be a random walk (strictly speaking we need the further assumption that ε is homoscedastic). His initial results did not appear to conflict with this assumption. Subsequent research has however suggested that there are two puzzles which appear to conflict with (this restrictive version of) the theory. Both are extremely well- covered in Deaton Ch 3, so I shall only sketch the main findings here.

6.1 “Excess Sensitivity”

Hall ran regressions which seemed to show that lags of labour income had no predictive power for consumption. This finding has been contested in subse- quent research, with quite a lot of evidence that lags of income do have some

predictive power. One possible explanation of this is the existence of credit market constraints. Campbell and Mankiw (1990), for example, set up a simple model in which there is an exogenous limit to borrowing; and some fraction λ of the population is at this limit, and therefore is constrained to consume

• ut of current income - hence inducing Keynesian-type consumption patterns.

Their estimates of λ were statistically significant, and appeared to indicate that anything up to half of consumers were subject to credit constraints.

6.2 “Excess Smoothness”

The random walk assumption does not rule out a correlation of consumption with eg income in the same period - it is a story about predictability, given past

• information. In fact the evidence suggests that this contemporaneous correla-

tion may be too small. It is often assumed that forward-looking consumption should be “smooth”; but this very much depends on what is driving the in- come process. Suppose for example that income follows a simple autoregressive process: Yt = (1 − ρ) b Y + ρYt−1 + ut (28) ut = NIID(0, σ2

Y )

then if expectations are rational, EtYt+i = b Y + ρi ³ Yt − b Y

´

. Substituting into (25), assuming, for simplicity, an infinitely lived consumer (hence κ = 1), and At = 0 gives Ct = b Y +

(

Θ 1 + Θ − ρ

) ³

Yt − b Y

´

Exercise 13: 1)Derive this! 2) Show, using this version of the equation how εt in (27) relates to ut in (28) 3) What will be the correlation between εt and ut? As ρ gets close to 1 (income is close to being a random walk - which appears not to be too far from the case), then something which looks very close to the Keynesian consumption function emerges. But it is due to the predictive power of current income for future income: a high value for ρ means, in effect, that current income and permanent income are virtually the same. Some empirical studies have found that consumption does not respond enough to innovations in current income, given its autoregressive properties: this is excess smoothness. This feature is particularly marked if the change in output is positively serially correlated, for which there is some evidence, since in this case the immediate

impact of a shock to income is magnified in the longer term (for more detail see Deaton, Ch 4). However, as Deaton shows, this empirical feature is highly dependent on the assumed process for Y. §

§Cochrane(“How big is the random walk in GDP” QJE 1988) suggests that the long-run

magnification of shocks may be grossly exaggerated if the assumed generating process of

• utput is driven only by short-run properties.

7 Reading/Preparation for Lecture 2

7.1 Exercises!

These should be your first priority. We shall go over these at the start of the next lecture

7.2 Reading

You may need to do some reading to help in answering the exercises from this lecture - as noted at the start, I recommend Deaton.

Next week’s lecture will cover topics where there is substantial overlap be- tween macro and finance: the consumption-based capital asset pricing model; stochastic discount factors; the equity premium puzzle. Suggested advanced reading for enthusiasts Cochrane Asset Pricing, Ch 1

• r

Campbell, Lo and MacKinlay, Econometrics of Financial Markets Ch 8 (al- though this an applied text, the coverage of the relevant theory is very clear)