Download THE DEMAND FOR CAPITAL

THE DEMAND FOR
CAPITAL

Firms demand capital as an input into the
production process
 When the firm hires a machine they pay the
renter of the machine some capital rental price
– call this rental price v
 The rental has to recoup the price (or value) of
the machine times the depreciation of the
machine and the price of the machine times the
opportunity cost of having money tied up in the
machine

v = Pd + Pr = P(d + r)
 v is then the market rental rate
 P = the value of the machine
 d = depreciation costs
 r = the interest rate that could be earned
on the money tied up in the machine

If the firm owns the machine, there is still
the depreciation costs of the machine as
well as the opportunity cost of having
funds tied up in the machine
 So again, these are related to the value of
the machine, and the cost of the capital
input is again v = P(d + r)
THE CASE OF NO
DEPRECIATION

If d = 0, then v = Pr is the market rental
rate
 Or, we can look at this in another way as
r = v/P --- the opportunity cost of the
funds tied up in the machine is the
market rental rate divided by the value of
the machine
PROFIT MAXIMIZATION
AND THE DEMAND FOR
CAPITAL

Our background in profit maximization
from microeconomic theory – or from
managerial economics suggests that a
firm will employ inputs and produce
outputs up to the point where marginal
revenue product = marginal cost
Profit maximization

But this just indicates that a firm will produce
up to the point where marginal revenue =
marginal cost
 If the firm is selling in a perfectly competitive
market, then marginal revenue product is just
equal to P•mp, which is Price of the product, P,
multiplied by the marginal product, mp of any
particular input, such as capital
Profit maximization

The marginal cost is the marginal cost of
the input, which in this case we have
designated as the market rental, v, of the
machine, or whatever capital item we are
using as input
 Hence, P•mpK = v, for K = capital
 We can just express this as PmpK = v
 mpK = the marginal product of capital
The firm’s profit
maximization problem

The firm uses input and produces output
via a production function, which here we
will just refer to as Q = f(L, K), where
f(L, K) is some function that combines
the inputs that we have called here labor,
L, and a capital item, K, and produces
output, Q --- There may be multiple
outputs, but we just restrict ourselves to
Q
The firm’s profit
maximization problem

If the firm is operating in a perfectly
competitive market environment, then
the firm’s problem is maximize profit, ,
given the production function and the
given market prices of the inputs
 Max  = Pf(L, K) – wL – vK
 Here, w is the market wage rate, and v is
the capital rental rate as discussed earlier
Finding the 1st order
conditions for maximization

We find the 1st-order conditions by taking
the derivative of profit, , with respect to
each of the inputs and setting this
derivative equal to zero --- again the zeroslope conditions
The 1st order conditions





/L = P f/ L – w = 0
/K = P f/ K – v = 0
Notice that f/ K is the change in production,
f(L, K) = Q, that the firm gets from a marginal
change in capital, K, and similarly for labor
f/ K = the marginal product of capital
Similarly f/ L = marginal product of labor
Solve the 1st order
conditions

Currently we are interested in capital, K
 /K = P f/ K – v = 0
 So P f/ K = v
 or PmpK = v -- marginal revenue
product = the rental price of capital
 Similarly, PmpL = w
The relationship of capital
rental to price of product
and marginal product of
capital

Since PmpK = v -- marginal revenue
product = the rental price of capital
 If the price of the product, Q, increases
given a level of marginal productivity of
the capital, then v = the rental price of
capital increases
 Similarly, if the marginal product of
capital increases, v increases
Demand for Capital

If we solve for capital, K, from the condition,
PmpK = v, we would get a relationship of
capital, K, to price of the product given as P,
and since another input is involved such as
labor here, the wage rate, w, and the own-price
of capital – the rental rate of capital, v
 This relationship is the demand for capital by
the firm
An Example

