Fundamental methods of mathematical economics

1
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) ! = !
!!!!!
2
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)
3
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.
4
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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