Financial Math Answers

5. Parexel Co. produces and sells cell phone batteries. Its production function is 𝒒 = 𝟑𝟐𝟏𝑲𝟎.𝟑𝑳𝟎.𝟑, while its cost function is 𝒄 = 𝟐𝟏𝟎𝑲 + 𝟗𝟎𝑳. Parexel can sell its batteries at 𝒑 = 𝟔𝟑.

a. Does this production function exhibit increasing, decreasing or constant returns to scale? Explain briefly.

[2]

b. Does this production function exhibit decreasing returns to capital? Explain briefly. [2]

c. Use implicit differentiation to find the marginal rate of technical substitution, 𝑴𝑹𝑻𝑺. [2]

d. Write down an expression for the isoquant if 𝒒 = 𝟐𝟎𝟒𝟖. [2]

e. Write down an expression for the labour elasticity of production, 𝒆𝑳. [2]

f. Which levels of 𝑳 and 𝑲 satisfy the the first-order and second-order conditions for the maximisation of

Parexel’s profit? [8]

g. Write down the levels of profit, revenue, cost and output at the profit maximising levels of 𝑲 and 𝑳. [2]

 If you wish to save $100,000 at the end of 5 years, how much will you need to invest each month if the interest rate is 12% compounded monthly?

Find fx(x,y), fy(x,y), fx(1,3), and fy(-2,4) for the given function. If

𝑧 = 𝑓(𝑥, 𝑦) = 3𝑥ଷ𝑦ଶ - 𝑥ଶ𝑦ଷ + 4𝑥 + 9

You give up a full-time salary of $45,000 a year to go to school for 2 years. The total cost of going to school is $30,000. If you want to be able to recover your investment in 5 years or less, what is the minimum salary you would need to earn upon earning your degree?

The inverse demand for table salt is 𝒑 = 𝟐𝟎𝟎 , while the inverse supply of table salt is

𝒒𝒅 +𝟏

𝒑 = 𝟏𝟎 + 𝟐𝒒𝒔.

a. Find the equilibrium price of table salt before AND after the imposition of a 40% ad valorem tax on the

consumers of table salt. [2]

b. Describe the distribution of the burden (incidence) of this ad valorem tax between consumers and

producers. [2]

c. Find and interpret the price elasticity of supply (𝒆𝒔) at the after-tax equilibrium price and quantity. [2]

Suppose the short-run cost function of Tyrell Corporation is 𝒄 = 𝟓𝟎𝟎𝟎 + 𝟏𝟎𝟎𝒒 + 𝟖𝒒^𝟐. 

a.  Find the level/s of  that satisfies/satisfy the first-order and second-order conditions for the minimisation

of Tyrell’s average cost (𝒂𝒄). [4]

Show that 𝒂𝒄 equals marginal cost (𝒎𝒄) at the level/s of  obtained in (a). [2

Jennifer opted for the plan that will pay her $ 52,000 annually for 15 years. The cash will be paid at the end of each year

Required

A. Assuming that the interest will remain at 6.8 % constant over the whole period , how much will the insurance company disburse at the end of the 15 years of annuity payment to Jennifer? [ordinary annuity]

B. If the payment of the cash flow is done in the beginning of the period , what will be the sum for the insurance company will disburse? [interest rate changes to 7.5%]

C. If Jennifer is supposed to invest a sum at the beginning of the retirement in the annuity fund of the insurance , calculate the sum she has to give to the company if

a. Payment is received at the end of each period and the interest rate is 4.8%

b. Payment is done at the beginning of each period and the interest rate is 5.8%

[ Annual payment = $52,000 and period is 15 years]

Mr LEO’s father has made a will to give him an amount of $500,000 in 10 years .So Leo wants to borrow an amount from the bank to set up a fashion boutique in the nearest town. The business planner told him that he does not expect the present discount rate to change in these 10 years. The following information is given regarding the discount rate

  GDP growth of the country

1.2%

Beta

1.8

Index in January 2020

16,570

Index in December 2020

17,590

Inflation

2.5%

 REQUIRED

How much should he borrow from the bank now if he is planning to pay back the loan with the money he will get in 10 years of business return?

Suppose Maria gets R13240.10 after investing a lump for 5 years.The bank quoted 11.5% simple interest.How much did Maria invest at the beginning?

A firm estimates that it can sell Q units of its product with an advertising expenditure of x thousand dollars where

Q = Q(x)=x² + 600x + 25

i) Over what level of advertising expenditure is the number of units of

product sold increasing?

ii) Over what level of advertising expenditure is the number of units of product sold decreasing?