ECN 505 – Mathematical Economics Problem Set 1 Spring 2017

ECN 505 – Mathematical Economics Problem Set 1 Spring 2017

This problem set is due Wednesday, February 8, in class. Please submit a legible paper copy of your write-up and staple multiple pages together. Please indicate your name at the top of the first page of your submission. You are allowed to work together with your classmates, but each of you is expected to submit a complete and independent write-up. Please justify all your answers.

Merely stating the final result without any explanation is not sufficient for full credit, even if your answer is correct.

Problem 1: A power company can develop a hydroelectric project at one of two capacity levels—one megawatt or two megawatts—with costs of $1 billion and $1.75 billion, respectively. Construction costs are incurred immediately in either case. If the smaller capacity is chosen, capacity cannot be increased later. The company can sell one megawatt capacity for a net revenue of $200 million per year, starting three years from now and continuing forever. An additional buyer is willing to purchase a second megawatt (if developed) for $300 million per year, starting eleven years from now, in perpetuity. Find the net present value (NPV) of both capacity options with a 5% interest rate. Determine whether the company should invest in 0, 1, or 2 megawatts.

Problem 2: (a) Compute the limit as n approaches infinity of the sequence:

\(a_n = \frac{8n^4 + 6n^3}{2n^2 + n} \; \text{and} \; \frac{5 2n^4 + 13n^3}{20n^2} \; \text{(as in the original problem)}.\)

(b) Consider the sequence \(a_t = \frac{1}{7^t}\) for \(t = 1, 2, 3, \dots\). Compute the limit of the series \(s_n = \sum_{t=1}^n a_t\) as \(n \to \infty\).

Problem 3: Two policy proposals combine income-support with taxation. Let \(x\) be earned income before taxes/transfers, and \(y\) be income after taxes and transfers.

  • Plan A: If \(x=0\), receive \$6,000 support; no support if \(x>0\). Income above \$6,000 is taxed at 20% on excess.
  • Plan B: Everyone gets \$6,000; income above this is taxed at 40%.

(a) Derive formulas for \(y\) as a function of \(x\) for each plan; graph these functions.

(b) Which plan is more favorable to taxpayers, and how does preferences depend on \(x\)?

Problem 4: Two firms produce identical goods, setting prices \(p_1\) and \(p_2\). The lower price wins the entire market; if prices are equal, market is shared equally. Demand function: \(y=50p\); cost per unit: \$10.

  • (a) For \(p_2 > 10\), find firm 1's profit \(\pi_1(p_1)\) as a function of \(p_1\). Sketch the profit function and identify any discontinuities.
  • (b) Determine the equilibrium prices and quantities sold in the Bertrand model.

Sample Paper For Above instruction

This paper explores key economic decision-making processes through several analytical problems. It begins with a net present value (NPV) calculation for a hydroelectric project, followed by an analysis of sequences and series. It then evaluates two taxation and income transfer policies and concludes with a theoretical examination of Bertrand price competition in a duopoly market. The goal is to demonstrate mastery in applying mathematical tools to economic scenarios, providing insights into investment decisions, growth patterns, policy impacts, and strategic firm behavior.

Problem 1: NPV of Hydroelectric Investment

A utility company's decision to develop a hydroelectric project involves evaluating the financial viability of different capacity options, considering initial costs and future revenue streams. The costs for one and two megawatt capacities are $1 billion and $1.75 billion, respectively. The revenue prospects depend on market agreements: the first megawatt is sold at a perpetual annual revenue of $200 million, starting three years from now, and another buyer offers $300 million annually for the second megawatt starting in eleven years.

To compute the NPVs, we utilize the standard formula for present value of perpetuities, PV = C / r, where C is annual cash flow and r is the discount rate (5%). For revenue starting after a delay, the present value incorporates a discount factor for the delay: PV = C / r * (1/(1+r)^t), where t is the delay in years.

For the first capacity (1 MW):

PV of revenue starting in 3 years = \$200 million / 0.05 (1 / (1.05)^3) ≈ \$200 million / 0.05 0.8638 ≈ \$3.455 billion.

For the second capacity, since revenue starts in 11 years:

PV of second buyer’s revenue = \$300 million / 0.05 (1 / (1.05)^11) ≈ \$6 billion 0.5820 ≈ \$3.492 billion.

Adding the initial costs and subtracting the present value of revenues yields the NPVs.

For the single megawatt:

NPV = -\$1 billion + \$3.455 billion ≈ \$2.455 billion.

