Total Revenue and Demand Elasticity Exercise

A firm’s total revenue function is given by:

$$TR(Q) = -0.5Q^3 – 2Q^2 + 100Q$$

Question

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

Solutions

(1) Finding the Quantity that Maximizes Total Revenue

The total revenue function is:

$$TR(Q) = -0.5Q^3 – 2Q^2 + 100Q$$

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

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

Solving for $Q$ using the quadratic formula:

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

Approximating:

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

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

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

The price elasticity of demand formula using total revenue is:

$$E_D = \frac{dTR/dQ}{TR/Q} – 1$$

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

$$|E_D| = 1$$

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