The following graph shows the market for sprockets in a local hardware store.

a) identify whether the price of sprockets from $4 - $3 is elastic, unit elastic, or inelastic. Explain.

b) what is the price of sprockets that maximize the hardware store’s revenue? Explain.

c) The price of sprockets is $1. if the hardware store wants to increase its revenue, how should it change the price? Explain.

d) using the midpoint formula, calculate the price elasticity of demand for sprockets when price increases from $2 to $3. Identify the coefficient as elastic, unit elastic, or inelastic.

e) using the midpoint formula, calculate the price elasticity of demand for sprockets when price decreases from $5 to $4. Identify the coefficient as elastic, unit elastic, or inelastic.

f) based on the scenario in part E, how does the decrease in price affect the hardware store’s revenue?

It is correct to state that the price for sprockets from $4 to $3 is inelastic. See the explanation below.

The formula for price elasticity is:
Price Elasticity of Demand = Percentage change in quantity / Percentage change in price.

Percentage change in quantity (for price drop of 4$ to $3) =  increase ÷ original number × 100.

= (2/18) * 100

= 11.11%

Percentage change in price drop from 4$ to $3) =  decrease ÷ original number × 100.

= (1/4) * 100

= 25%

Hence

Price Elasticity of Demand = 11.11%/25%

PED = 0.4444

Hence the demand is inelastic for the change in price from $4 to $3. This is because the price Elasticity is less than 1.

Because we don't have the function of the graph, we must find the revenue for all the given points.

Recall that Revenue is given as:

Price x Quantity; Hence,

From the above, it is clear that the price that maximizes the revenue the store is $4. This is because, that yields the highest revenue.

The mid-point formula is given as:

Price Elasticity = A/B where:

A = [(Q2-Q1)/((Q2+Q1)/2)]

B = [(P2-P1)/((P2+P1)/2)]

Q2 = Final Quantity = 18

Q1 = Initial Quantity = 20

P2 = Final Price = $3

P1 = Initial Price = $2

A = [(18-20)/(18+20)/2]]

= -0.02631578947

B = [(3-2)/((2+3)/2)]

= 0.4

PED = -0.026/0.4

PED = -0.065

From the foregoing, the Coefficient is Negative and it is also less than Zero. It means that the good is inferior and the demand for it is inelastic.

Using the same formula above we have:
A = [(Q2-Q1)/((Q2+Q1)/2)]

B = [(P2-P1)/((P2+P1)/2)]

Q2 = Final Quantity = 16

Q1 = Initial Quantity = 10

P2 = Final Price = $4

P1 = Initial Price = $5

A = [(16-10)/(16+10)/2]]

= 0.11538461538

B =  [(4-5)/(4+5)/2]]

= -0.05555555555

PED = 0.11538461538/-0.05555555555

PED = -2.08

The coefficient here is also negative but above 1. It means that the good is inferior and the price and quantity demanded are inversely related.

However, the value (2) is greater than 1, thus, the demand is elastic. This  means there is a strong reaction to quantity demanded when price increases or decreases.  

It is to be noted that decreasing the price as given in part E will trigger a drop in sales and a directly proportional reduction in profit. The reason for this is because, as seen in the analysis for E above the product is inferior.

When people perceive a product as inferior, a drop in price will create a substitutionary effect where they opt for a substitute good that is slightly higher.

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