Download Value of the Firm: Infinite Time Horizon

26. Capital and Long Run Cost in Excel
1. Use an excel spreadsheet to show the value of the firm for an infinite time horizon
and for time horizons of various years.
2. Use an excel spreadsheet to calculate the marginal revenue product of capital for a
given level of output. Also calculate the present value of the marginal revenue
product of capital.
3. Use an excel spreadsheet to calculate the implicit rental rate give the price of a
capital good, the interest rate, and the life of the capital good.
4. Use an excel spreadsheet to find the long run cost function, the profit maximizing
quantity and price, the revenue, costs, and profits, and the amount of labor and
capital to be used.
Value of the Firm: Infinite Time Horizon
Finding the value of the firm for an infinite time horizon is easy—assuming constant
future net cash flows. In the first cell, type in NCF = and then in the next cell, type in
equal and go to the profit sheet and select the profit. In the row below, put in a label for
the interest rate, R = and then type in the interest rate. In the example we are using 5%.
And then the result is simple. Type in a label like “Infinite time horizon” and in the
row below put VOF = and in the cell next to it, divide the profit by the interest rate.
Value of the Firm: Finite Time Horizon
Calculating the value of the firm for a finite time horizon is simple. By using the
“fill” function, you can find the value of the firm for any time horizon you choose.
Label the first column “n” for the number of years in the future you will include. In the
row below, type in 1.
In the next column, label it (1+r)^n. In the row below, type in the formula, starting
with “=” and using the interest rate “r” from the infinite time horizon problem. Be sure
to anchor it with a dollar sign. Then use the “n” from the first column. That shouldn’t
be anchored.
The third column will calculate the present value of the net cash flow for each near.
Just label it PVNCF. In the first row of that column, take the NCF from the infinite
horizon problem and divide it by the number in the previous column. Be sure to anchor
the NCF with a dollar sign, but not the value for (1+r)^n. The final column will be the
sum of the present value. Label it “sum.” To start, just type in equal and select the
PVNCF in the previous column.
Now, for the next row. In the first column, start with an “=” and click on the “1” in
the row above and type in “+ 1”. When you hit enter, you should get a “2”.
Copy and paste the formulas for the “(1+r)^n” and the “PRNVF” into the next row.
In the “sum” column, you need another formula. Start with “=” and then click on the
cell in the row above, type in “+” and then click on the cell in the same row and previous
column, that is the PVNCF for the second year. No need for anchoring.
Now, you have a spreadsheet that shows the value of the firm for a 2 year time
horizon. You read it in the last column in the row that has “2” in the first column. For
a one year time horizon, you have the result in the last column in the row that has “1” in
the first column.
To get all of the years you want, just highlight the last row (the one with 2 on it
Initially.) Go to the bottom right hand corner of the highlighted area and put your cursor
on the small box. Push down the left mouse button and drag down as far as you need to
go to find the time horizon you want.
Demand for a Capital Good
To find the demand for a capital good, copy the long run production function from
the labor sheet. Just start at the top of a new page, type in “=” and go to the labor page
and click on the cell that has “Q=” and hit enter. Drag over until you have the complete
production function. You will also need the marginal revenue function (which is on both
the labor and the revenue sheets) and the function that gives labor as a function of
quantity. That’s on the labor sheet. Finally, you need the profit maximizing quantity.
That will be on the profit sheet. Label it “profit maximizing quantity,” and like the rest,
type in equal in the next cell, go to the page where it is, click on it with the mouse and hit
enter. You will also need the amount of capital, the price of capital, the rental on
capital, and the interest rate.
Now that you have all the relevant information on the new sheet, you need to
calculate the marginal product of capital function. Label it MPk = and in the next cell
type in “=” click on the coefficient on the production function, type in “*” and then click
on the exponent on K. Then, in the next cell, type in L^ and in the next cell click on the
exponent on K, then in the next cell click on the exponent on K type in – and then type in
one. (Hey, you should be able to figure this stuff out.)
You need to find the quantity of labor that goes with the profit maximizing output.
Label it “L=” and use the function to find the amount of labor that goes with the profitmaximizing output.
Next, use the given amount of capital and the labor just discovered and the
marginal product of capital function to find the marginal product of capital.
