ECON2121-WE01
INTERMEDIATE METHODS FOR ECONOMICS AND FINANCE
2021
SECTION A
1. Sbragia Ltd is a Durham based company. It earns a before-tax profit of £100,000. Sbragia's shareholders have agreed to donate x percent of the after-tax company profits to the British Heart Foundation. The company must also pay a state tax of 5 percent of its profits (after the donation) and a federal tax of 40 percent of its profits (after the donation and the states taxes paid). Answer the following questions using the relevant theory:
a. Write and solve the system of linear equations to compute the company's donation D, state taxes S and federal taxes F when x can vary. (60 marks)
b. Assume that shareholders have voted for a 10 percent donation. What is the net cost of the charitable contribution? (25 marks)
c. If federal income taxes are deduced from state taxes, how much is a 10 percent donation to the British Heart Foundation? Comment. (15 marks)
2. Answer the following questions using the relevant theory:
a. Solve the minimization problem for the function f(x, y) = −x − 3y subject to x² + ay² = 10 and sketch your answer. (60 marks)
b. If a = 1, what is the problem solution? Now suppose that a increase by 1%. Estimate the new optimal value off after the increase in a via the Envelope Theorem. (25 marks)
c. Check that the result obtained in (b) is correct via the direct method. (15 marks)
SECTION B
3. Solve the following differential equations
a. ̇(x) = x − (t2 + t + 1)x2 (35 marks)
b. = 1 + 2√|x| (30 marks)
c. ̇(x) = max{√|x|, x2} (35 marks)
In each case also i) discuss the local existence and uniqueness of a particular solution around (t,x)=(0,0); ii) discuss the maximum interval of existence of the solution(s); iii) draw the integral curves to show your findings.
4. Consider a world with two types of countries, 1 and 2. Each country produces output by using human capital and a linear technology:
yi = Ahi
where hi and yi indicate respectively the human capital and output in a type-i country while A is a positive parameter indicating the level of technology.
Now suppose that the type-1 countries are the leading economies (ℎ1 > ℎ2) where the stock of human capital follows
h1 = h0 egt
with ℎ0 indicating the initial human capital stock and g the growth rate. On the other hand, the stock of human capital in type-2 countries follows the differential equation
ḣ2 = gh2(1)−θℎ1(θ)
with θ ∈ (0,1) .
a. Provide an economic explanation of the differential equation describing the evolution of ℎ2 . (10 marks)
b. Since the growth rate of human capital in country 2 is h2/h2 = g(h2/h1)θ will type-2 countries eventually catch-up with type-1 countries in terms of output? (20 marks)
c. Find the solution of the differential equation and explain the effect of a change in h0 , g and θ on h2 . (35 marks)
d. Draw the integral curves. Based on your finding, explain whether the predicted output dynamics may explain the empirical evidence described in the below figure. (35 marks)