### Demand Function and Elasticity#### (a) Proof of Constant Elasticity for a Given Demand Function**Problem Statement:**Show that for a demand function of the form \[ D(p) = \frac{c}{p^n} \]where $ c $ and $ n $ are positive constants, the elasticity is constant. *(Hint: Determine $ D'(p) $ and substitute into the formula for $ E(p) $.)***Solution:**1. **Determine the derivative $ D'(p) $:** The demand function is given by \[ D(p) = \frac{c}{p^n} \] Taking the first derivative with respect to $ p $: \[ D'(p) = \frac{d}{dp} \left( \frac{c}{p^n} \right) \] \[ D'(p) = \frac{d}{dp} \left( c \cdot p^{-n} \right) \] \[ D'(p) = c \cdot (-n) \cdot p^{-n-1} \] \[ D'(p) = -nc \cdot p^{-n-1} \]2. **Substitute into the elasticity formula $ E(p) $:** The formula for elasticity $ E(p) $ is: \[ E(p) = \frac{p}{D(p)} \cdot \frac{dD(p)}{dp} \] Substituting $ D(p) $ and $ D'(p) $ into this formula: \[ E(p) = \frac{p}{\frac{c}{p^n}} \cdot (-nc \cdot p^{-n-1}) \] Simplifying the expression: \[ E(p) = \frac{p \cdot p^n}{c} \cdot (-nc \cdot p^{-n-1}) \] \[ E(p) = \frac{p^{n+1}}{c} \cdot (-nc) \cdot p^{-n-1} \] \[ E(p) = (-nc) \cdot \frac{p^{n+1-n-1}}{c} \] \[ E(p)

The elasticity of demand is the sensitiveness or responsiveness of the quantity demanded to the…

(a) Show that for a demand function of the form (p) =p, where c and n are positive constants, the elasticity is constant. (Hint: Determine D'(p) and substitute into the formula for E(p).) (b) What type of demand function has elasticity equal to 1 for every value of p?

(a) Show that for a demand function of the form (p) =p, where c and n are positive constants, the elasticity is constant. (Hint: Determine D'(p) and substitute into the formula for E(p).) (b) What type of demand function has elasticity equal to 1 for every value of p?

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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