Reference no: EM132896596

ECON10192 Introduction to Mathematical Economics - The University of Manchester

Part A: Please answer any TWO of Questions 1-3.

1.(a)What is a sequence? Describe (formally and in your own words) what it means for such a sequence to converge to a limit z ∈ R.
(b) Compute the first five elements of the sequence {z_n}^∞_n=1, where

zn = 2n²/(1 + 3n + 2n²)

(c) Prove the sequences {x_n}^∞_n=1 and {y_n}^∞_n=1 converge, where

x_n = 2n/1+n and y_n = 2n/1+2n

Evaluate lim x_n and lim y_n.

(d) Using your answer to part (c), and carefully stating and describing any additional results you use, show that lim z_n = 1.

2.(a) Explain in your own words what it means for a function f : R |→ R to be k-times continuously differentiable (or C^k ) at a point x₀ ∈ R. (b)Consider the function f : R i→ R, where

f (x) = { 1/2(x+1)²    if x < 1,
           {    2x            if x ≥ 1.

Plot this function in a graph.

(c) Prove that f is C¹ at x₀ = 1.
(d) Prove whether f is C^k at x₀ = 1 for any k > 1. Is it C^k for all k > 1 everywhere else? Explain your answer.

3.(a) Define what it means for a set U ⊆ R to be compact.

(b) Prove that the set given by U = [ 0, 1) ∪ [ 2, 3] is not compact.

(c) Consider the function g : U i→ R where

g(x) = max {1 - (x - 1)², 1/2 x- 1} .

Plot this function in a graph.

(d) In your own words, explain what it means for x₀ ∈ U to be a global maximum of the function g. Prove that g does not have a global maximum. Discuss this fact in the light of your answer to part (b).

Part B:

professor chooses how much time to spend on three activities: research, teaching, and administrative duties. The amount of time spent conducting research is written z, the amount of time spent teaching y, and the amount of time spent attending to administrative duties x. The professor has a total amount of time t > 1, and has preferences over these three activities represented by a utility function u : R₊³i→ R, with

u(x, y, z) = ln x + ln y + a ln z, where a > 0

is a fixed parameter. The time spent conducting research must be at least 1 or else the professor will be fired; y and z cannot be negative.

(a) Write down the professor's constrained optimization problem.

(b) Prove that the professor's objective function is concave and that the professor's constraint set is convex.

(c) Why will a non-degenerate constraint qualification (NDCQ) hold?

(d) Write down the Lagrangian L for this problem. Compute the first- order conditions. Write down the associated complementary slack- ness, feasibility, and non-negative multiplier conditions.

(e) Briefly explain why the solution to the equations in your answer to part (d) will generate the solution to the problem in part (a).

(f) find the optimal amount of time spent on each activity (x∗, y∗, z∗) when t > a + 2. What happens when t ≤ a + 2?

(g) Derive the marginal utility of time as a function of t and plot it in a graph. Is this function continuous at t = a + 2? Is it differentiable?

(h) Show that the multiplier associated with the constraint that at least time 1 must be spent conducting research is given by

max {0, (a + 1)/(t-1) - 1}.

Interpret this multiplier both when t > a + 2 and when t ≤ a + 2.