Chapter 5: Problem 4

Why is the supply curve with constant unitary elasticity a straight line?

Short Answer

Expert verified

The supply curve with constant unitary elasticity is a straight line because the percentage change in the quantity supplied is always equal to the percentage change in price, regardless of the price level. By applying constant unitary elasticity and integrating the equation, we obtain the equation for a straight line: \(P = kQ_s + C\), where \(k\) and \(C\) are constants. This demonstrates that when the supply curve has constant unitary elasticity, it takes the form of a straight line.

Step by step solution

Defining constant unitary elasticity of supply

The constant unitary elasticity of supply means that the percentage change in the quantity supplied is always equal to the percentage change in price, regardless of the price level. Mathematically, this can be represented as: \[ \frac{\Delta Q_s}{Q_s} = \frac{\Delta P}{P} \] Where \(Q_s\) represents the quantity supplied, \(P\) represents the price, and the \(\Delta\) symbol represents a change in the specified variable.

Expressing elasticity with quantity and price

The elasticity of supply can also be expressed as a function of quantity and price: \[ E_s = \frac{\Delta Q_s}{Q_s} \times \frac{P}{\Delta P} \] Where \(E_s\) represents the elasticity of supply.

Applying constant unitary elasticity

As we are given that the supply curve has a constant unitary elasticity of 1, that means: \[ 1 = \frac{\Delta Q_s}{Q_s} \times \frac{P}{\Delta P} \] Rearrange this equation to isolate \(\Delta Q_s\): \[ \Delta Q_s = \frac{Q_s \times \Delta P}{P} \]

Integration to find the supply curve

Integrate both sides of the equation with respect to their corresponding variables, \(Q_s\) and \(P\), to find the supply curve: \[ \int\Delta Q_s = \int\frac{Q_s}{P}\Delta P \] \[ Q_s = \int\frac{Q_s}{P}\Delta P + C \] Here, C is the constant of integration.

Forming the straight-line equation

Solve the above equation for P: \[ P = \frac{\int\frac{Q_s}{P}\Delta P + C}{Q_s} \] Since the constant unitary elasticity of supply implies the percentage change in the quantity supplied is equal to the percentage change in price, the terms in the equation must maintain a constant proportion. That means the numerator should be proportional to the denominator and the equation can be simplified to: \[ P = kQ_s + C \] Where \(k\) is another constant. This is the equation for a straight line, which explains why the supply curve with constant unitary elasticity is a straight line.