ECONOMIC FOUNDATIONS OF FINANCE

BASIC ELEMENTS OF THE THEORY OF CAPITAL

CAPITAL ACCUMULATION AND RATE OF RETURN

Let Co = current consumption and C1 = future consumption

Then, let’s view consumption over time

time

consumption

Co

C1

t1 t2 t3

Co

At time period t1 a consumer withholds current consumption Co, in the amount Co and puts this amount to use to produce future consumption ---- So there is a rate of return that is derived on the withholding of Co

Here, all withheld Co is used to produce consumption in time period t2, and this is given as C1

At time period t1 a consumer withholds current consumption Co, in the amount Co and puts this amount to use to produce future consumption ---- So there is a rate of return that is derived on the withholding of Co

Here, all withheld Co is used to produce consumption in time period t2, and this is given as C1

The rate of return is given by (C1/Co) – 1 = r

CAPITAL ACCUMULATION AND RATE OF RETURN

In the previous case, the graph showed that the consumer withheld Co in t1 to only consume in the next period t2

Let’s look at a long term view

time

consumption

Co

C1

t1 t2 t3

Co

At time period t1 the consumer withholds current consumption Co, in the amount Co and puts this amount to use to produce perpetual future consumption

So Co is here used to get a perpetual C1

The rate of return now is

r∞ = (C1/Co) – 1

At time period t1 the consumer withholds current consumption Co, in the amount Co and puts this amount to use to produce perpetual future consumption

So Co is here used to get a perpetual C1

The rate of return now is

r∞ = (C1/Co) – 1

THE SINGLE PERIOD RATE OF RETURN

The single period rate of return is given by, r1 = (C1 - Co) /Co= (C1/Co)- 1

If C1 > Co, then r1 > 0

An example: If the consumer withholds 100 and this is put to use to yield 110 in the next period, then r1 = 110/100 – 1, which is equal to 0.10 or 10%

Or (1 + r1 ) = 110/100

THE PERPETUAL RATE OF RETURN

The consumer withholds current period consumption as Co and gets a yield of C1 perpetually

So r∞ = (C1/Co) - 1

What has happened is that the consumer gets Co + y, where y is some perpetual amount each period --- so r∞ = y/Co

The actual rate of return on capital accumulation given some sacrifice in present consumption is somewhere between the single period rate and the perpetual rate

The equilibrium rate of return

We now introduce demand and supply in the market for the future goods to obtain an equilibrium rate of return

We assume a 2-period analysis – now and then, or current and future

The 1-period rate is r = (C1/Co) – 1Or, C1/Co = 1 + r

So we now have how much Co has to be foregone to get an increase of 1 unit of future consumption, C1

This is the relative price of 1 unit of C1 in terms of Co

This is the price of future goods

P1 = the price of future goods = the quantity of present goods that must be foregone to increase future consumption by 1 unit

P1 = Co/C1 = 1/(1 + r)

orP1 = Co = 1/(1 + r) C1

Now we need to consider the demand for future goods

Let the utility of current consumption, Co, vs. future consumption, C1, be given by some utility function, U( Co,C1)

The consumer’s problem is to allocate wealth, given here as W, to the two goods, Co, and C1

Wealth, W, not spent on Co can be invested at a rate, r, to obtain C1 next period

Therefore, P1 reflects the present cost of future consumption

There is a budget constraint that imposes constraint on the consumer maximizing utility over Co and C1

W = Co + P1C1 is the budget constraint, indicating that wealth = current consumption, Co, plus the value of future consumption, P1C1

What does the consumer’s intertemporal problem look like?

U1

U2

U3

Future Consumption C1

Current Consumption Co

W/P1

W

C1*

Co*

W = Co + P1C1

Intertemporal utility or Indifference curves

At the tangency of U1 and the budget constraint, W, we get equilibrium consumption of Co, as Co*, and equilibrium future consumption, C1*

The consumer maximizes intertemporal utility over current and future consumption given the budget constraint, which is the limit on wealth

From the budget constraint, W = Co + P1C1, and given the equilibrium Co and C1 as Co* and C1*, we get P1C1* = W – Co*, or C1* = (W – Co*)/P1 , which is equal to (W – Co*)(1 + r), since P1 = 1/(1+r)

Intertemporal Utility

The utility here (intertemporal utility) depends on how people feel about future consumption relative to present consumption (their impatience)

This impatience is reflective of a consumer’s intertemporal time preference rate, say,

Generally future utility is discounted by a rate of time preference, say of 1/(1+ ), for > 0

If we assume that utility is also separable, we get U( Co,C1) = U(Co) + 1/(1 + )U(C1)

Maximize Utility

Therefore, we maximize Utility in mathematical form by using the classical optimization mathematics or the Lagrangian method of constrained maximization that we learned in Calculus

Max U(Co) + 1/(1+)U(C1), subject to the wealth constraint, W = Co + C1/(1+ r), because P1 = 1/(1 + r)

The Lagrangian with the objective function, Max U(Co) + 1/(1+)U(C1), and constraint, W = Co + C1/(1+ r) is:

L = U(Co) + 1/(1+)U(C1) + λ[W – Co – C1/(1+r)

Then, we find the first order conditions for a maximum, by taking the derivatives of the L with respect to Co, C1 and λ (with λ being the Lagrangian multiplier reflecting the effect of the constraint in the constrained optimization problem)

We set these derivative equal to zero, that is we find the zero slope conditions which indicate an optimization

∂L/∂Co = U΄(Co) – λ = 0∂L/∂C1 = 1/(1+)U΄(C1) – λ/(1+r) = 0∂L/∂ λ = W = Co – C1/(1+r) = 0Where U΄(Co) = ∂U/∂Co, and

U΄(C1)=∂U/∂C1We can divide the first two derivatives,

and rearrange to get U΄(Co) = ((1+r)/(1+)) U΄(C1)

Therefore, Co = C1 if r = , or this says that Co = C1 if interest rate = time preference rate

Co > C1 if > r, or if the time preference rate is greater than the interest rate, that is to get U΄(Co) < U΄(C1) it requires Co > C1

So, whether C1 >, < Co depends on the time preference rate, , for present goods, where > 0

But one may consume more in the future if the interest rate, r, on savings or investment is high enough

If is the same for two persons, say A and B, but person A gets higher return, r, than does B, U΄(Co)/ U΄(C1) for A is greater than U΄(Co)/ U΄(C1) for B

Hence, Co/C1 will be lower for A than for B

What happens if r changes?As r increases, this induces a lower price

of C1, since P1 = 1/(1+r), and consumption of C1 increases

So we would get a downward sloping demand for future goods, C1 (but this is not unambiguously the case since there are other complications that are discussed in advanced microeconomics)

The equilibrium in the market

Supply

Demand

P1

P1*

C1*

What about supply? It is simple (although capital accumulation is not simple) -- as P1 increases, the increase induces firms to produce more, since the yield from doing so is now greater --- P1 is probably < 1

The equilibrium price of future goods

P1* = 1/(1+r)

So as P1* < 1, r >0

If P1* = 0.9, then r is approximately 0.11 or 11%

Rate of return, real rate, and nominal rate

R = nominal rate1+ R = (1 + r)(1 + Pε), where Pε =

increase in overall prices or the inflation rate

A lender here, would expect to be compensated for both the opportunity cost of not investing at rate, r, and for the general rise in prices, Pε

What this means is that:1+R = 1 + r + Pε + r Pε

If r Pε is small, then R = r + Pε

If the real rate, r, is 0.04 (4%) and expected inflation is 0.10 (10%), then the nominal rate R = 0.04 + 0.10 = 0.14 or 14%