Answer to Question #200282 in Microeconomics for Vision

Question

Make use calculus to prove that the price elasticity of demand is a constant everywhere

along the demand curve whose demand function is , where A is a positive constant

and p is the market pricMake use calculus to prove that the price elasticity of demand is a constant everywhere

along the demand curve whose demand function is , where A is a positive constant

and p is the market price.e.

Answer 1

The price elasticity of demand measures the sensitivity of quantity demanded to price. It tells us the percentage change in quantity demanded when price changes by 1%.

This is a more useful measure of the responsiveness of demand to price.

suppose the price changes from P to P+ $\Delta$ P causing the quantity demanded to change from

Q = g(P) to Q + $\Delta$ Q, the percentage change in price is $\frac {100\Delta P}{P}$ and the percentage change in quantity is $\frac {100\Delta Q}{Q}$

Substituting this in the equation for elasticity, we obtain

-$\frac {\Delta Q}{Q}$ /$\frac {\Delta P}{P}$ =-$\frac {P}{Q}$ $\frac {\Delta Q }{\Delta P}$

Taking the limit of this expression as P approaches zero gives us the calculus definition of the price elasticity of demand which is denoted by:

$\epsilon$ =- $\frac {P}{Q}$ $\frac {dQ}{dP}$

The value of price elasticity is normally positive because according to the law of demand, the derivative of the demand function will be negative.

If the form of the demand function is : Q = a P^-c

where a and c are positive constants, the elasticity of demand is c.

This is the only class of demand function for which the elasticity is constant.

An expression for the price elasticity of demand can be obtained by returning to the inverse demand function P= f(Q)

According to the inverse function rule,

$\frac {(dP)}{(dQ)}$=1/$\frac {dQ}{dP}$

and so

$\epsilon$ = -$\frac {P}{Q}$ /$\frac {dP}{dQ}$

= - $\frac {f(Q)}{Qf Q}$

In an example where a commodity faces the inverse demand function

P= 8000 - 80Q

the elasticity of demand will be given by;

$\epsilon$ = -$\frac {8000-80 Q}{Q \times -80}$ =$\frac {100}{Q}$ - 1

The elasticity expressed in terms of price is:

Q = $\frac {8000- P}{80}$

So,

$\epsilon$ = -$\frac {P}{Q}$ $\frac {dQ}{dP}$

= -$\frac {80 P}{8000 - P}$ $\times$ - $\frac {1}{80}$

=$\frac {P}{8000- P}$

Each of the two expression for $\epsilon$ shows that it falls as we move to the right along the demand curve, increasing Q and reducing P. This is for every linear demand function as is the result that $\epsilon$ approaches zero as P approaches zero and $\epsilon$ approaches infinity as P approaches its maximum value where Q=0.