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Chapter 17
CAPITAL MARKETS
Lee, Junqing Department of Economics , Nankai University
CONTENTS Capital Determination of Rate of Return The Firm’s Demand for Capital Present Discounted Value Optimal Resource Allocation Over Time
Capital
Lee, Junqing Department of Economics , Nankai University
Capital
The capital stock of an economy is the sum total of machines, buildings, and other reproducible resources in existence at a point in time these assets represent some part of the
economy’s past output that was not consumed, but was instead set aside for future production.
Present “sacrifice” for future “gain”
Lee, Junqing Department of Economics , Nankai University
Rate of Return
Time
Consumption
t1 t2 t3
c0
s
At period t1, a decision is made to hold s from current consumption for one period
s is used to produce additional output only in period t2
xOutput in period t2 rises by x
Consumption returns to its long-run level (c0) in period t3
Lee, Junqing Department of Economics , Nankai University
Rate of Return
The single period rate of return (r1) on an investment is the extra consumption provided in period 2 as a fraction of the consumption forgone in period 1
11
s
x
s
sxr
Lee, Junqing Department of Economics , Nankai University
Rate of Return
Time
Consumption
t1 t2 t3
c0
s
At period t1, a decision is made to hold s from current consumption for one period
s is used to produce additional output in all future periods
yConsumption rises to c0 + y in all future periods
Lee, Junqing Department of Economics , Nankai University
Rate of Return
The perpetual rate of return (r) is the permanent increment to future consumption expressed as a fraction of the initial consumption foregone
s
yr
Lee, Junqing Department of Economics , Nankai University
Rate of Return
When economists speak of the rate of return to capital accumulation, they have in mind something between these two extremes a measure of the terms at which consumption
today may be turned into consumption tomorrow
Determination of Rate of Return
Lee, Junqing Department of Economics , Nankai University
Rate of Return andPrice of Future Goods
Assume that there are only two periods The rate of return between these two periods
(r) is defined to be
10
1
c
cr
Rewriting, we get
rc
c
1
1
1
0
Lee, Junqing Department of Economics , Nankai University
Rate of Return andPrice of Future Goods
The relative price of future goods (p1) is the quantity of present goods that must be foregone to increase future consumption by one unit
rc
cp
1
1
1
01
Lee, Junqing Department of Economics , Nankai University
Demand for Future Goods
An individual’s utility depends on present and future consumption
U = U(c0,c1)
and the individual must decide how much current wealth (w) to devote to these two goods
The budget constraint is
w = c0 + p1c1
Lee, Junqing Department of Economics , Nankai University
Utility Maximization
c0
c1
w
w/p1
w = c0 + p1c1
U0
U1
The individual will maximize utility by choosing to consume c0* currently and c1* in the next period
c0*
c1*
Lee, Junqing Department of Economics , Nankai University
Utility Maximization
The individual consumes c0* in the present period and chooses to save w - c0* to consume next period
This future consumption can be found from the budget constraint
p1c1* = w - c0*
c1* = (w - c0*)/p1
c1* = (w - c0*)(1 + r)
Lee, Junqing Department of Economics , Nankai University
Intertemporal Impatience
Individuals’ utility-maximizing choices over time will depend on how they feel about waiting for future consumption
Assume that an individual’s utility function for consumption [U(c)] is the same for both periods but period 1’s utility is discounted by a “rate of time preference” of 1/(1+) (where >0)
Lee, Junqing Department of Economics , Nankai University
Intertemporal Impatience
This means that
0 1 0 1( , ) ( )1
1( )U c c U c U c
Maximization of this function subject to the intertemporal budget constraint yields the Lagrangian expression
r
ccwccU
1),( 1
010L
Lee, Junqing Department of Economics , Nankai University
Intertemporal Impatience
The first-order conditions for a maximum are
0)(' 00
cUc
L
01
)('1
11
1
rcU
c
L
01
10
r
ccw
L
Lee, Junqing Department of Economics , Nankai University
Intertemporal Impatience
Dividing the first and second conditions and rearranging, we find
)('1
1)(' 10 cU
rcU
Therefore, if r = , c0 = c1
if r < , c0 > c1
if r > , c0 < c1
' 0
'' 0
U
U
Lee, Junqing Department of Economics , Nankai University
Effects of Changes in r
If r rises (and p1 falls), both income and substitution effects will cause more c1 to be demanded (same direct) unless c1 is inferior (unlikely)
This implies that the demand curve for c1 will be downward sloping
Lee, Junqing Department of Economics , Nankai University
Effects of Changes in r
c0
c1
w
w/p1
w = c0 + p1c1
U0
U1
