Chapter 17, Problem 10E

a)

To determine

To describe: The net present costs of the two programs.

Answer to Problem 10E

Net present cost in first and second alternatives are $\$431.35$ and $\$3,174.1$ respectively. Total discounted cost of second alternative is greater than the first alternative

Explanation of Solution

Present value of cost of first alternative,

Present value of cost incurred in first year $(PV)=200\left(\frac{1}{{\left(1+r\right)}^n}\right)$

" $r$ " is the discount rate

" $n$ " is the number of years

Preset value in the first year is,

  $\begin{array}{l}(PV)=200\left(\frac{1}{{\left(1+r\right)}^n}\right)\\ =200\left(\frac{1}{1.1}\right)\\ =200\left(.909\right)\\ =181.8\end{array}$

So, the cost incurred in the first year = $181.8$

Present value of cost incurred in second year $(PV)=200\left(\frac{1}{{\left(1+r\right)}^n}\right)$

" $r$ " is the discount rate

" $n$ " is the number of years

Preset value in the second year is,

  $\begin{array}{l}(PV)=200\left(\frac{1}{{\left(1+.10\right)}^2}\right)\\ =200\left(\frac{1}{1.331}\right)\\ =200\left(.751\right)\\ =150.2\end{array}$

So, the cost incurred in the second year = $150.2$

Present value of cost incurred in third year $(PV)=100\left(\frac{1}{{\left(1+r\right)}^n}\right)$

" $r$ " is the discount rate

" $n$ " is the number of years

Preset value in the third year is,

  $\begin{array}{l}(PV)=100\left(\frac{1}{{\left(1+.10\right)}^3}\right)\\ =100\left(\frac{1}{1.464}\right)\\ =100\left(.683\right)\\ =68.3\end{array}$

So, the cost incurred in the third year = $68.3$

Present value of cost incurred in fourth year $(PV)=50\left(\frac{1}{{\left(1+r\right)}^n}\right)$

" $r$ " is the discount rate

" $n$ " is the number of years

Preset value in the fourth year is,

  $\begin{array}{l}(PV)=50\left(\frac{1}{{\left(1+.10\right)}^4}\right)\\ =50\left(\frac{1}{1.610}\right)\\ =50\left(.621\right)\\ =31.05\end{array}$

So, the cost incurred in the fourth year = $31.05$

  $\begin{array}{l}TotalDicountedRate=\left[\begin{array}{l}DiscountedCostOfFirstYear\\ +DiscountedCostOfFirstYear\\ +DiscountedCostOfThrirdYear\\ +DiscountedCostOfFourthYear\end{array}\right]\\ =181.8+150.2+68.3+31.05\\ =\$431.35\end{array}$

Present value of cost of second alternative,

Present value of cost incurred in first year $(PV)=700\left(\frac{1}{{\left(1+r\right)}^n}\right)$

" $r$ " is the discount rate

" $n$ " is the number of year

Preset value in the first year is,

  $\begin{array}{l}(PV)=700\left(\frac{1}{\left(1+.10\right)}\right)\\ =700\left(\frac{1}{1.1}\right)\\ =700\left(.909\right)\\ =636.3\end{array}$

So, the cost incurred in the first year = $636.3$

Present value of cost incurred in second year $(PV)=1800\left(\frac{1}{{\left(1+r\right)}^n}\right)$

" $r$ " is the discount rate

" $n$ " is the number of year

Preset value in the second year is,

  $\begin{array}{l}(PV)=1800\left(\frac{1}{{\left(1+.10\right)}^2}\right)\\ =1800\left(\frac{1}{1.331}\right)\\ =1800\left(.751\right)\\ =1351.8\end{array}$

So, the cost incurred in the second year = $1351.8$

Present value of cost incurred in third year $(PV)=1,100\left(\frac{1}{{\left(1+r\right)}^n}\right)$

" $r$ " is the discount rate

" $n$ " is the number of year

Preset value in the third year is,

  $\begin{array}{l}(PV)=1,100\left(\frac{1}{{\left(1+.10\right)}^3}\right)\\ =1,100\left(\frac{1}{1.464}\right)\\ =1,100\left(.683\right)\\ =751.3\end{array}$

