Chapter 11, Problem 24PS

Economic rents Taxes are a cost, and, therefore, changes in tax rates can affect consumer prices, project lives, and the value of existing firms. The following problem illustrates this. It also illustrates that tax changes that appear to be “good for business” do not always increase the value of existing firms. Indeed, unless new investment incentives increase consumer demand, they can work only by rendering existing equipment obsolete.

The manufacture of bucolic acid is a competitive business. Demand is steadily expanding, and new plants are constantly being opened. Expected cash flows from an investment in a new plant are as follows:

Assumptions:

1. 1. Tax depreciation is straight-line over three years.
2. 2. Pretax salvage value is 25 in year 3 and 50 if the asset is scrapped in year 2.
3. 3. Tax on salvage value is 40% of the difference between salvage value and depreciated investment.
4. 4. The cost of capital is 20%.
5. a. What is the value of a one-year-old plant? Of a two-year-old plant?
6. b. Suppose that the government now changes tax depreciation to allow a 100% writeoff in year 1. How does this affect the value of existing one- and two-year-old plants? Existing plants must continue using the original tax depreciation schedule.
7. c. Would it now make sense to scrap existing plants when they are two rather than three years old?
8. d. How would your answers change if the corporate income tax were abolished entirely?

Summary Introduction

To determine: values of one year old plant and two year old plant.

Explanation of Solution

Here, value in the sense, present value.

Calculation of present value of one-year old plant:

    $\begin{array}{c}\text{PV}=\frac{{\text{C}}_{\text{1}}}{{\left(1+\text{r}\right)}^1}+\frac{{\text{C}}_{\text{2}}}{{\left(1+\text{r}\right)}^2}\\ =\frac{\$43.33}{1.20}+\frac{\$58.33}{{1.20}^2}\\ =\$76.62\end{array}$

Hence, the present value of one-year old plant is $76.62.

Calculation of present value of two-year old plant:

                                                                        $\begin{array}{c}\text{PV}=\frac{{\text{C}}_{\text{1}}}{{\left(1+\text{r}\right)}^1}\\ =\frac{\$58.33}{1.20}\\ =\$48.61\end{array}$

Hence, the present value of two-year old plant is $48.61.

Summary Introduction

To determine: Effect of government on taxes on one and two year old plant and existing plants should continue using the schedule of  original tax depreciation.

Explanation of Solution

Given the industry is competitive, the investments in new plant producing the product must yields zero NPV.

Calculation of Revenues (R) at which the NPV of new plant is zero:

Hence,

  $\begin{array}{c}\text{NPV}=\frac{\text{Cash}\text{flows}}{{\left(1+\text{r}\right)}^{\text{i}}}\\ =-\$100+\frac{\left(\$0.6\text{R}-\$30+\$40\right)}{1.20}+\frac{\left(\$0.6\text{R}-\$30\right)}{{1.20}^2}+\frac{\left(\$0.6\text{R}-\$30+\$15\right)}{{1.20}^3}\end{array}$

By solving this, the value of R= $95.88.

By using the value of R we can calculate the present values of existing one and two years old plants.

Calculation of present value of one and two-year old plants:

$\begin{array}{c}\text{PV of one year old plant}=\frac{{\text{C}}_{\text{1}}}{{\left(1+\text{r}\right)}^1}+\frac{{\text{C}}_{\text{2}}}{{\left(1+\text{r}\right)}^2}\\ =\frac{\$40.86}{1.20}+\frac{\$55.86}{{1.20}^2}\\ =\$72.84\end{array}$

Hence, the present value of one-year old plant is $72.84.

$\begin{array}{c}\text{PV of two year old plant}=\frac{{\text{C}}_{\text{n}}}{{\left(1+\text{r}\right)}^{\text{n}}}\\ =\frac{\$55.86}{1.20}\\ =\$46.55\end{array}$

Hence, the present value of two-year old plant is $46.55

Summary Introduction

To determine: Net salvage value of two year old plant.

Explanation of Solution

Calculation of salvage value:

$\begin{array}{c}\text{Salvage}\text{value}=\text{P}-\left(\text{I}\times \text{Y}\right)\\ =\$50-\left[\left(0.40\right)\times \left(\$50-\$33.33\right)\right]\\ =\$43.33\end{array}$

The salvage value is $43.33.

Summary Introduction

To determine: Values the corporate tax rates are abolished entirely.

Explanation of Solution

Solving for zero NPV:

$\begin{array}{c}\text{NPV for 0 year}=\frac{\text{Cash}\text{flows}}{{\left(1+\text{r}\right)}^{\text{i}}}\\ =-\$100+\frac{\left(\text{R}-\$50\right)}{1.20}+\frac{\left(\text{R}-\$50+\$25\right)}{{1.20}^2}\\ +\frac{\left(\text{R}-\$50+\$25\right)}{{1.20}^3}\\ \text{R}=\$90.60\end{array}$

By using the revenues of $90.60

Calculation of present value of one-year old plant:

$\begin{array}{c}\text{PV of one year old plant}=\frac{{\text{C}}_{\text{1}}}{{\left(1+\text{r}\right)}^1}+\frac{{\text{C}}_{\text{2}}}{{\left(1+\text{r}\right)}^2}\\ =\frac{\$40.60}{1.20}+\frac{\$65.60}{{1.20}^2}\\ =\$79.40\end{array}$

Hence, the present value of one-year old plant is $79.40.

Calculation of present value of two-year old plant:

$\begin{array}{c}\text{PV of two year old plant}=\frac{{\text{C}}_2}{{\left(1+\text{r}\right)}^{\text{n}}}\\ =\frac{\$65.60}{{1.20}^{}}\\ =\$54.67\end{array}$

Hence, the present value of two-year old plant is $54.67