The individual will maximize utility by choosing to consume c0* currently and c1* in the next period
c0*
c1*
Lee, Junqing Department of Economics , Nankai University
Effects of Changes in r
The sign of c0/p1 is ambiguous the substitution and income effects work in
opposite directions Thus, we cannot make an accurate prediction
about how a change in the rate of return affects current consumption
Lee, Junqing Department of Economics , Nankai University
Supply of Future Goods
An increase in the relative price of future goods (p1) will likely induce firms to produce more of them because the yield from doing so is now greater this means that the supply curve will be
upward sloping
Lee, Junqing Department of Economics , Nankai University
Equilibrium Price ofFuture Goods
c1
p1
S
D
c1*
p1*
Equilibrium occurs at p1* and c1*
The required amount of current goods will be put into capital accumulation to produce c1* in the future
Lee, Junqing Department of Economics , Nankai University
Equilibrium Price ofFuture Goods
We expect that p1 < 1 individuals require some reward for waiting
(uncertainty) capital accumulation is “productive”
sacrificing one good today will yield more than one good in the future (trees , wine and cheese)
Lee, Junqing Department of Economics , Nankai University
The Equilibrium Rate of Return
The price of future goods is
p1* = 1/(1+r) Because p1* is assumed to be < 1, the rate
of return (r) will be positive p1* and r are equivalent ways of measuring
the terms on which present goods can be turned into future goods
Lee, Junqing Department of Economics , Nankai University
Rate of Return & Real and Nominal Interest Rates
Both the rate of return and the real interest rate refer to the real return that is available from capital accumulation
The nominal interest rate (R) is given by
rate) inflation expected 1)(1(1 rR
)1)(1(1
eprR
Lee, Junqing Department of Economics , Nankai University
Rate of Return & Real and Nominal Interest Rates
Expansion of this equation yields
ee prprR 11
small, is that Assuming
epr
eprR
The Firm’s Demand for Capital
Lee, Junqing Department of Economics , Nankai University
The Firm’s Demand for Capital
In a perfectly competitive market, a firm will choose to hire that number of machines for which the MRP is equal to the market rental rate
Lee, Junqing Department of Economics , Nankai University
Determinants of Market Rental Rates
Consider a firm that rents machines to other firms
The owner faces two types of costs: depreciation on the machine
assumed to be a constant % (d) of the machine’s market price (p)
the opportunity cost of the funds tied up in the machine rather than another investment
assumed to be the real interest rate (r)
Lee, Junqing Department of Economics , Nankai University
Determinants of Market Rental Rates
The total costs to the machine owner for one period are given by
pd + pr = p(r + d) If we assume the machine rental market is
perfectly competitive, no long-run profits can be earned renting machines the rental rate per period (v) will be equal to the
costs
v = p(r + d)
Lee, Junqing Department of Economics , Nankai University
Nondepreciating Machines
If a machine does not depreciate, d = 0 and
v/p = r An infinitely long-lived machine is equivalent
to a perpetual bond and must yield the market rate of return
Lee, Junqing Department of Economics , Nankai University
Ownership of Machines
Firms commonly own the machines they use
A firm uses capital services to produce output these services are a flow magnitude
It is often assumed that the flow of capital services is proportional to the stock of machines
Lee, Junqing Department of Economics , Nankai University
Ownership of Machines
A profit-maximizing firm facing a perfectly competitive rental market for capital will hire additional capital up to the point at which the MRPk is equal to v under perfect competition, v will reflect both
depreciation costs and the opportunity costs of alternative investments
MRPk = v = p(r+d)
Lee, Junqing Department of Economics , Nankai University
Theory of Investment
If a firm decides it needs more capital services that it currently has, it has two options: hire more machines in the rental market purchase new machinery - inverse to real rate of
interest called investment
MRPk = v = p(r+d)
Present Discounted Value
Lee, Junqing Department of Economics , Nankai University
Present Discounted Value
When a firm buys a machine, it is buying a stream of net revenues in future periods it must compute the present discounted value of
this stream Consider a firm that is considering the
purchase of a machine that is expected to last n years it will provide the owner monetary returns in
each of the n years
Lee, Junqing Department of Economics , Nankai University
Present Discounted Value
The present discounted value (PDV) of the net revenue ( marginal revenue products) flow from the machine to the owner is given by
nn
r
R
r
R
r
RPDV
)1(...
