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Question: Simplify exp (2 ln 𝐴 + 3 ln 𝐡 βˆ’ 4 ln 𝐢 + 5 ln 𝐷6), where 𝐴, 𝐡, 𝐢, 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 𝐡 − 4 ln 𝐢 + 5 ln 𝐷6), where 𝐴, 𝐡, 𝐢, 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 π‘₯𝑦 = 12? What would a contour map of the function 𝑓(π‘₯, 𝑦) ≡ π‘₯𝑦 look like?

  1. One suitable contour height for 𝑅(𝑃, 𝑄) is β„Ž = 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




By gathering berries for 𝑑 hours, Robinson Crusoe can obtain 𝑋(𝑑) kg of



berries, where 𝑋(𝑑) = 5√𝑑. How many kg of berries will he obtain in his fifth



hour of labour?



Robinson works continuously, starting at 8:00 am. At noon, what is his



marginal product of labour-time?






National income π‘Œ currently stands at $20 tn/yr (trillion dollars per year) and consumption 𝐢 = 𝐢(π‘Œ) at $16 tn/yr. If marginal consumption 𝐢'(π‘Œ) = 0.7,



use the Small Increment Formula to obtain an approximation to the level of



consumption if π‘Œ rises by $0.2 tn/yr.



The general price level 𝑃 is rising over time according to the formula 𝑃(𝑑) =



𝑃0(exp(𝛼𝑑) + 𝑏𝑑), where 𝑃0, 𝛼, and 𝑏 are positive parameters. Calculate:



(i)         The rate of increase of prices, 𝑃̇ at 𝑑;



(ii)        The rate of growth of prices 𝑃^ at 𝑑. Then



(iii)       Evaluate both 𝑃̇  and 𝑃^ when 𝑑 = 0.


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


Q5        Consider the optimization problem whose formal statement is

min 𝑓(π‘₯), where 𝑓(π‘₯) ≡ 3π‘₯4 − 4π‘₯3 − 12π‘₯2 + 10 [P]


  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''(π‘₯), decide which critical points, if any, are strict local minima of 𝑓(π‘₯).
  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 π‘ˆ(𝐴, 𝐡) = 𝐴𝐡. He loses 𝑏 bananas, reducing his banana holding to 𝐡 − 𝑏. How many extra apples π‘Ž will he need as compensation, to restore him to his former level of utility?

(b)        From consuming 𝐴 apples and 𝐡 bananas, Ms Jones gets utility π‘ˆ(𝐴, 𝐡) =

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



Q7        My utility function over apples (A) and bananas (B) is π‘ˆ(𝐴, 𝐡) = 𝐴2𝐡. 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, 𝐡, 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 {π‘₯: π‘₯ ∈ ℝ(+) ∧ (π‘₯2 < 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 𝑑 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 ℝ(+), and why is this set important in economic applications?
  2. From income π‘Œ dollars per year where 0 ≤ π‘Œ ≤ 106, and leisure 𝐿 hours per day, where 0 ≤ 𝐿 ≤ 24, I get utility π‘ˆ(π‘Œ, 𝐿), which is a non-negative real number. State the domain and codomain of π‘ˆ, and use these to express π‘ˆ in colon/arrow notation. HINT. The domain and codomain are always sets.


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