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ECON1003 S1 2023

PROBLEM SET 4

1. Find stationary points and characterise them for the following functions:

a. f(x) = x3 – 3x

b. f(x) = x2 – x – 2

2. A firm’s total revenue function is given by TR = 90Q - 3Q2.

a. Find the value of Q for which TR is maximised, hence calculate the maximum TR.

b. Write down the equations of the average revenue and marginal revenue functions. Describe how AR and MR change before and after the maximum of TR.

3. A firm’s total cost function is given by the equation,

TC = Q 3 − 30Q 2 + 3500Q + 900

(a) Show, by differentiation, that TC has neither a maximum nor a minimum value. (b) Find the point of inflection.

4. For the total cost function,

TC = Q 3 − 15Q 2 + 480Q + 900

(a) Write down the equations for MC, AFC and AVC.

(b) Find the values of Q for which MC and AVC are minimized.

(c) Plot the graph of MC, AFC and AVC on the same diagram.  Comment.

5. The production function, Q = 36L 2 − 3L 3 , describes the number of electronic components assembled per day in terms of the number of employees (L) hired.

(a) Write down the equations of the marginal product of labour ( MPL ) and average product of labour (APL).

(b) Calculate the units of labour at which the APL are maximized, and plot both graphs on the same diagram.  Comment on any important features of the graphs.