)1(1 221
If the PDV exceeds the price of the machine, the firm should purchase the machine
Lee, Junqing Department of Economics , Nankai University
Present Discounted Value
In a competitive market, the only equilibrium that can prevail is that in which the price is equal to the PDV of the net revenues from the machine
Thus, market equilibrium requires that
1 22
...1 (1 ) (1 )
nn
RR RPDV
r rP
r
Lee, Junqing Department of Economics , Nankai University
Simple Case
Suppose that machines are infinitely long-lived and the MRP (Ri) is the same in every year
Ri = v in a competitive market Therefore, the PDV from machine
ownership is
...)1(
...)1(1 2
nr
v
r
v
r
vPDV
Lee, Junqing Department of Economics , Nankai University
Simple Case
This reduces to
...
)1(
1...
)1(
1
1
12 nrrr
vPDV
1
1
r
rvPDV
rvPDV
1
Lee, Junqing Department of Economics , Nankai University
Simple Case
In equilibrium P = PDV so
rvP
1
or
P
vr
Lee, Junqing Department of Economics , Nankai University
General Case
We can generate similar results for the more general case in which the rental rate on machines is not constant over time and in which there is some depreciation
Suppose that the rental rate for a new machine at any time s is given by v(s)
The machine depreciates at a rate of d
Lee, Junqing Department of Economics , Nankai University
General Case
The net rental rate of the machine will decline over time
In year s the net rental rate of an old machine bought in a previous year (t) would be
v(s) e -d(s-t)
Lee, Junqing Department of Economics , Nankai University
General Case
If the firm is considering the purchase of the machine when it is new in year t, it should discount all of these net rental amounts back to that date
The present value of the net rental in year s discounted back to year t is
e -r(s-t) v(s)e -d(s-t) = e(r+d)tv(s)e -(r+d)s
Lee, Junqing Department of Economics , Nankai University
General Case
The present discounted value of a machine bought in year t is therefore the sum (integral) of these present values
dsesvetPDV sdr
t
tdr )()( )()(
In equilibrium, the price of the machine at time t [p(t)] will be equal to this present value
dsesvetp sdr
t
tdr )()( )()(
Lee, Junqing Department of Economics , Nankai University
General Case
Rewriting, we get
Differentiating with respect to t yields:
t
sdrtdr dsesvetp )()( )()(
t
tdrtdrsdrtdr etvedsesvedrdt
tdp )()()()( )()()()(
)()()()(
tvtpdrdt
tdp
Lee, Junqing Department of Economics , Nankai University
General Case
This means that
l g
( ) ( ) ( )
( ) ( ) (
(
( ))
)
capita ain
dp t
dtdp t
v t r d p t
r d p t v tdt
dp(t)/dt represents the capital gains that accrue to the owner of the machine
Lee, Junqing Department of Economics , Nankai University
Cutting Down a Tree
Consider the case of a forester who must decide when to cut down a tree
Suppose that the value of the tree at any time t is given by f(t) [where f’(t)>0 and f’’(t)<0] and that l dollars were invested initially as payments to workers who planted the tree
Lee, Junqing Department of Economics , Nankai University
Cutting Down a Tree
When the tree is planted, the present discounted value of the owner’s profits is
PDV(t) = e-rtf(t) - l The forester’s decision consists of choosing
the harvest date, t, to maximize this value
0)()(')(
tfretfedt
tdPDV rtrt
Lee, Junqing Department of Economics , Nankai University
Cutting Down a Tree
Dividing both sides by e-rt,
f’(t) – r f(t)=0 Therefore,
)(
)('
tf
tfr
Note that l drops out (sunk cost) The tree should be harvested when r is
equal to the proportional growth rate of the tree
The proportional rate
of growth of the tree
Lee, Junqing Department of Economics , Nankai University
Cutting Down a Tree
Suppose that trees grow according to the equation
tetf 4.0)(
If r = 0.04, then t* = 25(before 25 ,don’t cut ; but after 25, cut)
If r rises to 0.05, then t* falls to 16
ttf
tf 2.0
)(
)('
Optimal Resource Allocation Over Time (omitted)
Lee, Junqing Department of Economics , Nankai University
Optimal Resource Allocation Over Time
Two variables are of primary interest for the problem of allocating resources over time the stock being allocated (k)
the capital stock a control variable (c) being used affect
increases or decreases in kthe savings rate or total net investment
Lee, Junqing Department of Economics , Nankai University
Optimal Resource Allocation Over Time
Choices of k and c will yield benefits over time to the economic agents involved these will be denoted U(k,c,t)
The agents goal is to maximize
T
dttckU0
),,(
where T is the decision time period
Lee, Junqing Department of Economics , Nankai University
Optimal Resource Allocation Over Time
There are two types of constraints in this problem the rules by which k changes over time
),,( tckfkdt
dk
