# The Value of the Firm, Production in the Long Run, and the Demand

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24. The Value of the Firm

Questions:
1. Since economists focus on maximizing profits, they ignore the need to take the long view when managing a firm.
Discuss. (Value of firm, formula used to calculate it, why interest is included, net cash flow vs. profit, shifting profit
or net cash flow between years.)

Problems
2. Be able to calculate the value of the firm given appropriate information.

Value of the Firm
In the long run, all inputs can be adjusted, including fixed inputs like capital--machines, buildings, and
equipment. The basic principle that the marginal revenue product of the input must equal the price (or rental) of the
input remains true, but complications abound. Most capital goods last for many periods. Instead of looking to
maximize profits in a single period, it is necessary to make decisions that will influence profits in many periods. The
concept that allows us to do that is maximizing the value of the firm.
The value of a firm is equal to the present value of the expected future net cash flow (NCF) that the firm
will generate. The general formula is simple:

VOF =      1 +           2         +           3               + .......... ...+                  n
(1+ r 1 ) (1+ r 1 )(1+ r 2 ) (1+ r 1 )(1+ r 2 )(1+ r 3 )                   (1+ r 1 )(1+ r 2 )(1+ r 3 )...(1 + r n )

If we assume that the interest rate remains constant, that simplifies to:

VOF =       1 +  2 +  3 + .......... ...+  n
1      2      3                   n
(1+ r) (1+ r) (1+ r)              (1+ r)

Where  is the net cash flow from a particular year, r is the interest rate, and n is the number of years the firm
will exist. What is the difference between profit and net cash flow? Net cash flow takes revenues minus costs
without any concern for depreciation of assets. (If the purchase if financed, then the relevant cost would be the loan
payments.)
It seems plausible enough that profits (or net cash flows) should be summed to determine the value of the firm,
but why is the interest rate taken into account? The key reason is opportunity cost. If a firm makes a decision to
sacrifice short run profits to increase profitability in the long run, it is shifting money into the future. From the point
of view of the individual firm and its owners, if one is willing to shift money into the future, one can earn interest on
that money in other employments. For the decision to shift profitability to be beneficial to the owners, it must
generate future money at least as well as other opportunities to earn interest.
So actions that reduce the profits this year but increase them enough next year or at anytime in the lifetime of the
firm can increase the value of the firm.

Suppose a firm is only going to exist for three years. What would be its value if its profit was \$5,015,000 and
the interest rate is 5%?
2

5,015,000          5,015,000       5,015,000
VOF =                1
+           2
+             3
(1+ .05 )   (1+ .05 )                  (1+ .05 )
VOF = 13,657,088.87

Few firms, however, will last only three years, especially if they are making a profit. The
long the firm lasts, the more valuable it will be.

10 years    38,724,500.67
25 years    70,681,132.00
50 years    91,553,466.18
100 years   99,537,269.65
150 years   100,233,487.07

Notice, however, that the changes in the value of the firm become less significant as the distant future is
considered. The difference between 3 and ten years is very significant as is the difference between 10 and 25 years
and even the 21 million betweeen 25 and 50 years can’t be ignored. But the difference between 50 years and 100
years is pretty small and between 100 and 150 years amounts to less than 1% of the value of the firm.

Suppose the firm is expected to last forever?    An infinite horizon is easiest to calculate as long as the NCF and
interest rates are expected to remain the same.

VOF = NCF/r

VOF = \$5,015,000/.05
= 100,300,000.00

The reason is simple, \$100,300,000 earns about \$5,015,000 per year at 5% interest. Notice how the profits
anticipated from 150 years to forever have next to no impact on the value of the firm. In fact, less that 10% of the
value of the firm is based upon profits expected beyond 50 years.
Unfortunately, this isn’t really all that useful, since the whole point of this tool is to look at what happens to the
value of the firm when some action lowers profits in some years and raises them in other years. That means that the
profits aren’t constant. For most problems, we are interested in how changes in the Net Cash Flow in near term
years will impact the value of the firm.

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