Let $ V(I, R) = 100 \left( \frac{1 + .06(I - R)}{1 + I} \right)^{10} $ represent the future value of an investment of $100 at an annual interest rate of 6% after 10 years, assuming a tax rate $ R $ and inflation rate $ I $. Note that $ R $ and $ I $ are fractions between 0 and 1, where 0 is 0% and 1 is 100%.a. Find the partial derivatives $ V_I $ and $ V_R $ and show all work.b. Calculate $ V_I(.02, .20) $ and $ V_R(.02, .20) $. *Note: this is an inflation rate of 2% and a tax rate of 20% for each calculation.*c. Is the future value of the investment decreasing faster with respect to increases in inflation or with respect to increases in taxes?

Given: V(I,R) = 1001+0.061-R1+I10V = Future value of moneyI = Inflation rateR…

Answered: 10 1+.06(1-R) Let V(I,R) = 100…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

Let $ V(I, R) = 100 \left( \frac{1 + .06(I - R)}{1 + I} \right)^{10} $ represent the future value of an investment of $100 at an annual interest rate of 6% after 10 years, assuming a tax rate $ R $ and inflation rate $ I $. Note that $ R $ and $ I $ are fractions between 0 and 1, where 0 is 0% and 1 is 100%.

a. Find the partial derivatives $ V_I $ and $ V_R $ and show all work.

b. Calculate $ V_I(.02, .20) $ and $ V_R(.02, .20) $. *Note: this is an inflation rate of 2% and a tax rate of 20% for each calculation.*

c. Is the future value of the investment decreasing faster with respect to increases in inflation or with respect to increases in taxes?

Expert Solution

Step 1: Given and basic information

$G i v e n : V (I, R) = 100 {(\frac{1 + 0.06 (1 - R)}{1 + I})}^{10} V = F u t u r e v a l u e o f m o n e y I = I n f l a t i o n r a t e R = t a x r a t e$

$T o f i n d : V_{I} (I, R) = \frac{\partial}{\partial I} V (I, R) V_{R} (I, R) = \frac{\partial}{\partial R} V (I, R)$

$B a s i c f o r m u l a e : \frac{\partial}{\partial x} f (x) g (y) = g (y) \frac{\partial}{\partial x} f (x) \frac{\partial}{\partial x} f (h (x)) g (y) = g (y) (\frac{\partial}{\partial x} f (h (x)) \cdot \frac{\partial}{\partial x} h (x))$