For both megawatts:

NPV = -\$1.75 billion + \$3.455 billion + \$3.492 billion ≈ \$5.197 billion.

Thus, investing in two megawatts provides a higher NPV, encouraging the company to develop the full capacity.

Problem 2: Sequences and Series

(a) The sequence is:

\[

a_n = \frac{8n^4 + 6n^3}{2n^2 + n} \quad \text{and} \quad \frac{5 2n^4 + 13n^3}{20n^2}.

\]

By dividing numerator and denominator by \(n^4\), the limits can be determined:

\[

\lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{8 + 6/n}{2/n + 1/n^2} \to 4.

\]

Similarly for the second term, the limit is found to be 0.5.

(b) For the series \(s_n = \sum_{t=1}^n a_t\), the sum converges if the sequence \(a_t\) tends to zero. Since \(a_t \to 0.5\), the series diverges to infinity.

Problem 3: Income Support and Tax Policies

Plan A offers a fixed support of \$6,000 to individuals with no income, and imposes a 20% tax rate on income above \$6,000. Mathematically:

\[

y =

\begin{cases}

6,000, & x=0 \\

x - 0.2(x - 6,000), & x > 6,000

\end{cases}

\]

which simplifies to \( y = 6,000 \) if \(x \leq 6,000\), and \( y = 0.8x + 1,200 \) if \(x > 6,000\).

Plan B provides a \$6,000 basic income to all, with income taxed at 40%:

\[

y = 6,000 + (1 - 0.4)(x) = 6,000 + 0.6x.

\]

Graphing both functions reveals that for low \(x\), Plan A yields a flat level of \$6,000 support. For higher \(x\), plan B results in higher net income, making it more favorable to higher earners.

From the taxpayer's perspective, Plan A benefits those with income close to or less than \$6,000, while Plan B favors higher earners due to the flat basic income and lower tax rate. The choice depends on individual income levels, with low-income taxpayers preferring Plan A and higher-income taxpayers favoring Plan B.

Problem 4: Bertrand Competition

In this duopoly, firms set prices \(p_1, p_2\). The losing firm sells zero units; if both set the same price, they split the market. Market demand at price \(p\): \(y=50p\); costs are constant at \$10 per unit.

(a) For fixed \(p_2 > 10\), profit of firm 1:

\[

\pi_1(p_1) =

\begin{cases}

( p_1 - 10 ) \times 50 p_1, & p_1

\frac{1}{2}( p_1 - 10 ) \times 50 p_1, & p_1 = p_2.

\end{cases}

\]

The profit sharply drops to zero if \(p_1 > p_2\). The graph shows a continuous rise up to just below \(p_2\), then discontinuity at \(p_1=p_2\).

(b) The Nash equilibrium occurs at the price marginally above cost, \(p^* = 10\). Both firms set prices just above 10, selling half the market demand \(y=50 \times 10 = 500\) units each. They earn just above zero profit, as any higher price invites the other to undercut.

Conclusion

This analysis demonstrates how mathematical economics tools—NPV calculations, limits of sequences, utility functions, and game theory—provide quantitative insights into strategic investment, growth, taxation, and pricing decisions. Such frameworks are vital for policymakers and firms to optimize economic outcomes under various constraints and competitive environments.

References

  • Brealey, R. A., Myers, S. C., & Allen, F. (2020). Principles of Corporate Finance (13th ed.). McGraw-Hill Education.
  • Varian, H. R. (2010). Intermediate Microeconomics: A Modern Approach (8th ed.). W. W. Norton & Company.
  • Sraffa, P. (1960). Production of Commodities by Means of Commodities. Cambridge University Press.
  • Samuelson, P. A., & Nordhaus, W. D. (2010). Economics (19th ed.). McGraw-Hill Education.
  • Frank, R. H., & Bernanke, B. S. (2015). Principles of Economics (6th ed.). McGraw-Hill Education.
  • Mas-Colell, A., Whinston, M. D., & Green, J. R. (1995). Microeconomic Theory. Oxford University Press.
  • Cournot, A. (1838). Researches into the Mathematical Principles of the Theory of Wealth. Dover Publications.
  • Tirole, J. (1988). The Theory of Industrial Organization. MIT Press.
  • Luenberger, D. G. (1969). Optimization by Vector Space Methods. Wiley-Interscience.
  • Perloff, J. M. (2016). Microeconomics (7th ed.). Pearson.

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