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ECON1003 S1 2023

PROBLEM SET 3

1. Find the derivatives of the following functions.

(a) f(x) = xi +4x-8

(b) f(x) =(e)

(c) f(x) = (x2 + )(x + 1)

(d) f(x) =

(e) f(x) = +x-4

2. A firm has the following total cost and total revenue function TC = 1/3Q3 – 9Q2 +200Q +5050, TR = Q(120 – 10Q)

Find (i) marginal cost, (ii) marginal revenue (iii) average cost (iv) average revenue.

3. Find the first, second and third derivatives of the following total cost function:  TC = q3/5 – 8Q2 +5Q/2 +180

4. The demand function for a monopolist is P = 90 – 3Q.

(a) Derive expressions for TR, MR and AR.  Evaluate TR, MR and AR at Q = 15. Hence, describe, in words, the meaning of each function at Q = 15.

(b) Show that the demand function, AR and price are identical.

(c) Show that the slope of the MR function is twice that of the AR function.

(d) Calculate the value of Q at which MR is zero.  Calculate the value of AR and TR when MR is zero. Calculate the maximum number of units which a sensible monopolist should sell.

(e) Graph TR, MR and AR on the same diagram.  Confirm the answers to part (d) graphically.

5. Consider the function q = f(P). (i) What is the shape of the function if f ’(P) > 0 for P < P* and f ’(P) < 0 for P > P* and f ”(P) < 0 for all P > 0? (ii) What if f ’(P) > 0 and f ”(P) > 0 for all P > 0.

6.

(a) Derive a general expression for the price elasticity of demand for the function Q = 84P−1 (Q anp d > 0). Describe verbally how demand changes in response to price changes for this function.

(b) Evaluate the point elasticity , d ,  at d = 15 and at d = 30