Question

In 2018, the Chicago Transit Authority (CTA) faced a large budget shortfall, due to state-wide budget cuts. In an effort to balance their budget, CTA raised the fare for one train ride from $2.25 to $2.50 on January 1, 2018. In 2017, prior to the fare hike, annual ridership was about 225 (measured in millions). CTA also conducted a study that year and found that the price elasticity of demand was ???? = −0.5. a) Estimate a linear market demand function for train rides in Chicago. Graph the (inverse) market demand curve and show the price/quantity point where the price of a ride is $2.25. What is the price elasticity of demand at that point? Did the increase in fare increase or decrease revenue in the short run, based on the information provided? Can you justify your answer without actually finding the new revenue? b) What is the predicted number of riders (in millions) in 2018? Show this price/quantity point on your graph. Suppose we also know that in 2018, gas prices were low and Uber and Lyft availability increased in Chicago. Would this information change your predicted ridership for 2018? If so, demonstrate this change on your graph, and explain why your graph is consistent with the information given. (Note: Do not try to find a new value for ridership; just describe whether it would be higher or lower than your original projection).

Answer 1

Answer: The fare before the price hike is let, P₁ = $2.25

fare after the Price hike is P₂ = $2.50

The annual ridership Q1 = 225 millions

Price elasticity of Demand PED = -0.5

a). Linear market demand function for train rides in Chicago:

The form of demand function will be Q = a - bP

we can find the new annual ridership from Elasticity of demand

Since PED = % change in quantity /% change in price

-0.5 = (Q₂ - Q₁) /Q₁ / (P₂ - P₁ ) / P₁

=>   (Q₂ - 225) /225 / (0.25 / 2.25)

=> -0.5 = ((Q₂ - 225) /225) * 2.25 / 0.25

On solving the equation for Q₂

Q₂ = 212.5 million.

For demand function Q = a - bP , before the price hike

=> Q₁ = a - b *P₁

=> 225 = a- b * 2.25

=> a - 2.25 *b - 225 =0-------------------------------------equation (1)

Similarly for demand function after price hike

212.5 = a - b * 2.50

=> a- 2.50 *b - 212.5 =0 ------------------------------------equation (2)

Solving both the equation

a = 337.5

b = 50

Hence the linear equation for demand :

Q = 337.5 - 50P

and the inverse demand function is:

P = 6.75 - 0.02 Q

The price elasticity of demand at that point is -0.5

The increase in fare increase or decrease revenue in the short run, based on the information provided:

after price hike Revenue R = Q*P

R = 212.5 * 2.50

R= 531.25

Before price hike R = 225 *2.25 =506.25

Hence the revenue increased after price hike.

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b). Predicted riders in in 2018 = Q₂ which we have found in part A

Q₂ = 212.5 million

If the gas price is lower in 2018, it will affect the Demand for train and the demand curve will shift to left .

Due to the Uber and lyft taxis are available and only if they offer low prices then the ridership in train will decrease.