initial and terminal values for k
k(0) = k0
k(T) = kT
Lee, Junqing Department of Economics , Nankai University
Optimal Resource Allocation Over Time
To find a solution, we will convert this dynamic problem into a single-period problem and then show how the solution for any arbitrary point in time solves the dynamic problem as well we will introduce a Lagrangian multiplier (t)
the marginal change in future benefits brought about by a one-unit change in k
the marginal value of k at the current time t
Lee, Junqing Department of Economics , Nankai University
A Mathematical Development
The total value of the stock of k at any time t is given by (t)k
The rate of change in this variable is
kkdt
dK
dt
dk
dt
ktd )(
The total net value of utility at any time is given by
kktckUH ),,(
Lee, Junqing Department of Economics , Nankai University
A Mathematical Development
The first-order condition for choosing c to maximize H is
0),,(
c
k
c
tckU
c
H
Rewriting, we get
0
c
k
c
U
Lee, Junqing Department of Economics , Nankai University
A Mathematical Development
For c to be optimally chosen: the marginal increase in U from increasing c is
exactly balanced by any effect such an increase has on decreasing the change in the stock of k
Lee, Junqing Department of Economics , Nankai University
A Mathematical Development
Now we want to see how the marginal valuation of k changes over time need to ask what level of k would maximize H
Differentiating H with respect to k:
0),,(
k
k
k
tckU
k
H
Lee, Junqing Department of Economics , Nankai University
A Mathematical Development
Rewriting, we get
k
k
k
U
Any decline in the marginal valuation of k
must equal the net productivity of k in
either increasing U or increasing
k
Lee, Junqing Department of Economics , Nankai University
A Mathematical Development
Bringing together the two optimal conditions, we have
0
c
k
c
U
c
H
0
k
k
k
U
k
H
These show how c and should evolve over time to keep k on its optimal path
Lee, Junqing Department of Economics , Nankai University
Exhaustible Resources
Suppose the inverse demand function for a resource is
p = p(c)
where p is the market price and c is the total quantity consumed during a period
The total utility from consumption is
c
dxxpcU0
)()(
Lee, Junqing Department of Economics , Nankai University
Exhaustible Resources
If the rate of time preference is r, the optimal pattern of resource usage will be the one that maximizes
T
rt dtcUe0
)(
Lee, Junqing Department of Economics , Nankai University
Exhaustible Resources
The constraints in this problem are of two types: the stock is reduced each period by the level
of consumption
ck
end point constraints
k(0) = k0
k(T) = kT
Lee, Junqing Department of Economics , Nankai University
Exhaustible Resources
Setting up the Hamiltonian
kcUekkUeH rtrt
)()(
yields these first-order conditions for a maximum
0
c
Ue
c
H rt
0
k
H
Lee, Junqing Department of Economics , Nankai University
Exhaustible Resources
Since U/c = p(c),
e-rtp(c) = The path for c should be chosen so that the
market price rises at the rate r per period
Lee, Junqing Department of Economics , Nankai University
CONTENTS Capital Determination of Rate of Return The Firm’s Demand for Capital Present Discounted Value Optimal Resource Allocation Over Time
Lee, Junqing Department of Economics , Nankai University
Important Points to Note:
Capital accumulation represents the sacrifice of present for future consumption the rate of return measures the terms at
which this trade can be accomplished
Lee, Junqing Department of Economics , Nankai University
Important Points to Note:
The rate of return is established through mechanisms much like those that establish any equilibrium price the equilibrium rate of return will be
positive, reflecting both individuals’ relative preferences for present over future goods and the positive physical productivity of capital accumulation
Lee, Junqing Department of Economics , Nankai University
Important Points to Note:
The rate of return (or real interest rate) is an important element in the overall costs associated with capital ownership it is an important determinant of the market
rental rate on capital (v)
Lee, Junqing Department of Economics , Nankai University
Important Points to Note:
Future returns on capital investments must be discounted at the prevailing real interest rate use of present value provides an
alternative way to study a firm’s investment decisions
Lee, Junqing Department of Economics , Nankai University
Important Points to Note:
Capital accumulation (and other dynamic problems) can be studied using the techniques of optimal control theory these models often yield competitive-type
results
Chapter 17
CAPITAL MARKETS
END
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