Questions:
You may assume that electricity prices remain constant, and inflation can be ignored. Assume the power generated by the solar panels is the same yearround.
 Which is the cheapest solar system type that would provide 50% of the family electricity consumption?
 Assuming that the solar system performs as advertised, what is the family’s quarterly bill from its energy provider? Assume there are 91 days in a quarter.
 Using an annual discount rate of 5.15% and ignoring inflation, what is the present value of 15 years’ worth of electricity bills?
 Other than buying solar panels, what is the quarterly cost of the next best alternative, which still provides 50% renewable energy?
 What is the present value of 30 years’ worth of this solution? Again, ignore inflation, and use a discount rate of 5.15%.
 Conditional on using at least 50% renewable energy, what is the net present value (NPV) of purchasing the solar panel system named in question (1)? Hint: to answer this question you will need to consider the present value of renewable energy sources and the present value residual electricity bills along with any associated installation costs.
 Interpret your result from question (6). Is buying solar panels a good idea? Explain your answer.
 Under the same assumptions as above, assume the family is not committed to purchasing renewable energy (i.e. the family is happy with consuming nonrenewable energy from the grid.) What is the NPV of purchasing solar panels now?
 Will the family in question (8) purchase the solar panel system named in question (1)?
 Some governments have offered subsidies to consumers to install solar panels in their homes. Why?
Answers:
Section A: Population Growth
 The exponential population growth graph exhibits a Jshaped curved that is given as follows:
Figure 1: Exponential population growth graph
The curve shows the rate of increase in population over time. In the given graph, P_{0} is the yintercept and represents the population level at time (t) = 0.
 The given formula for population growth is:
Rearranging the formula to solve for r, we get:
Taking natural log for both sides, we get:
 The annual average population growth is calculated as follows:
Here, 2011 = 0 and 2015= 4
China
Here, P_{t} = 1371 (population in 2015)
P_{0} = 1344 (population in 2011)
t = time = 4
= log () = log (1.02) = (0.0086) = 0.00215 0.002
India
Here, P_{t} = 1311 (population in 2015)
P_{0} = 1247 (population in 2011)
t = time = 4
log () = log (1.051) = (0.0216) = 0.0054 0.005
Hence, the annual growth rate of population for China is 0.002 or 0.2% and for Indi
a is 0.005 or 0.5%.
 Calculation of population for year 2030 (19) and 2037 (26) for China and India using:
China
 Population in 2030
1344 × 1.038731 = 1396.055
 Population in 2037
= = 1344 × 1.053376 = 1415.737
India
 Population in 2030
= = 1247 × 1.099659 = 1371.275
 Population in 2037
= = 1247 × 1.138828 = 1420.119
Thus, the estimated population for China and India using the general formula of is given as follows:
Country 
General Formula 
2030 
2037 
China 