So, the cost incurred in the third year = $751.3$

Present value of cost incurred in fourth year $(PV)=700\left(\frac{1}{{\left(1+r\right)}^n}\right)$

" $r$ " is the discount rate

" $n$ " is the number of year

Preset value in the second year is,

  $\begin{array}{l}(PV)=700\left(\frac{1}{{\left(1+.10\right)}^4}\right)\\ =700\left(\frac{1}{1.610}\right)\\ =700\left(.621\right)\\ =434.7\end{array}$

So, the cost incurred in the second year = $434.7$

  $\begin{array}{l}TotalDicountedRate=\left[\begin{array}{l}DiscountedCostOfFirstYear\\ +DiscountedCostOfFirstYear\\ +DiscountedCostOfThrirdYear\\ +DiscountedCostOfFourthYear\end{array}\right]\\ =636.3+1351.8+751.3+434.7\\ =\$3,174.1\end{array}$

Total discounted cost of second alternative is greater than the first alternative.

Economics Concept Introduction

Introduction: The discount rate is the rate at which a discounted cash flow (DCF) model will measure the real value of potential cash flows. This helps to assess whether future cash flows from a project or investment are more than the investment required to fund the existing project or amount.

b)

To determine

To describe:

The cost and benefit analysis of two programs.

Answer to Problem 10E

Based on the cost and benefit analysis second alternative is less as compare to first one.

Explanation of Solution

Discountedprofit from first alternative,

Present value of cost incurred in first year $(PV)=50\left(\frac{1}{{\left(1+r\right)}^n}\right)$

" $r$ " is the discount rate

" $n$ " is the number of year

Preset value in the first year is,

  $\begin{array}{l}(PV)=50\left(\frac{1}{{\left(1+.10\right)}^n}\right)\\ =50\left(\frac{1}{1.1}\right)\\ =50\left(.909\right)\\ =45.45\end{array}$

So, the cost incurred in the first year = $45.45$

Present value of cost incurred in second year $(PV)=100\left(\frac{1}{{\left(1+r\right)}^n}\right)$

" $r$ " is the discount rate

" $n$ " is the number of year

Preset value in the second year is,

  $\begin{array}{l}(PV)=100\left(\frac{1}{{\left(1+.10\right)}^2}\right)\\ =100\left(\frac{1}{1.331}\right)\\ =100\left(.751\right)\\ =75.1\end{array}$

So, the cost incurred in the second year = $75.1$

Present value of cost incurred in third year $(PV)=250\left(\frac{1}{{\left(1+r\right)}^n}\right)$

" $r$ " is the discount rate

" $n$ " is the number of year

Preset value in the third year is,

  $\begin{array}{l}(PV)=250\left(\frac{1}{{\left(1+.10\right)}^3}\right)\\ =250\left(\frac{1}{1.464}\right)\\ =250\left(.683\right)\\ =170.75\end{array}$

So, the cost incurred in the third year = $170.75$

Present value of cost incurred in fourth year $(PV)=100\left(\frac{1}{{\left(1+r\right)}^n}\right)$

" $r$ " is the discount rate

" $n$ " is the number of year

Preset value in the fourth year is,

  $\begin{array}{l}(PV)=100\left(\frac{1}{{\left(1+.10\right)}^4}\right)\\ =100\left(\frac{1}{1.610}\right)\\ =100\left(.621\right)\\ =62.1\end{array}$

So, the cost incurred in the fourth year = $62.1$

  $\begin{array}{l}TotalDicountedRate=\left[\begin{array}{l}DiscountedCostOfFirstYear\\ +DiscountedCostOfFirstYear\\ +DiscountedCostOfThrirdYear\\ +DiscountedCostOfFourthYear\end{array}\right]\\ =45.45+75.1+170.75+62.1\\ =\$353.4Thosand\end{array}$

Discounted profit of second alternative,

Present value of cost incurred in first year $(PV)=150\left(\frac{1}{{\left(1+r\right)}^n}\right)$

