Call/WhatsApp/Text: +44 20 3289 5183

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

05 Nov 2023,9:58 PM

# 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?

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

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

 Q3 (a) By gathering berries for 𝑡 hours, Robinson Crusoe can obtain 𝑋(𝑡) kg of berries, where 𝑋(𝑡) = 5√𝑡. How many kg of berries will he obtain in his fifth hour of labour? (b) Robinson works continuously, starting at 8:00 am. At noon, what is his marginal product of labour-time? Q4 (a) National income 𝑌 currently stands at \$20 tn/yr (trillion dollars per year) and consumption 𝐶 = 𝐶(𝑌) at \$16 tn/yr. If marginal consumption 𝐶'(𝑌) = 0.7, use the Small Increment Formula to obtain an approximation to the level of consumption if 𝑌 rises by \$0.2 tn/yr. (b) The general price level 𝑃 is rising over time according to the formula 𝑃(𝑡) = 𝑃0(exp(𝛼𝑡) + 𝑏𝑡), where 𝑃0, 𝛼, and 𝑏 are positive parameters. Calculate: (i)         The rate of increase of prices, 𝑃̇ at 𝑡; (ii)        The rate of growth of prices 𝑃^ at 𝑡. Then (iii)       Evaluate both 𝑃̇  and 𝑃^ when 𝑡 = 0.

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.

1. 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’.

Q9.       (a)         What is meant by [2,3)? Explain informally, then formally, using bound- variable notation.

1. What is meant by (+), and why is this set important in economic applications?
2. From income 𝑌 dollars per year where 0 ≤ 𝑌 ≤ 106, and leisure 𝐿 hours per day, where 0 ≤ 𝐿 ≤ 24, I get utility 𝑈(𝑌, 𝐿), which is a non-negative real number. State the domain and codomain of 𝑈, and use these to express 𝑈 in colon/arrow notation. HINT. The domain and codomain are always sets.

https://apaxresearchers.com/storage/files/2023/11/05/9667-laS_21_58_01_questions.pdf

This Question Hasn’t Been Answered Yet! Do You Want an Accurate, Detailed, and Original Model Answer for This Question?