1396.055 
1415.737 
India 

1371.275 
1420.119 
Table 1: Estimated population (in millions)
 To identify the time taken when the population for both the countries will be equal, the population equation for both China and India need to be equated and solved for t:
Equation (a) for China is given as:
Equation (b) for China is given as:
Equating equation a and b, we get:
Solving the above equation for t:
1.078 =
1.078 =
Putting natural log on both sides, we get:
log (1.078) = 0.003t
= 10.8442720082 10.8 years
 To identify the time in which the population will double itself, the population figures at t = 0 are doubled and the equation is solved for t. Thus,
China
Population in 2011 = 1344
Doubling the values and putting in the general equation, we get:
Solving the above equation for t, we get:
log (2) = 0.002t
t = 0.3010/0.002 = 150.51 or 150 years 6 months
India
Population in 2011 = 1247
Doubling the values and putting in the general equation, we get:
Solving the above equation for t, we get:
log (2) = 0.005t
t = 0.3010/0.005 = 60.20 or 60 years 2 months
Thus, the population of China will take 150.5 years to double in size, while India’s population will double in approximately 60 years 2 months.
Assumptions:
 The population is doubled at t = 0. Thus, for the purpose of this question, population for China and India in the 2011 has been taken.
 The growth rate remains constant.
 No, if a country’s population follows an exponential curve indefinitely, that population would not be sustainable as it will cause a strain on the natural resources thereby decreasing the carrying capacity of environment. The depletion of resources critical for survival of individuals will limit their consumption thereby affecting the survival. Further, limited resources will encourage competition between the members of the population. Thus, lack of critical resources will affect the death rate and birth rates thereby threatening the sustainability of increased population. In addition, increase in population will also lead to increase in waste accumulation thereby causing disturbances in natural order of the ecosystem and affecting health survival of individuals.
Section B: Malthusian Disaster
 The given graph of world population depicts a geometric growth in the population as the increase in population across generation is geometric rather than constant.
 The expression for population at time (t) with initial population is a_{0}and growth rate (r) is given as:
P_{t} = a_{o }(1 + r)^{t}
 The expression for food supply when initial supply is b tones per year and increase in supply is m tones is given as:
S_{t} = b + mt
Where, S = food supply
b = initial quantity or y intercept
m = change in supply
t = time
 Using the equations formulated in above questions and substituting the given values, the general equations for population growth and food supply are:
Population Growth: P_{t } = 1000 (1+ 0.105)^{t}
Food Supply: S_{t} = 2100 + 160t
Section 2
Solving the above equation for values t = 0, 10, 20, 30,40 and 50, we get:
t 
Population 
Food Supply 
0 
1000 
2100 
10 
2714.081 
3700 
20 
7366.235 
5300 
30 
19992.557 
6900 
40 
54261.416 
8500 
50 
147269.869 
10100 
Table 2: Population and Food Supply at different time intervals
Figure 2: Population and Food Supply Graph
From the figure shown above, it can be seen that rate of growth in population is much higher as compared to increase in food supply. While the equation for food supply is depicted by a straight line, the population equation is a Jshaped curve that shows greater increase in population as compared to change in time.
 Assuming that each person consumes 1 tonne of food in an year, the year of food shortage can be identified by equating the equations for population growth and food supply and solving it for t. It is shown as follows:
Population growth = Food supply
1000 (1+ 0.105)^{t } = 2100 + 160t
Solving for t, it is found that the population will start experiencing food shortage after 15 years.
 Assuming that each individual consumes 1 tonne of food in an year, the year in which the population’s demand for food exceeds 19,000 per year is given as:
1000 (1+ 0.105)^{t } = 19,000
Solving the above equation for t, we get:
t log (1.105) = log (19)
t × 0.0433 = 1.27875
t = 1.27875/0.0433 = 29.9
Thus, the year in which the population’s demand for food exceeds 19,000 per year is 30^{th} year.
 Assuming that growth rate is 1.01 and that each individual consumes 1 tonne of food in an year the year in which the population’s demand for food exceeds 19,000 per year is given as:
1000 (1+ 0.101)^{t } = 19,000
Solving the above equation for t, we get:
t log (1.101) = log (19)
t × 0.041787 = 1.27875
t = 1.27875/0.041787 = 30.60 or 30 years 7 months
Thus, after 30 years and 7 months, the population demand for food will exceed 19,000 tonnes.
 Yes, if the food grows arithmetically and population grows geometrically, the food supplies will always run out. This is because the populations demand will eventually exceed supply of food thereby causing shortage of food. This relation was describes by Thomas Malthus as Malthusian Disaster that is described as a situation where the population in a particular region exceeds so much that it results into food shortage and starvation. While the modern economists criticized the theory on the grounds that advanced technology has enabled to increase food supply to a larger extent and family planning has decreased birth rates, the theory still holds true. This is because, even though the birth rate has decreased, it still remains higher that the replacement rate thereby leading to increase in consumption and strain of natural resources. Additionally, the world is experiencing depletion in natural resources like fossil fuels, forest areas and clean drinking water as a result of greater than before consumption to support rising population. Thus, if the advanced technology that helps to increase the output per unit of input, does not find ways to economize on natural resources, the earth is headed towards Malthusian Disaster.
Section 3: Solar Panels Investment
 The cheapest solar system type that would provide 50% of the family electricity consumption is 4kW that costs $14,000 and produces 15.6 kWh electricity per day.
 The family consumers 15kWh per day. Hence, in a quarter, its total consumption of electricity is 1365kWh. Out of this, 50% will be used from solar panels and the rest (682.5 kWh) will be taken from the electricity grid. Thus, keeping in view the alternatives available to the family, the family’s quarterly bill is as follows:
cost 
Nonrenewable 
50% Renewable 
100% Renewable 
Alternative source 
$183.183 
$193.6935 
$204.204 
Supply Charge 
$ 62.8628 
$ 62.8628 
$ 62.8628 
Total Cost 
$246.05 
$256.56 
$267.07 
Table 3: Quarterly Electricity Bill from Energy Provider
 The cost of electricity for the next 15 years is as follows:
Cost 
Energy consumption 
Nonrenewable 
50% Renewable 
100% Renewable 
Solar Energy 
2730 kWh 
$0 
$0 
$0 
Alternative source 
First 1000 kwh 
$268.40 
$283.80 
$299.20 

