Question

both 3 and 4 need to be answered

3. Those in favour of immediate reductions in fossil fuel consumption in favour of higher cost renewable energy to reduce global warming, which is expected to begin creating damages 1 years in the future, typically use a very low discount rate, i.e., close to zero. Those who are critical of immediate reductions in fossil fuel consumption typically use a discount rate closer to 3%. (a) Express this problem in terms of a net present value formula, where is the cost today of higher energy sources and B is the damages forgone beginning T periods in the future. Express your discounting in terms of continuous compounding and assume that once the benefits begin occurring, they occur forever. Letr be the discount rate. Assume T is a fixed known date. (b) Show that if the costs of renewable energy are a one time cost of installation today, how an increase in the discount rate affects the net present value of immediate reductions. (c) Show how an increase in T, the number of years before benefits of clean energy take place, affects the net present value. 4. Edna has found your dream vehicle that it is priced at $30,000. She can arrange for financing from the bank with 0 down and monthly payments at the end of each month) at an annu- alized interest rate of 12% (r = 0.12), with a term of 60 months. If she buys it, it is hers. Alternatively, the dealer offers to lease it to her for four years. The terms of the lease are monthly payments (end of month) of $400 for 48 months plus a down payment of $4,000. At the end of the lease the car ownership reverts to the dealer. (a) If she finances the car, what is Edna's monthly payment? Hint: first calculate Edna's monthly interest rate, I'm; then find her monthly payment. Remember that her first payment doesn't occur until the end of the first month! (b) Find the market value of the car at the end of year four which would make Edna indif- ferent between leasing and buying.

Answer 1

a)The discount rate is the expression of time preference. Higher the value of discount rate implies that present consumption is preferred more than future consumption

Given that benefit B will start T years later and then after it will begin occurring continuously.

Present value of benefit occuring T years later in continous compounding is \(\mathrm{Be}^{\mathrm{r} T}\) where \(r\) is discount rate.

If this occurs forever, the total present value will be as calculated

$$ \int_{T}^{\infty} B e^{-\tau T} d t $$

\(B\left[e^{-\tau T]_{T}^{\infty} /-r}\right.\)

which is \(\mathrm{Be}^{-\mathrm{r} T} / \mathrm{r}\)

Present discounted value \(=\) Benefit - cost

\(=\mathrm{Be}^{-\mathrm{rT}} / \mathrm{r}-\mathrm{C}\)

\(=\left(\mathrm{B} / \mathrm{re}^{\mathrm{rt}}\right)-\mathrm{C}\)

b) if the rate of discounts rises, the net present value decreases, as can be seen from the above relation of present discounted value and discount rate. Net present value is inversely proportional to re \(^{\mathrm{rt}}\)

c) similarly since the present benefit is inversely related to \(\mathrm{T}\)

As net present benefit \(=\mathrm{Be}^{-\mathrm{r} T} / \mathrm{r}\) which is equal to \(\mathrm{B} / \mathrm{re}^{\mathrm{rt}}\)

Net present benefit is negatively related to number of years before benefit of Clean energy take place.

As T increases, the net present value decreases.