ECON131 Quantitative Methods in Economics Business and Finance

You may assume that electricity prices remain constant, and inflation can be ignored. Assume the power generated by the solar panels is the same year-round.

1. Which is the cheapest solar system type that would provide 50% of the family electricity consumption?
2. 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.
3. Using an annual discount rate of 5.15% and ignoring inflation, what is the present value of 15 years’ worth of electricity bills?
4. Other than buying solar panels, what is the quarterly cost of the next best alternative, which still provides 50% renewable energy?

5. What is the present value of 30 years’ worth of this solution? Again, ignore inflation, and use a discount rate of 5.15%.
6. 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.
7. Interpret your result from question (6). Is buying solar panels a good idea? Explain your answer.
8. Under the same assumptions as above, assume the family is not committed to purchasing renewable energy (i.e. the family is happy with consuming non-renewable energy from the grid.) What is the NPV of purchasing solar panels now?
9. Will the family in question (8) purchase the solar panel system named in question (1)?
10. Some governments have offered subsidies to consumers to install solar panels in their homes. Why?

Section A: Population Growth

1. The exponential population growth graph exhibits a J-shaped 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, P₀ is the y-intercept and represents the population level at time (t) = 0.

2. The given formula for population growth is:

Re-arranging the formula to solve for r, we get:

Taking natural log for both sides, we get:

3. The annual average population growth is calculated as follows:

Here, 2011 = 0 and 2015= 4

China

Here, P_t = 1371 (population in 2015)

P₀ = 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)

P₀ = 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

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 J-shaped curve that shows greater increase in population as compared to change in time.

5. 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.

6. 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.

7. 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.

8. 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

1. 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.
2. 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:

Table 3: Quarterly Electricity Bill from Energy Provider

3. The cost of electricity for the next 15 years is as follows:

Table 4: Present Values of electricity bill

4. 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.

Table 5: Quarterly cost of next best alternative

5. The present value for 30 years of the solution is given as follows:

*For calculating costs, it is assumed that solar panels were re-installed after 15 years.

Table 5: Present Values of electricity bill

6. 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:

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

7. 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.
8. Assuming that the family uses non renewable energy source for electricity, the NPV of buying solar panels is calculated as follows:

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

9. 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.
10. 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.