> 1000 kWh 
485.265 
511.907 
538.549 
Supply Charge 
At $0.6908/day 
$251.45 
$251.45 
$251.45 
Yearly Cost 

$1,005.12 
$1,047.16 
$1,089.20 
15 years cost 

$15,076.74 
$15,707.37 
$16,338.00 
PVIF 
At 5.15% 
11.275 
11.275 
11.275 
Present Value 

$169,990.28 
$177,100.63 
$184,210.98 
Table 4: Present Values of electricity bill
 Other than buying solar panels, the quarterly cost of next best alternative that for the family is $3008.8663 that comes from 50% renewable electricity grid.
cost 
50% Renewable Energy 
Quarterly consumption 
1365kWh 
First 1000Kwh 
$283.8 
>1000 kWh 
$108.0035 
Supply Charge 
$ 62.8628 
Total Cost 
$454.6663 
Table 5: Quarterly cost of next best alternative
 The present value for 30 years of the solution is given as follows:
Cost 
Energy consumption 
Nonrenewable 
50% Renewable 
100% Renewable 
Solar Energy 
2730 kWh 
$0 
$0 
$0 
Alternative source 
First 1000 kwh 
$268.40 
$283.80 
$299.20 

> 1000 kWh 
485.265 
511.907 
538.549 
Supply Charge 
$0.6908/day 
$251.45 
$251.45 
$251.45 
Yearly Cost 

$1,005.12 
$1,047.16 
$1,089.20 
30 years cost 

$30,153.49 
$31,414.75 
$32,676.01 
PVIF 
5.15% 
16.11 
16.11 
16.11 
PV 

$485,772.66 
$506,091.56 
$526,410.46 
*For calculating costs, it is assumed that solar panels were reinstalled after 15 years.
Table 5: Present Values of electricity bill
 The NPV is the difference of the total spending and the income generated from a project. In case of the current situation, the outflows refer to the annual electricity bill and installation of solar panels. However, inflow would be considered as the savings in electricity bill as a result of using solar energy along with 50% renewable energy. Thus, the net inflow will be calculated as:
Particulars 
Current bill without solar energy 
Bill with solar energy 
Total consumption 
5460 kWh 
5460 kWh 
Solar energy 
 
$0 
Alternative source 
$283.8 
$283.80 

1319.714 
511.907 
Supply Charge 
$251.45 
$251.45 
Yearly Cost 
$1854.97 
$1,047.16 
15 years cost 
$27,824.478 
$15,707.37 
PVIF At 5.15% 
11.275 
11.275 
Present Value 
$313,720.989 
$177,100.63 
Table 6: Calculation of present value of electricity bill
Net Inflow = Savings = value of electricity bill without solar energy – electricity bill with solar energy
= $136,620.359
NPV = Saving – Initial Investment = $136,620.359  $14,000 = $122,620.359
 Yes, buying solar panels is a good idea because the family will be able to save $122,620.359 in its electricity bill over the period of 15 years by using solar energy.
 Assuming that the family uses non renewable energy source for electricity, the NPV of buying solar panels is calculated as follows:
Particulars 
Current bill without solar energy 
Bill with solar energy 
Total consumption 
5460 kWh 
5460 kWh 
Solar energy 
 
$0 
Alternative source 
$268.4 
$268.40 

1251.03 
485.265 
Supply Charge 
$251.45 
$251.45 
Yearly Cost 
$1770.88 
$1,005.12 
15 years cost 
$26,563.2 
$15,076.74 
PVIF At 5.15% 
11.275 
11.275 
Present Value 
$299,500.08 
$169,990.28 
Table 7: Calculation of present value of electricity bill
Net Inflow = Savings = value of electricity bill without solar energy – electricity bill with solar energy = $299,500.08  $169,990.28
= $129,509.8
NPV = Saving – Initial Investment = $129,509.8  $14,000 = $115,509.8
 Yes, the family in question 8 will purchase the solar panel system as the NPV is positive indicating a savings in electricity bill amounting to $115,509.8.
 Some governments have offered subsidies to consumers to install solar panels in their homes to encourage people to install solar panels as an alternative source of energy and move them towards environment friendly energy consumption and support the industry by attracting customers. As such, homeowners and business owners are given subsidies, rewards and rebates for switching to solar energy consumption.
This problem has been solved.
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