Chapter 14, Problem 14.6P

a

To determine

To plot: Graphical representation of scotch value function.

Explanation of Solution

Given the scotch value function in time period $t$ as

  $V=100t-6t^2$

The graph of this has been drawn below wherein the time period (t) is measured on the x-axis and value of the scotch (V) is measured on the y-axis.

  

The graph of the scotch value function exhibits a parabolic shape indicating that with the increase in time period, t, the value of the scotch, V, also increases but only up to point a. Thereafter the value of the scotch starts declining if it is kept for further time period.

Economics Concept Introduction

Introduction:

Nominal interest rate is the monetary value of returns for the investment done.

Real interest rate is the nominal interest rate adjusted to inflation.

b)

To determine

To calculate: Value of t at which barrel of scotch is most valuable.

Answer to Problem 14.6P

The scotch is most valuable when it is sold at $t=8.33$

Explanation of Solution

Given Information:

The scotch value function in time period $t$ as $V=100t-6t^2$

Value of scotch = $(100-12t)/V$

The barrel of scotch will be most valuable at point a (see graph above) because it is at this point the scotch value curve reaches its maximum height. In order to determine the value of t at point a we need to obtain the first order condition $(F.O.C)$ of the scotch value function and then set it to zero. Thus,

  $V=100t-6t^2$

  $\begin{array}{l}\frac{dV}{dt}=100-12t=0\\ -12t=-100\\ t=\frac{100}{12}\\ t=8.33\end{array}$

Hence, the scotch is most valuable when it is sold at $t=8.33$

Economics Concept Introduction

Introduction:

Nominal interest rate is the monetary value of returns for the investment done.

Real interest rate is the nominal interest rate adjusted to inflation.

c)

To determine

To calculate: Time period for immediate sale of the distiller should bottle the scotch

Answer to Problem 14.6P

The distiller should bottle the scotch for immediate sale at $t=6.67$ .

Explanation of Solution

Given Information:

The scotch value function in time period $t$ as $V=100t-6t^2$

Value of scotch = $(100-12t)/V$

Rate of interest = 5%

If the real interest rate is 5 percent, then the distiller should bottle the scotch for sale when the real interest rate of interest is equal to the proportional rate of growth of the scotch value. Thus,

  $r=\frac{100-12t}{V}$

Substituting the value of r and value for V,

  $0.05=\frac{100-12t}{100t-6t^2}$

  $0.3t^2-17t+100=0$

Using the quadratic equation formula,

  $t=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

  $t=\frac{-(-17)\pm \sqrt{{17}^2-4(0.3)(100)}}{2(0.3)}$

  $t=\frac{17\pm 13}{0.6}$

So that,

  $t=50$ And $t=6.67$

Since, $t=50$ is extraneous to the function, therefore, the distiller should bottle the scotch for immediate sale at $t=6.67$ .

Economics Concept Introduction

Introduction:

Nominal interest rate is the monetary value of returns for the investment done.

Real interest rate is the nominal interest rate adjusted to inflation.

d)

To determine

To calculate: Time period for immediate sale of the distiller should bottle the scotch when interest rate is 10%

Answer to Problem 14.6P

The distiller should bottle the scotch for immediate sale at $t=5.32$ .

Explanation of Solution

Given Information:

The scotch value function in time period $t$ as $V=100t-6t^2$

Value of scotch = $(100-12t)/V$

Rate of interest = 10%

If the real interest rate changes to 10 percent, then

  $r=\frac{100-12t}{V}$

Substituting the value of r and value for V, we have

  $0.10=\frac{100-12t}{100t-6t^2}$

  $0.10(100t-6t^2)=100-12t$

  $0.6t^2-22t+100=0$

Using the quadratic equation formula,

  $t=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

  $=\frac{-(-22)\pm \sqrt{{(22)}^2-4(0.6)(100)}}{2(0.6)}$

  $=\frac{22\pm 15.62}{1.2}$

So that,

  $t=31.35$ And $t=5.32$

Since, $t=31.35$ is extraneous to the function, therefore, the distiller should bottle the scotch for immediate sale at $t=5.32$ .

Economics Concept Introduction

Introduction:

Nominal interest rate is the monetary value of returns for the investment done.

Real interest rate is the nominal interest rate adjusted to inflation.