Suppose, f(L, K) is given by the production
function, f(L, K) = L0.6K0.3
 Let’s concentrate on capital
 The profit maximization condition is then given
by {that is, find the marginal product of
capital} P[0.3L0.6K0.3 – 1] = v
 [0.3L0.6K0.3 – 1] is the marginal product of
capital – by taking the K derivative of f(L, K)
in this example
Notice that 0.3L0.6K0.3 – 1 = 0.3Q/K, where
Q/K is just average product
 Why is this so? L0.6K0.3 = Q, and K-1 is
just 1/K
 So, we have P(0.3)Q/K = v = PmpK

Using P[0.3L0.6K0.3 – 1] = v as given before
we can solve for K
 But, output Q and labor, L, are jointly
determined with capital K
 So we have to solve all the first order
conditions simultaneously


If we do just that, and it takes several
algebraic steps to do so we end up with
the demand for capital
 K = (0.6/w)(0.6/)(0.3/v)(0.4/)P(1/)
 Where  = 1 – 0.6 – 0.3
 The demand for labor in this case is
similar, L = (0.6/w)(0.7/)(0.3/v)(0.3/)P(1/)
Our example demand for
capital
K = (0.6/w)(0.6/)(0.3/v)(0.4/)P(1/)
 Looking at the example we have, and in
general, the demand for capital as
derived from the profit maximization
process is a function of the price of other
inputs, like w = wage here, the price of
the product, P, and the rental rate of
capital v (the own-input price)

K = (0.6/w)(0.6/)(0.3/v)(0.4/)P(1/)
 As you can see ∂K/∂v < 0, or as the price
of capital increases, then demand for
capital decreases
 ∂K/∂P > 0, so as the price of the product
increases, then there is increased demand
for the capital input to produce more
output in response to the price increase

Demand for capital under
conditions of cost minimization
subject to a given output

In this case, price of the output, P, is given and
the firm is attempting to minimize the cost of
producing a set target output, Q = Qo
 Again, we could go through the constrained
optimization to find the demand for capital in
this case, which is the conditional derived
demand for capital
Conditional derived demand
for capital

Suffice it to say that in this case, the
demand for capital is a function of the
prices of other inputs, like labor, etc., the
rental price of capital and the output
level and not the price of output
Given the previous empirical
example of Q = L0.6K0.3

Demand for capital in this case is given
as:
 K = (0.6/w)-(0.6/ө)(0.6/v)(0.6/ө)Q(1/ө)
 Where ө = 0.6 + 0.3
 Again, demand for K decreases as the
rental value of capital, v, increases
Deriving the imputed value of
capital in production

We can use the relationship of the marginal
revenue product being equal to the rental price
of capital in equilibrium to derive the imputed
value (return) of capital in the firm’s
production process
 PmpK = v
 So price of the product multiplied by the
marginal product of capital gives us the
imputed (shadow value) return to capital
An example


Suppose we use the same example we
have been using for the production
function as Q = L0.6K0.3
Marginal product of capital we found to be
0.3Q/K = 0.3(average product of capital)
 But we need PmpK, so we need a price of the
output P
Impute the return, v, on capital

If price of the output is given as $1.50
 Average product of capital is given as $2.49, or
$2.49 per $1
 Then the imputed return on capital is given by
PmpK = P(0.3)Q/K = 1.5(0.3)(2.49) = 1.12, or
capital earns an imputed rate of (1 + r) = 1.12
or 12% in this particular production process
and level of employment of capital in the
process

This rate would be close to what we call
the Internal Rate of Return (IRR) in
finance
 It is the rate of return the firm is getting
out of its employment of capital in the
firm’s business
Compare with the market rate

We could use the comparison rule in finance to
see how this rate compares with the market
rate
 Suppose the firm has borrowed its funds from a
bank, and the bank charged 11% for operating
funds (like an operating loan)
 The firm can meet the payment of 11% because
it derives a 12 % return on the capital

Of course, the firm is not at the profit
maximization level of capital use in the
production process, since 12% > 11%, i.e.,
marginal revenue product of capital is greater
than the marginal cost of capital
 The firm could employ more capital, drive the
marginal product of capital down, so that the
imputed return is equal to the market rate and
maximize profit

Here, we have just used the simple tools
and theory that we learned in
microeconomics and/or managerial
economics to derive the imputed value of
capital and to make the rate of return to
market rate comparison