Pricing Part4

ACCA F5 - Performance Management

Mathematically we know,

TR = P (average price) * Q (quantity)

Differentiating TR with respect to output Q

d(TR)/d(Q) = d(P*Q)/d(Q),    Using product rule of differentiation

d(TR)/d(Q)= P(dQ/dQ) + Q (dP/dQ)   By definition d(TR)/d(Q) = MR

MR = P + Q (dP/dQ)

MR = P ( 1 + (dP/dQ)*(Q/P))

MR = P (1 + 1/E_P) …………………….

E_P = (∆Q/Q) / (∆P/P)   where, ∆Q = Q₂ - Q₁ and ∆P = P₂ - P₁

Calculation for above table: E_P = (6/1)/(-60/180) = -18 , ∆Q =7-1 and ∆P = 120-180

Price function can be mathematically presented as, P = AR = a - b Q ………………3

where, a is highest price, q is quantity and b is slope(i.e. b = dP/dQ)

Now,

MR = P ( 1 + (dP/dQ)*(Q/P)) MR = P - b Q (because EP is negative) MR = a - b Q - b Q MR = a - 2 b Q

Alternatively, MR = P + Q (dP/dQ) MR = a - b Q - b Q MR = a - 2 b Q

Price for 1^st unit P1 MR = 180 and AR = 180

 MR = 200 (projecting to zero unit) and AR = 190(projecting price to zero unit)

Price for 7^th unit P7 MR = 60 and AR = 120

i.e. Average price/revenue P for 7^th unit 120 = 190 - b 7uints   OR, b=10

 Gives, Price function: AR = 190 - 10Q  …………………….3(a)

For Marginal price/revenue P for 7^th unit 60 = 200 - c 7uints   OR, c=20 = 2b

Gives, Price function: MR = 180 - 2*10 Q ……………………4(a)

MR = a -2 b Q  ……………………………4

Gathering formulas together:

Total cost = Fixed cost + Unit variable cost * Production unit	Y = a + b X
Within identified production range marginal and variable costs are equal	MC = VC
Price Elasticity of demand	EP= (∆Q/∆P)*(P/Q)
Marginal revenue	MR = P (1 + 1/EP)
Price function can be mathematically presented as (Average Price)	P = AR = a - b Q
Marginal revenue	MR = a -2 b Q

CIMA Article: Management Accounting - Decision Management