Some useful de…nitions with applications

In order to be able to discuss results in subsequent sections easily, we review some de…ni-tions here that will be used frequently in later parts of this book. We are mainly interested in the intertemporal elasticity of substitution and the time preference rate. While a lot of this material can be found in micro textbooks, the notation used in these books di¤ers of course from the one used here. As this book is also intended to be as self-contained as possible, this short review can serve as a reference for subsequent explanations. We start with the

Marginal rate of substitution (MRS)

Let there be a consumption bundle (c1; c₂; ::::; c_n) :Let utility be given by u (c1; c₂; ::::; c_n) which we abbreviate to u (:) : The MRS between good i and good j is then de…ned by

M RS_ij(:) @u (:) =@ci

@u (:) =@c_j: (2.2.7)

It gives the increase of consumption of good j that is required to keep the utility level at u (c1; c2; ::::; cn) when the amount of i is decreased marginally. By this de…nition, this amount is positive if both goods are normal goods - i.e. if both partial derivatives in (2.2.7) are positive. Note that de…nitions used in the literature can di¤er from this one.

Some replace ’decreased ’by ’increased’(or - which has the same e¤ect - replace ’increase’

by ’decrease’) and thereby obtain a di¤erent sign.

Why this is so can easily be shown: Consider the total di¤erential of u (c1; c₂; ::::; c_n) ; keeping all consumption levels apart from ci and cj …x. This yields

du (c₁; c₂; ::::; c_n) = @u (:)

@c_i dc_i+ @u (:)

@c_j dc_j:

The overall utility level u (c1; c₂; ::::; c_n) does not change if

As a reminder, the equivalent term to the MRS in production theory is the mar-ginal rate of technical substitution M RT Sij(:) = _{@f (:)=@x}^{@f (:)=@xi}

j where the utility function was replaced by a production function and consumption ck was replaced by factor inputs xk. On a more economy-wide level, there is the marginal rate of transformation M RTij(:) =

@G(:)=@yi

@G(:)=@yj where the utility function has now been replaced by a transformation function G (maybe better known as production possibility curve) and the yk are output of good k.

The marginal rate of transformation gives the increase in output of good j when output of good i is marginally decreased.

(Intertemporal) elasticity of substitution

Though our main interest is a measure of intertemporal substitutability, we …rst de…ne the elasticity of substitution in general. As with the marginal rate of substitution, the de…nition implies a certain sign of the elasticity. In order to obtain a positive sign (with normal goods), we de…ne the elasticity of substitution as the increase in relative consump-tion ci=cj when the relative price pi=pj decreases (which is equivalent to an increase of p_j=p_i). Formally, we obtain for the case of two consumption goods ij

d(ci=cj) d(pj=pi)

pj=pi

ci=cj. This de…nition can be expressed alternatively (see ex. 6 for details) in a way which is more useful for the examples below. We express the elasticity of substitution by the derivative of the log of relative consumption ci=c_j with respect to the log of the marginal rate of substitution between j and i,

ij

d ln (c_i=c_j)

d ln M RS_ji: (2.2.8)

Inserting the marginal rate of substitution M RSji from (2.2.7), i.e. exchanging i and j in (2.2.7), gives

The advantage of an elasticity when compared to a normal derivative, such as the MRS, is that an elasticity is measureless. It is expressed in percentage changes. (This can be best seen in the following example and in ex. 6where the derivative is multiplied by ^p_c^j=pi

i=cj:) It can both be applied to static utility or production functions or to intertemporal utility functions.

The intertemporal elasticity of substitution for a utility function u (ct; c_t+1) is then simply the elasticity of substitution of consumption at two points in time,

t;t+1

u_ct=u_ct+1 c_t+1=c_t

d (c_t+1=c_t)

d u_ct=u_ct+1 : (2.2.9)

Here as well, in order to obtain a positive sign, the subscripts in the denominator have a di¤erent ordering from the one in the numerator.

The intertemporal elasticity of substitution for logarithmic and CES utility functions For the logarithmic utility function Ut= ln c_t+ (1 ) ln c_t+1from (2.2.1), we obtain an intertemporal elasticity of substitution of one,

t;t+1 = ^ct=_c¹

It is probably worth noting at this point that not all textbooks would agree on the result of “plus one”. Following some other de…nitions, a result of minus one would be obtained. Keeping in mind that the sign is just a convention, depending on “increase”or

“decrease” in the de…nition, this should not lead to confusions.