You need the marginal revenue, which is found with the marginal revenue
function and the profit maximizing quantity.
Finally, use the marginal revenue and the marginal product of capital to find the
marginal revenue product of capital. Label it with “MRPk=”
If the Marginal Revenue product of capital is greater than the capital rental rate,
then renting another capital good adds to profit. If the marginal revenue product of
capital is less than the capital rental rate, then one less unit of capital should be rented to
add to profit.
Buying Capital Goods
To find the present value of a the marginal product of a capital good, you need to
add a bit more to the spreadsheet. Just go to the value of the firm page and copy the first
two rows (and labels) of the section where you calculate the value of the firm for a finite
time horizon. Paste that on your capital page. It will look like a mess, but that’s O.K.
First, change the labels for two columns. The NCF column should instead be MRPk.
In the row below that column heading, type in equal and click on the MRPk above.
Next, you need to change the interest rate. Go to the column labeled (1+r)^n and
change the formula in the cell below so that the interest rate listed above on the page is
used in the formula. That should fix things. In the column for PRVNCF change that to
PVMRPk. It should be fine. The n stands for the number of years the capital good will
last. And you highlight the last row and drag down to the proper number of years. The
“sum” column gives you the present value of the marginal revenue product of capital.
Look at the number of years the capital good will last, and then the last column labeled
“sum” needs to be compared to the amount you must pay.
Implicit Capital Rental
If you have the price of a capital good, the interest rate, and the number of years the
capital good will last, you can calculate the implicit rental on the capital good. On
another page of the spreadsheet, put the price of the capital good (labeled Pk,) and the
interest rate (labeled R) on the top. Then go back to the page were you calculated the
value of the firm for a finite time horizon. Again, copy the labels and the first two rows.
Paste onto the new page. Delete the label for NCF and type in “1” in the first row and
second row. (When you copy it, you will have a column full of ones.) Change the
formula for (1+r)^n so that it references the interest rate above. Add another column.
Label it Kr. Then type in “=” and select the price of the capital good. Be sure to anchor
it with a dollar sign. Then type in “/” and select the number in the previous column.
(That’s the one labeled “sum”) Highlight the bottom row and drag down to get to the
number of years the capital good will last. The implicit rental is given in the last
column in the row that represents the life of the capital good.
Long Run Cost Function
The final page on the spreadsheet is used to find the long run cost function. Start
with an equal and go back to your demand for capital page and copy over the long run
production function. You need the wage rate, which you can copy over from the labor
page and the rental rate on capital. Type that in. You will need the price function from
the revenue page and the marginal revenue function from that same page.
Next calculate the marginal product of capital and the marginal product of labor.
That will be the next two rows. Then find the capital to labor ratio. Label that K/L and
it is equal to the wage rate divided by the capital rental rate. The next row should be
capital. Just put K= in the first cell, type in equal and click on the cell above to repeat
the capital to labor ratio, and then in the next cell put in L.
In the next row, you must solve find quantity as a function of labor. You must
multiply the coefficient on the production function by the capital to the labor ratio taken
to the power on capital in the production function. Remember, start with equal and
click on the various numbers—don’t type the numbers in. This number is multiplied by
“L”.
The next step is to find labor as a function of output. Start with “L=”, then in the
next cell put =1/ and select the number in the cell above. Then in the next cell put Q.
The next row finds capital as a function of output. Start with “K=” and then in the
next cell type “=”, select the capital to labor ratio and type “*” and select the number in
the cell above. In the next cell put “Q”
Now, for the total cost function. Start with “TC=.” In the next cell start with = and
then click on the wage and then on the number in the cell between the L = and the Q, then
type in “+” and click on the rental for capital and type in “*” and select the number in the
cell between the K = and the Q. And then in the next cell type in Q.
List the MC, which just requires that you type in MC= and then in the next cell just
select the number between the TC= and the Q. Average cost is the same. Type in AC=
and type in = and select the cell above.
Now to find the profit maximizing output. Start with Q= and then use the
marginal revenue function and the marginal cost to find the output. Figure it out.
Find price using the price function, revenue by multiplying price and quantity, and
profit by subtracting cost from revenue. Find the amount of capital to use from by using
the profit maximizing quantity with the capital function you found above. Do the same
to find the amount of labor.