" $r$ " is the discount rate

" $n$ " is the number of year

Preset value in the first year is,

  $\begin{array}{l}(PV)=150\left(\frac{1}{\left(1+.10\right)}\right)\\ =150\left(\frac{1}{1.1}\right)\\ =150\left(.909\right)\\ =136.35\end{array}$

So, the cost incurred in the first year = $136.35$

Present value of cost incurred in second year $(PV)=225\left(\frac{1}{{\left(1+r\right)}^n}\right)$

" $r$ " is the discount rate

" $n$ " is the number of year

Preset value in the second year is,

  $\begin{array}{l}(PV)=225\left(\frac{1}{{\left(1+.10\right)}^2}\right)\\ =225\left(\frac{1}{1.331}\right)\\ =225\left(.751\right)\\ =168.97\end{array}$

So, the cost incurred in the second year = $168.97$

Present value of cost incurred in third year $(PV)=250\left(\frac{1}{{\left(1+r\right)}^n}\right)$

" $r$ " is the discount rate

" $n$ " is the number of year

Preset value in the third year is,

  $\begin{array}{l}(PV)=250\left(\frac{1}{{\left(1+.10\right)}^3}\right)\\ =250\left(\frac{1}{1.464}\right)\\ =250\left(.683\right)\\ =170.75\end{array}$

So, the cost incurred in the third year = $170.75$

Present value of cost incurred in fourth year $(PV)=300\left(\frac{1}{{\left(1+r\right)}^n}\right)$

" $r$ " is the discount rate

" $n$ " is the number of year

Preset value in the second year is,

  $\begin{array}{l}(PV)=300\left(\frac{1}{{\left(1+.10\right)}^4}\right)\\ =300\left(\frac{1}{1.610}\right)\\ =300\left(.621\right)\\ =186.3\end{array}$

So, the cost incurred in the second year = $186.3$

  $\begin{array}{l}TotalDicountedRate=\left[\begin{array}{l}DiscountedCostOfFirstYear\\ +DiscountedCostOfFirstYear\\ +DiscountedCostOfThrirdYear\\ +DiscountedCostOfFourthYear\end{array}\right]\\ =136.35+168.97+170.75+186.3\\ =\$662.37Thousand\end{array}$

Benefit-cost ratio of the first alternative

  $\text{Benefit-cost Ratio=}\frac{\text{present value of benefits}}{\text{present value o}f\cos ts}$

  $\begin{array}{l}=\frac{353.4}{431.35}\\ =0.819\end{array}$

Benefit-cost ratio of the second alternative

  $\text{Benefit-cost Ratio=}\frac{\text{present value of benefits}}{\text{present value o}f\cos ts}$

  $\begin{array}{l}=\frac{662.37}{3174.1}\\ =0.208\end{array}$

Based on the cost and benefit analysis second alternative is less as compare to first one.

Economics Concept Introduction

Introduction:

The discount rate is the rate at which a discounted cash flow (DCF) model will measure the real value of potential cash flows. This helps to assess whether future cash flows from a project or investment are more than the investment required to fund the existing project or amount.

c)

To determine

To describe:

The ambiguous choice between the two alternatives and the factors are likely to weigh on the ultimate choice.

Answer to Problem 10E

Government can adopt the second alternative.

Explanation of Solution

From the costs and the benefits analysis, it is indicated that the government must choose the first alternative instead of second ones. But the further insight into the lives saved and injuries prevented is required. The first alternative shows that expenditure reduces from $\$1000$ thousand to $\$50$ thousand in the 4^th year, the lives saved all fall from 35 to 25. Likewise, injuries prevented fall 300 to 150.

On the other hand, in second alternatives, the expenditure reduced from $\$1,100$ thousand to $\$700$ in 4^th year. The saved lives increase from 100 to 125. The prevented injuries remainthe same at 900. So the marginal outcomes of extra expenditure in the second alternative are encouraging. Therefore, the government can adopt the second alternative.

Economics Concept Introduction

Introduction:

The discount rate is the rate at which a discounted cash flow (DCF) model will measure the real value of potential cash flows. This helps to assess whether future cash flows from a project or investment are more than the investment required to fund the existing project or amount.