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Fundamental Methods of Mathematical Economics

(ECO 3410)

Additional Assignments for Extra Credits

1.Use four-step process calculating derivative to prove

!(!!)

!” = !!!!!

2. Find the derivative of each function

(a) ! ! = 8 !! + 3 ! − 14

(b) !(!) = (3! − 10)!!

(c) ! ! = !!!!

(d) ! ! = ln !!

(e) ! ! = (2! + 4)!

(f) ! ! = !!!!

!!!!

(g) ! ! = log !!

(h) ! ! = 10 !! + 7 ! !

(i) ! ! = 1 − 0.5!!” ∗ 5! !!

(j) ! = !

!!!!!

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3. Production Strategy

A monopoly company has the following price-demand

equation:

! = 10000 − 1000!

!ℎ!”! !ℎ! ! !” !ℎ! !”#$%&'(#) !”#$”#

!”# ! !” !ℎ! !”#$% !” !”#$%&’

And its financial department also gives the cost function

as

Total Cost = Fixed Costs + Variable Costs,

Note: fixed costs is always $7000 and variable costs is

$2 per unit of production

(a) Find the domain of the function defined by the

price-demand equation

(b) Define the marginal revenue function and profit

function

(c) Calculate the optimal production level of x when

MR = MC

(d) If the company decides to produce the good

when p = AR at the optimal production level of x,

what is the price of good? ( Note: AR is the average

revenue)

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4. Profit Maximization (Perfect Competition)

In the model of perfect competition, all firms are pricetakers

since they treat price as a market-determined

constant. Firm Perfcomp’s total revenue function is

!” ! = ! ∗ !,

in which P equals the output price. Assume that P = 24

and the total cost function is

!” ! = !! − 7.5!! + 36! − 10

(a) Determine the firm’s profit function

(b) Determine the level of output at which Firm

Perfcomp should produce in order to maximize

profits

(c) Determine the price at the level of output at

which Firm Perfcomp produces

(d) Confirm that this quantity represents maximum

profits for the firm by using the second-order

condition.

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5. Profit Maximization (Monopolist)

Firm Perfcomp is acquired by a bigger firm called

MonopoCorp. Overtime, MonopoCorp successfully

acquires all other widget-producing firms and becomes

the monopolist in the market. MonopoCorp’s average cost

for producing Q quantity goods are 12,

!” ! = 12,

The inverse market demand for widgets is a linear

function of Q quantity,

! ! = 36 − 4!

(a) Determine the MonopoCorp’s profit function.

(b) Determine the level of output at which Firm

MonopoCorp should produce in order to maximize

profits.

(c) Determine the price at the level of output at

which Firm MonopoCorp produces.

(d) Confirm that this quantity represents maximum

profits for the firm by using the second-order

condition.

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