ECON2121-WE01
INTERMEDIATE METHODS FOR ECONOMICS AND FINANCE
2023
SECTION A
1. Solve the following differential equations
a. (30 marks)
b. ̇(x) = x − (t2 + t + 1)x2 (30 marks)
c. (40 marks)
In each case also i) discuss the local existence and uniqueness; ii) discuss the maximum interval of existence of the solution(s) (HINT: in 1a and 1b you may find it easier to restrict your analysis to the cases with an arbitrary constant equal to 1 and -1); iii) draw the integral curves to show your findings.
2. Consider the following version of the Solow model:
where 0 < S1 < S2 < 1. Answer the following questions
a. Using a phase diagram show how the number of equilibrium points depends on the threshold k(̅) , and find their value. (40 marks)
b. Show both analytically and geometrically the stability properties of the equilibria. (30 marks)
c. Provide an economic interpretation of the results and explain whether and under which parameter conditions a poverty trap can emerge in this model. (30 marks)
SECTION B
3. Consider the following system of linear equations:
Discuss the consistency of this system.
If the system is consistent, explain which solving method can be applied in this case and find the solution. (100 marks)
4. Consider a firm which uses two inputs of production - capital (K) and labour (L). Market prices of these inputs are Pk = 10 and PL = 5. The firm seeks to minimize the total cost TC = PkK + PLL subject to the following constraint on output:
Q = 20K + 15L − (K2 + L2 ) = 55
a. Find the solution of this constrained optimisation problem using the Lagrangian. (40 marks)
b. Check the solution for a minimum using the bordered Hessian. Compute the minimum total cost. (60 marks)