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 t 1 t 2 t 3 Co At time period t 1 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 t 2, and this is given as  C1 At time period t 1 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 t 2, 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 t 1 to only consume in the next period t 2 Let’s look at a long term view time consumption  Co  C1 t 1 t 2 t 3 Co At time period t 1 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 t 1 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, r 1 = (  C1 -  Co) /  Co= (  C1/  Co)- 1 If  C1 >  Co, then r 1 > 0 An example: If the consumer withholds 100 and this is put to use to yield 110 in the next period, then r 1 = 110/100 – 1, which is equal to 0.10 or 10% Or (1 + r 1 ) = 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) – 1 Or,  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

P 1 = the price of future goods = the quantity of present goods that must be foregone to increase future consumption by 1 unit P 1 =  Co/  C1 = 1/(1 + r) or P 1 =  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, P 1 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 + P 1 C1 is the budget constraint, indicating that wealth = current consumption, Co, plus the value of future consumption, P 1 C1

What does the consumer’s intertemporal problem look like? U1U1 U2U2 U3U3 Future Consumption C1 Current Consumption Co W/P 1 W C1* Co* W = Co + P 1 C1 Intertemporal utility or Indifference curves At the tangency of U 1 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 + P 1 C1, and given the equilibrium Co and C1 as Co* and C1*, we get P 1 C1* = W – Co*, or C1* = (W – Co*)/P 1, which is equal to (W – Co*)(1 + r), since P 1 = 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 P 1 = 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) = 0 Where U΄(Co) = ∂U/∂Co, and U΄(C1)=∂U/∂C1 We 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) C1

So, whether C1 >, 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 P 1 = 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 P 1 P 1 * C 1 * What about supply? It is simple (although capital accumulation is not simple) -- as P 1 increases, the increase induces firms to produce more, since the yield from doing so is now greater --- P 1 is probably < 1

The equilibrium price of future goods P 1 * = 1/(1+r) So as P 1 * 0 If P 1 * = 0.9, then r is approximately 0.11 or 11%

Rate of return, real rate, and nominal rate R = nominal rate 1+ R = (1 + r)(1 + P ε ), where P ε = 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, P ε

What this means is that: 1+R = 1 + r + P ε + r P ε If r P ε is small, then R = r + P ε If the real rate, r, is 0.04 (4%) and expected inflation is 0.10 (10%), then the nominal rate R = = 0.14 or 14%