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# Question: Simplify exp (2 ln π΄ + 3 ln π΅ β 4 ln πΆ + 5 ln π·6), where π΄, π΅, πΆ, π· 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 π΄ + 3 ln π΅ − 4 ln πΆ + 5 ln π·6), where π΄, π΅, πΆ, π· are all positive quantities.

Q2        Revenue π(π, π) = ππ is a function of two variables: price π and quantity π. Sketch a contour map of π in the region 0 ≤ π ≤ 10, 0 ≤ π ≤ 20.

HINTS: (i) What mathematical shape is exemplified by the equation π₯π¦ = 12? What would a contour map of the function π(π₯, π¦) ≡ π₯π¦ look like?

1. One suitable contour height for π(π, π) is β = 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 π‘ hours, Robinson Crusoe can obtain π(π‘) kg of berries, where π(π‘) = 5√π‘. 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 π currently stands at \$20 tn/yr (trillion dollars per year) and consumption πΆ = πΆ(π) at \$16 tn/yr. If marginal consumption πΆ'(π) = 0.7, use the Small Increment Formula to obtain an approximation to the level of consumption if π rises by \$0.2 tn/yr. (b) The general price level π is rising over time according to the formula π(π‘) = π0(exp(πΌπ‘) + ππ‘), where π0, πΌ, and π are positive parameters. Calculate: (i)         The rate of increase of prices, πΜ at π‘; (ii)        The rate of growth of prices π^ at π‘. Then (iii)       Evaluate both πΜ  and π^ when π‘ = 0.

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

Q5        Consider the optimization problem whose formal statement is

min π(π₯), where π(π₯) ≡ 3π₯4 − 4π₯3 − 12π₯2 + 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 π''(π₯), decide which critical points, if any, are strict local minima of π(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(π΄, π΅) = π΄π΅. He loses π bananas, reducing his banana holding to π΅ − π. How many extra apples π will he need as compensation, to restore him to his former level of utility?

(b)        From consuming π΄ apples and π΅ bananas, Ms Jones gets utility π(π΄, π΅) =

√π΄π΅ + π΄ + 2π΅. What is Ms Jones’ marginal utility with respect to apples, in terms of π΄ and π΅ ?

Q7        My utility function over apples (A) and bananas (B) is π(π΄, π΅) = π΄2π΅. 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 π΄, π΅, explaining your working clearly. You may assume that π΄ ≠ 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₯ ∈ β(+) ∧ (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 π mean ‘The workforce is well-educated’. Let π‘ 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 β(+), and why is this set important in economic applications?
2. From income π dollars per year where 0 ≤ π ≤ 106, and leisure πΏ hours per day, where 0 ≤ πΏ ≤ 24, I get utility π(π, πΏ), which is a non-negative real number. State the domain and codomain of π, and use these to express π 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

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