Topic 1: Finance Theory Under Certainty (One period model) – Copeland, Weston, Shastri – Chapter 1
How do capital markets benefit society? Consider a one-period economy. Without capital markets and without production, individuals are constrained to either (1) consume their particular initial endowments, y, y, or (2) store 01
all of, or a portion of, their time 0 endowment (at a zero interest rate) for consumption at time 1.
Where current endowments are low, but where future endowments will be high, this could cause hardship. Capital markets allow these individuals to borrow against their anticipated future income (endowments). They borrow from other individuals who lend excess current income (endowments) at positive interest rates. Why does the interest rate have to be positive?
The investment/consumption decision depends on the individual's subjective preference for consumption across the different time periods, opportunities for investment in productive opportunities, and the market interest rate, r.
To analyze this decision, we will consider Irving Fisher's Theory of the Real Interest Rate (1930).
What we get from the analysis:
1) How capital markets lead to an efficient allocation of resources to investment projects;
2) A foundation for net present value rule;
3) Fisher Separation Theorem.
1) One period world (today and end of the period).
3) Perfect capital markets (i.e., no taxes, transaction costs, etc,)
4) Initial wealth endowment (y and y) 01
5) Individual preferences for consumption today (t = 0) vs. consumption at the end of the period (t = 1) are a
function of their utility function, U(C, C) and associated indifference curves 01
C0The slope of an individual’s indifference curve at a particular point (C, C) is denoted as . MRS01C1
？1~?~?；UCC；UCC(,)(,)C01010?(?(MRS！？We will see later that . Further assume that individuals C?(?(1；C；C?)?)01
~?~?；U(C,C)；U(C,C)0101?(?(prefer more consumption to less, and , and the marginal utility of ，0，0?(?(；C；C01?)?)
22~?~?；U(C,C)；U(C,C)0101?(?(consumption is negative, and . We will talk more about utility ，0，022?(?(；C；C01?)?)
theory in chapter three. What do these utility functions look like graphically?
6) Market determined interest rate for borrowing and lending, r. Let T(C,C) be the transformation function 01
relating t = 0 and t = +1 consumption opportunities. The lower the consumption this period, the higher the
investment (at interest rate r) and, therefore, the higher the consumption next period. The marginal rate of
C0transformation at a particular point, for example at (C, C), is denoted as . With borrowing and 01MRTC1
C0lending at a fixed interest rate r, , for all possible C,C. MRT！？(1？r)01C1
7) Firms with productive capacity, defined by the marginal rate of transformation and initial capital. Let
T(P,P) be the transformation function that relates dividend payments (production) today with dividend 01
payments (production) next period. The lower the dividend this period, the higher the t = 0 investment.
This results in a higher dividend payment (production) next period. First, a graphical presentation
In each case, note how the individual’s budget constraint (i.e., opportunity set) expands (or shrinks). Also, note how the individual’s utility from consumption changes as the budget constraint changes.
Case 1: Individual endowed y, y, no storage, no capital markets, no production. Note the amount of “utility” 01
the individual has at this point.
Case 2: Individual endowed y, y, storage, no capital markets, no production. The tangency point of the 01
individual’s indifference curve with the “storage” line indicates preferred location for individual.
A. Who would want to store?
B. How much better off is this individual?
C. Who wouldn’t want to store?
Case 3: Individual endowed y, y, storage, capital markets, no production. The “capital market line” reflects the 01
interest rate and has a constant slope of -(1+r).
A. What do the X and Y intercepts signify?
B. Let’s graphically identify an individual’s preferred location on the capital market line. As before,
notice that the individual’s indifference curve is tangent to the capital market line at this preferred
location. What does this mean? That is, at what rate is the individual indifferent to exchanging time 0
and time 1 dollars at this point?
C. Where would two different individuals, with difference time preferences, locate on the capital market
D. Important point: at their respective tangency points, what rate are these individuals indifferent to
exchanging time 0 and time 1 dollars?
E. What is the amount of lending / borrowing at time 0 for these two individuals?
F. How is consumption at time 1 affected by the decision to borrow or lend at time 0?
G. In sum, how does the existence of capital markets (i.e., the ability to borrow / lend at some positive
interest rate r) benefit individuals? Is anyone worse off with the introduction of capital markets? Is
everyone better off?
Case 4: Individual (100% owner of a firm) endowed with y, y, with productive investment opportunities, but 01
no capital markets. The curved line is called the “production possibility frontier” or “investment opportunity schedule.” Let’s assume that projects are infinitely divisible and independent, ordered from best (highest return) to worst (lowest return), resulting in smooth curved line. The initial endowment is the lower right point of the curve. Movement up and to the left is investment in the firm’s projects. How is this case different (the same as) case 3?
A. How do the different owners’ time preferences affect the investment decisions of their two firms?
B. What is the marginal rate of production at the owners’ preferred locations? What do we call the
marginal rate of production in an undergraduate finance class?
C. At their respective tangency points, what rate are these individuals indifferent to exchanging time 0
and time 1 dollars?
Case 5: Individual (100% owner of a firm) endowed y, y, with productive investment opportunities, and 01
A. How much better off is the consumer in case 5 than in case 4?
B. Examine the graph. Be able to point out:
Initial endowment (y, y) 01
Dividend payments by the firm (P, P) 01
Investment by the firm
Rate of return on the last “dollar” invested
PV and FV of consumer’s wealth at initial endowment
PV and FV of dividend payments
NPV of investment
Preferred location by consumer on the capital market line (and PV and FV of this preferred location)
Amount of borrowing or lending at time 0 (difference between C and P) and associated adjustment to 00
time 1 consumption (difference between C and P) 11