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Chapter 3: Problem 21

Current annual consumption of energy is 78 billion units and this is expected to rise at a fixed rate of \(5.8 \%\) each year. The capacity of the industry to supply energy is currently 104 billion units. (a) Assuming that the supply remains steady, after how many years will demand exceed supply? (b) What constant rate of growth of energy production would be needed to satisfy demand for the next 50 years?

Short Answer

Expert verified
Demand exceeds supply in 6 years. The energy production must grow at a rate of 4.56% annually for the next 50 years.

Step by step solution

01

Define the problem variables

The current annual consumption of energy is 78 billion units. The annual growth rate of consumption is 5.8%. The current supply of energy is 104 billion units.
02

Set up the consumption growth formula

The consumption after n years can be calculated using the compound growth formula: \[ E(n) = E_0 \times (1 + r)^n \]where \( E_0 \) is the initial consumption (78 billion units) and \( r \) is the growth rate (5.8% or 0.058).
03

Find the year when demand exceeds supply

To find when the consumption exceeds the supply (104 billion units), we need to solve for n in the equation: \[ 78 \times (1 + 0.058)^n > 104 \]This inequality can be solved as follows:\[ (1.058)^n > \frac{104}{78} \approx 1.333 \]Taking the natural logarithm of both sides gives:\[ n \times \text{ln}(1.058) > \text{ln}(1.333) \]Solving for n yields:\[ n > \frac{\text{ln}(1.333)}{\text{ln}(1.058)} \approx 5.04 \]So, the demand will exceed supply after 6 years (since we round up to the next whole number).
04

Determine the required growth rate of production

To find the constant rate of growth needed to satisfy demand for the next 50 years, we set up the equation:\[ S_f = S_0 \times (1 + g)^{50} \]where\( S_0 \) is 104 billion units, \( S_f \) is the final supply needed to match the final consumption, and \( g \) is the growth rate of production. After 50 years, the consumption will be:\[ E(50) = 78 \times (1.058)^{50} \approx 783.16 \text{ billion units} \]So, we need:\[ 104 \times (1 + g)^{50} = 783.16 \]Solving for \( g \):\[ (1 + g)^{50} = \frac{783.16}{104} \approx 7.53 \]Taking the 50th root of both sides, we get:\[ 1 + g = (7.53)^{\frac{1}{50}} \approx 1.0456 \]\[ g \approx 0.0456 \text{ or } 4.56\% \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Compound Growth Formula
To understand how energy consumption grows over time, we use the compound growth formula. It's a way to calculate future values based on a constant growth rate over a period. The formula is:
Energy Demand and Supply
Energy demand refers to the amount of energy consumed by users, while energy supply is the amount provided to meet this demand. In the exercise, the current demand is 78 billion units, and the supply is 104 billion units.
Natural Logarithm
A natural logarithm (ln) is a mathematical function that helps solve equations involving exponential growth or decay. It's based on the constant \( e \) (approximately 2.71828) and is particularly useful when working with growth formulas.
Growth Rate Calculation
Determining the necessary growth rate for energy production involves reverse engineering the compound growth formula. We need to find a growth rate that matches future demand with supply over a specific period.

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Most popular questions from this chapter

The value of an asset, currently priced at \(\$ 100000\), is expected to increase by \(20 \%\) a year. (a) Find its value in 10 years' time. (b) After how many years will it be worth \(\$ 1\) million?

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A firm decides to invest in a new piece of machinery which is expected to produce an additional revenue of \(\$ 8000\) at the end of every year for 10 years. At the end of this period the firm plans to sell the machinery for scrap, for which it expects to receive \(\$ 5000\). What is the maximum amount that the firm should pay for the machine if it is not to suffer a net loss as a result of this investment? You may assume that the discount rate is \(6 \%\) compounded annually.

A project requires an initial investment of \(\$ 12000\). It has a guaranteed return of \(\$ 8000\) at the end of year 1 and a return of \(\$ 2000\) each year at the end of years 2,3 and \(4 .\) Estimate the IRR to the nearest percentage. Would you recommend that someone invests in this project if the prevailing market rate is \(8 \%\) compounded annually?

If a principal, \(P\), is invested at \(r \%\) interest compounded annually then its future value, \(S\), after \(n\) years is given by $$ S=P\left(1+\frac{r}{100}\right)^{n} $$ (a) Use this formula to show that if an interest rate of \(r \%\) is compounded \(k\) times a year then after \(t\) years $$ S=P\left(1+\frac{r}{100 k}\right)^{k t} $$ (b) Show that if \(m=100 k / r\) then the formula in part (1) can be written as $$ S=P\left(\left(1+\frac{1}{m}\right)^{m}\right)^{r t / 100} $$ (c) Use the definition $$ \mathrm{e}=\lim _{m \rightarrow \infty}\left(1+\frac{1}{m}\right)^{m} $$ to deduce that if the interest is compounded with ever-increasing frequency (that is, continuously) then $$ S=P \mathrm{e}^{r / 100} $$
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