Chapter 17 Capital Markets

Transcription

1 Chapter 7 Capital Markets Capital stock is the total of all machines, buildings, and other manufactured, nonlabor resources that are in existence. It represents some part of the economy s output in the past that was not consumed but was set aside to be used to produce future goods for consumption. Present sacrifice (disutility) for future gain (utility) is an essential aspect of capital accumulation. Single-Period Rate of Return The amount earned on the set aside consumption. All that is set aside in t is consumed in t. x s 6 Single-period Rate of Return, r eg.,.6 s Where r = the single-period rate of return; s = amount set aside in t (decrease in consumption in t ); x = increase in consumption in t. r will be positive if x >s (if consumption in t increases by more than consumption in t decreases). x s x r, s s The above also implies x r s s, eg.,.6.6 Eg., 6 =.6() +

2 Perpetual Rate of Return If the amount set aside in t raises consumption in all subsequent periods (t t ), then r = y/s, where y is the permanent increase in consumption for 6 each time period. r will be > if y >. We usually refer to some idea between the single period and the perpetual rate when we speak of the rate of return to capital accumulation. Determination of Rate of Return The equilibrium r is established through the interaction of supply and demand. Demand: eg., For a two-period model, the rate of return is defined as: c c or r, c or c r, eg.,.94.6 To get one more unit of consumption in t, you give up.94 unit now in t. c / c is the amount of consumption that must be foregone in t to increase consumption in t by one unit. Thus, it is the price (or cos of future consumption in today s terms (in terms of today s goods). c The price of future goods is p.94. c r.6 It is the quantity of present goods that must be foregone to increase future consumption by one unit..6 r x s s x c s c

3 Demand for Future Goods Individual utility is a function of today s (t ) consumption (c ) and next period s (t ) consumption (c ); U = U(c, c ). Utility is maximized by allocating current wealth (W) to c and c. No income is acquired in t and all wealth is available in t (resources). The c c. r budget constraint is: W c pc c c In this constraint, p r =. It takes one unit of W to get one unit of c but only p unit of W to get one unit of c. c If all wealth is allocated to c, consumption will be W in t and zero in t. p decreases, r increases W W ( r) p c * If all wealth is allocated to c, consumption will be W/p = W(+r) in t and zero in t. That is, if all W is saved and invested, wealth in t will be W/p = W(+r). c * U 2 U WU c If we change r, which changes p, we can observe the change in c and develop a demand function for c. If r increases, p decreases and if r decreases, p increases because p = /+r. Therefore, if r increases (p decreases), c will increase because of the substitution and income (c normal) effects. Thus, the demand for c slopes downward. A decrease in p (increase r) will increase real wealth and cause substitution of c (increase) for c (decrease). Thus, income and substitution effects are both negative (unless c is inferior, which is unlikely). An increase in r (decrease in p ), causes more to be consumed in t. However, the cross-price effect, the effect of p on c, is not clear because income and substitution effects are opposite in sign. A decrease in p causes c to decrease from the substitution effect, but c p r low c increases because of the income effect (c r high? normal good). p c p D c

4 Supply of Future Goods If c is to be larger, firms must bid c away from consumers in t. Firms must pay higher and higher costs to induce consumers to give up more and more c. Thus, MC increases as c increases, so supply of future goods is upward sloping. p p * =/(+r*) S C D When there is no excess demand or excess supply of future goods (equilibrium), the required amount of current consumption will be foregone to yield c *. Individuals will only forego current consumption if C * some gain (r>) is achieved. Sacrificing one unit now will produce more than one unit in the future. Also individuals require some reward for waiting. Because p * <, r * >. Once p * is known, r * = (- p * )/ p *. r * is equivalent to the real interest rate because it is the opportunity cost of lending money (the income foregone from not investing in real capital). The nominal rate of interest is: R = r + p e, where p e is the rate of inflation.

