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)