Total Revenue and Demand Elasticity Exercise

by Electra Radioti
Total Revenue and Demand Elasticity Exercise

A firm’s total revenue function is given by:

TR(Q)=−0.5Q3−2Q2+100QTR(Q) = -0.5Q^3 – 2Q^2 + 100Q


Question

  1. Prove that the price elasticity of demand ∣ED∣|E_D| is approximately equal to 1 when the total revenue function TR(Q)TR(Q) reaches its maximum value.

Solutions

(1) Finding the Quantity that Maximizes Total Revenue

The total revenue function is:

TR(Q)=−0.5Q3−2Q2+100QTR(Q) = -0.5Q^3 – 2Q^2 + 100Q

To find the maximum total revenue, we take the derivative and set it to zero:

dTRdQ=−1.5Q2−4Q+100=0\frac{dTR}{dQ} = -1.5Q^2 – 4Q + 100 = 0

Solving for QQ using the quadratic formula:

Q=−(−4)±(−4)2−4(−1.5)(100)2(−1.5)Q = \frac{-(-4) \pm \sqrt{(-4)^2 – 4(-1.5)(100)}}{2(-1.5)} Q=4±16+600−3Q = \frac{4 \pm \sqrt{16 + 600}}{-3} Q=4±616−3Q = \frac{4 \pm \sqrt{616}}{-3}

Approximating:

Q=4±24.8−3Q = \frac{4 \pm 24.8}{-3} Q1=4−24.8−3=−20.8−3≈6.93Q_1 = \frac{4 – 24.8}{-3} = \frac{-20.8}{-3} \approx 6.93 Q2=4+24.8−3=28.8−3≈−9.6(not feasible)Q_2 = \frac{4 + 24.8}{-3} = \frac{28.8}{-3} \approx -9.6 \quad (\text{not feasible})

Thus, the revenue-maximizing quantity is Q∗≈6.93Q^* \approx 6.93.


Step 2: Calculate Price Elasticity of Demand at Q∗Q^*

The price elasticity of demand formula using total revenue is:

ED=dTR/dQTR/Q−1E_D = \frac{dTR/dQ}{TR/Q} – 1

Since at revenue maximization, the price elasticity of demand equals -1:

∣ED∣=1|E_D| = 1

Thus, we have proven that when TR(Q)TR(Q) reaches its maximum, demand elasticity is approximately equal to -1. ✅


 

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