When we consider a utility function where instantaneous utility is not logarithmic but of CES type

Ut= c¹_t + (1 ) c¹_t+1; (2.2.10) the intertemporal elasticity of substitution becomes

t;t+1

Inserting this into (2.2.11) and cancelling terms, the elasticity of substitution turns out to be

This is where the CES utility function (2.2.10) has its name from: The intertemporal elasticity (E) of substitution (S) is constant (C).

The time preference rate

Intuitively, the time preference rate is the rate at which future instantaneous utilities are discounted. To illustrate, imagine a discounted income stream

x₀ + 1

1 + rx₁+ 1 1 + r

2

x₂+ :::

where discounting takes place at the interest rate r: Replacing income xtby instantaneous utility and the interest rate by , would be the time preference rate. Formally, the time preference rate is the marginal rate of substitution of instantaneous utilities (not of consumption levels) minus one,

T P R M RS_t;t+1 1:

As an example, consider the following standard utility function which we will use very often in later chapters,

U₀ = ¹_t=0 ^tu (c_t) ; 1

1 + ; > 0: (2.2.12)

Let be a positive parameter and the implied discount factor, capturing the idea of impatience: By multiplying instantaneous utility functions u (ct) by ^t, future utility is valued less than present utility. This utility function generalizes (2.2.1) in two ways: First and most importantly, there is a much longer planning horizon than just two periods. In fact, the individual’s overall utility U0 stems from the sum of discounted instantaneous utility levels u (ct) over periods 0; 1; 2; ... up to in…nity. The idea behind this objective function is not that individuals live forever but that individuals care about the well-being of subsequent generations. Second, the instantaneous utility function u (ct) is not logarithmic as in (2.2.1) but of a more general nature where one would usually assume positive …rst and negative second derivatives, u⁰ > 0; u⁰⁰ < 0.

The marginal rate of substitution is then M RS_t;t+1(:) = @U₀(:) =@u (c_t)

@U0(:) =@u (ct+1) = (1= (1 + ))^t

(1= (1 + ))^t+1 = 1 + : The time preference rate is therefore given by :

Now take for example the utility function (2.2.1). Computing the MRS minus one, we have

= 1 1 = 2 1

1 : (2.2.13)

The time preference rate is positive if > 0:5:This makes sense for (2.2.1) as one should expect that future utility is valued less than present utility.

As a side note, all intertemporal utility functions in this book will use exponential discounting as in (2.2.12). This is clearly a special case. Models with non-exponential or hyperbolic discounting imply fundamentally di¤erent dynamic behaviour and time inconsistencies. See “further reading” for some references.

Does consumption increase over time?

This de…nition of the time preference rate allows us to provide a precise answer to the question whether consumption increases over time. We simply compute the condition under which ct+1> c_t by using (2.2.4) and (2.2.5),

c_t+1> c_t, (1 ) (1 + r_t+1) W_t > W_t, 1 + r_t+1>

1 , r^t+1 > 1 +

1 , r^t+1 > :

Consumption increases if the interest rate is higher than the time preference rate. The time preference rate of the individual (being represented by ) determines how to split the present value Wt of total income into current and future use. If the interest rate is su¢ ciently high to overcompensate impatience, i.e. if (1 ) (1 + r) > in the …rst line, consumption rises.

Note that even though we computed the condition for rising consumption for our special utility function (2.2.1), the result that consumption increases when the interest rate exceeds the time preference rate holds for more general utility functions as well. We will get to know various examples for this in subsequent chapters.

Under what conditions are savings positive?

Savings are from the budget constraint (2.1.2) and the optimal consumption result (2.2.5) given by

s_t= w_t c_t = w_t w_t+ 1

1 + r_t+1w_t+1 = w 1

1 + r_t+1

where the last equality assumed an invariant wage level, wt = wt+1 w. Savings are positive if and only if

s_t > 0, 1 >

1 + r_t+1 , 1 + r^t+1 >

1 ,

r_t+1 > 2 1

1 , r^t+1 > (2.2.14)

This means that savings are positive if interest rate is larger than time preference rate.

Clearly, this result does not necessarily hold for wt+1 > w_t: