Economics Homework Discussion Questions

Economics Homework Discussion Questions

ECON7150 Assignment 1
All answers must be hand written clearly, scanned as one pdf document (scanners
are available at the Library) and submitted ONLY ONCE via Turnitin by 17th Sept 12
pm. Multiple submissions are NOT allowed, so make sure before submitting.
The total mark of 48 will be converted to make up 30% of the final exam mark.
Questions are based on material covered in lectures 1 to 6.
Make sure you show all steps, key formulae, and workings clearly.
Question 1 (8 marks)
A firm has a demand function
q p   100
and a total cost function
1 3 2 7 111 50
3
C q q q    
Find an expression in terms of q only for the firm’s
(a) revenue function
(b) marginal revenue function
(c) marginal cost function
(d) profit function
(e) Solve for the profit maximising level of output, explaining which is a local
minimum or maximum.
Question 2 (6 marks)
Let 𝑓(𝑖,𝑗) =
𝑖
𝑗
𝑗!
(a) Calculate 𝑓(2,3).
(b) Calculate ∑ 𝑓(2,𝑗)
2
𝑗=0
, ∑ 𝑓(𝑖, 3)
3
𝑖=1
.
(c) Calculate ∑ ∑ 𝑓(𝑖,𝑗)
𝑖
𝑗=0
3
𝑖=1
.
Question 3 (5 marks)
The point (x0, y0) = (1, 0) lies on the curve
2 3 1 2
y x
xe y x e

  
Show that (x1, y1) = (0, 4) lies on the tangent to the curve at (x0, y0).
Question 4 (10 marks)
(a) Find the third-order Taylor polynomial for the function 𝑓(𝑥) =
1
√1−𝑥
about 𝑥 = 0.
(b) Use your answer from part (a) to calculate the approximate value of 1
3√11
.
Round your answer to six decimal places.
(c) Find the domain, range, and inverse of 𝑓(𝑥) =
1
√1−𝑥
. State your answer using
interval notation.
Question 5 (10 marks)
A firm’s price in a perfectly competitive market is 1000. Its cost function is
3 2 C x x x x ( ) 0.01 3 1108 960     ,
where x  0 is the number of units produced and sold.
(a) Find an expression for the profit function (x) for x  0.
(b) Find all stationary points and determine the profit maximising level of output.
(c) Using a sign diagram, determine the intervals over which (x) is increasing
and decreasing.
(d) Determine the intervals over which (x) is concave and convex.
(e) Where is the point of inflection? Give an economic interpretation of the point
of inflection.
Question 6 (5 marks)
Given the demand function
aQ bP k    0
, where a, b and k are positive constants,
show that the price elasticity of demand is minus one when marginal revenue is zero.
Question 7 (4 marks)
Let
1
( ) , 0. x
f x e x  
(a) Compute f (x) and f (x).
(b) Determine the intervals where f is concave/convex.

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