mathematical economics

Consider a project that has the following cashflow: x0 = 1.25, x1 = 3, x2 = 3, x3 = 8.

(a) Define the present value P V (x, r) of this project as a function of an interest rate r. Produce 1-2 plots showing P V (x, r) as a function of r to illustrate its shape. [10 marks]

(b) Find the internal rate of return and prove that it is unique. [25 marks]

(c) Prove that the internal rate of return is quasi-concave for projects x0, x1, ...., xn with x0 > 0. Explain what statement needs to be proved and write out the formal proof [25 marks]