5 Firm s Demand for Capital The firm will use that amount of capital for which MRP k = v. This assumes perfect competition in the capital market. We discussed this at length earlier. The question now is, how is v determined? Assume a rental firm owns a machine with a current market price of p. It depreciates at d per period. Also assume the real interest rate is r. Then, the rental rate on the machine per period will have to be at least pd + pr. But, if the rental market is perfectly competitive, v = pd + pr (means profit = ), because in a perfectly competitive capital market, the rental rate will be bid down to v by competition among sellers of the machinery services. If v > pd + pr, sellers of the machinery services will be attracted and will bid v down to pd + pr (and Vice Versa for buyers). The rental rate per period on the owned machine equals depreciation plus interest foregone. More commonly, firms own rather than rent the machines that they use. However, if we view the firm as being able to rent out the machines that it owns, then v is the opportunity cost of using the machine in the firm s own operation. Thus, the cost of capital is still v (implicit cos. The amount of capital demanded is where MRP k = v = pd + pr, where v is the rental rate per period. This analysis assumes storage, insurance and taxes are zero. If d = (no depreciation; perpetual capital), v = pr or r = v/p or p = v/r.

6 Investment (purchasing/ownership) The firm increases the amount of capital it uses (unless it rents) by purchasing new capital equipment. Investment is inversely related to r (real interest rate) because a fall in r will reduce v. A lower v will lead to increased use of capital so that the relationship MRP k = v = pr is maintained. An increase in r causes an increase in v and a decrease in capital investment, while a decrease in r causes a decrease in v and an increase in capital investment. Present Discounted Value Present Discounted Value is an alternative way of viewing investment decisions; whether to purchase new capital or not. The earlier analysis said invest if MRP k > v = pd + pr. The fundamental notion of Present Discounted Value is that $ today is worth more than $ in the future. If $ is available today, it can be invested and be worth $(+r) one period from now. (Also, uncertainty about the future leads to discounting; Bird in Hand. ) Today s value of $ one period from now is the amount that $ would be worth today. The Present Discounted Value (PDV) of $ one period from now is PDV = $/(+r), or $ next period is worth $/(+r) today, or $ today is worth $(+r) tomorrow. This formula can be generalized to any number of dollars and any number of years. For two years, PDV($) = $/(+r) 2. For a one-time payment of $N nyears from now, PDV($N) = $N/(+r) n, where $N is the number of dollars and n is the number of years in the future. As n increases, PDV decreases and as r increases, PDV decreases.

7 Stream of Revenues When a firm invests in capital equipment, it foresees a series of years in the future, each containing a net revenue. So, the firm must compute the PDV of the net revenue stream. R R2 Rn PDV... 2 n r ( r) ( r) If the PDV of the machine s net revenue stream (net of all operating costs) is greater than the machine s current price, it will benefit the firm to invest in the machine; if PDV > p invest! If PDV < p, do not invest because the firm could earn more in the alternative investment represented by a rate of return of r. Thus, if PDV > p, many firms will invest in the machine and competition among buyers will drive the price (p) of the machine up until PDV = p. Also, if PDV < p, no firms will invest in the machine and competition among sellers will drive the price of the machine (p) down until PDV = p. Therefore, the only equilibrium in the long run in perfect competition is where PDV = p = v/r or v = pr = MRP k = rpdv assuming d =.

8 Proof that v and PDV give the same result Earlier we showed that v = pd + pr and v =pr (if d = ) would be the equilibrium rental rate on a machine. If the net revenue in any give period, R i, equals the rental rate in the period; v i = R i = MRP ki, (at optimality) then, for a machine that lasts forever (d=), If all v s are the same, PDVv... v 2 r ( r) ( r) ( r ) Because in equilibrium p = PDV, PDV = p = v/r. v v v So p r pr, r p v v2 PDV r ( r) 2 v... ( r) r v v r r which is the equilibrium shown above where d =. Two approaches give same outcome! v r.

9 General Case (Buy in period t; time is continuous; d ; and v i varies) p( PDV( t e ( rd ) t v( s) e ( rd ) s where: p( is the machine s purchase price in period t = PDV. e = 2.72 (the base of natural logarithms). r = real interest rate per period. d = rate of depreciation per period. v(s) = rental rate for the machine in period s if the machine were new (undepreciated). (s is the number of years over which the machine has been decaying. So, v(s) can vary over time and there can be some depreciation (d) which occurs at an exponential rate. Because the formula uses an integral, time is assumed to be continuous. ds Differentiate the above expression with respect to t to get, dp( dt ( r d) p( v(, so v( ( r d) p( dp(. dt This equation is the same as before except the last term represents capital gains accruing to the owner of the machine.