Q1 and Q2 - Week 1. From Pure Maths To The Maths Of Economics And Finance

Q1        Simplify exp (2 ln 𝐴 + 3 ln 𝐵 − 4 ln 𝐶 + 5 ln 𝐷⁶), where 𝐴, 𝐵, 𝐶, 𝐷 are all positive quantities.

Q2        Revenue 𝑅(𝑃, 𝑄) = 𝑃𝑄 is a function of two variables: price 𝑃 and quantity 𝑄. Sketch a contour map of 𝑅 in the region 0 ≤ 𝑃 ≤ 10, 0 ≤ 𝑄 ≤ 20.

HINTS: (i) What mathematical shape is exemplified by the equation 𝑥𝑦 = 12? What would a contour map of the function 𝑓(𝑥, 𝑦) ≡ 𝑥𝑦 look like?

2. One suitable contour height for 𝑅(𝑃, 𝑄) is ℎ = 50. Pick other contour heights using your own judgment.
3. Note that the question asks for a sketch, not an exact plot.

Q3 and Q4 - Week 2: Things Economists Do With Differentiation

Q5 - Week 3: Unconstrained Optimization With A Single Choice Variable

Q5        Consider the optimization problem whose formal statement is

min 𝑓(𝑥), where 𝑓(𝑥) ≡ 3𝑥⁴ − 4𝑥³ − 12𝑥² + 10 [P]

x

1. Write down the first-order condition for this problem.
2. Find any critical points of the problem.
3. By using the second derivative function 𝑓^''(𝑥), decide which critical points, if any, are strict local minima of 𝑓(𝑥).
4. Using the information obtained so far, solve the problem P.

Q6 and Q7 - Week 4: Many Variables. Constrained Optimization.

Q6        (a)         Mr Jones has utility function 𝑈(𝐴, 𝐵) = 𝐴𝐵. He loses 𝑏 bananas, reducing his banana holding to 𝐵 − 𝑏. How many extra apples 𝑎 will he need as compensation, to restore him to his former level of utility?

(b)        From consuming 𝐴 apples and 𝐵 bananas, Ms Jones gets utility 𝑈(𝐴, 𝐵) =

√𝐴𝐵 + 𝐴 + 2𝐵. What is Ms Jones’ marginal utility with respect to apples, in terms of 𝐴 and 𝐵 ?

Q7        My utility function over apples (A) and bananas (B) is 𝑈(𝐴, 𝐵) = 𝐴²𝐵. If apples cost

$6 per kg, and bananas cost $4 per kg, and I have $50 to spend. I wish to maximize utility, subject to the constraint that the value of my purchases equals by budget limit $50.

1. State my problem formally, as a problem of constrained optimization, and say what the objective function and choice variables are.
2. Write down the corresponding Lagrangean function.
3. Obtain first-order conditions, and state what other condition is needed to obtain a solution.
4. Obtain optimal values 𝐴^∗, 𝐵^∗, explaining your working clearly. You may assume that 𝐴^∗ ≠ 0 at the solution. You may also assume that the second- order conditions for this constrained problem are satisfied.

Q8 and Q9 - Week 5: Logic, Sets, and Functions.

Q8        (a)         Express the set {𝑥: 𝑥 ∈ ℝ₍₊₎ ∧ (𝑥² < 16)} using interval notation.

2. Are the following three statements logically equivalent? Use the symbols and methods of propositional logic to find out, and explain.
   1. Without a well-educated labour force, the economy cannot thrive.
   2. If there is a well-educated labour force, the economy can thrive.
   3. If the economy can thrive, the labour force must be well-educated.

HINT. Let 𝑒 mean ‘The workforce is well-educated’. Let 𝑡 mean ‘The economy can thrive’.