Math Problem - Car park revenue

[BadBoy]

A vacant lot has opened up next door to my work place that was turned into a temporary car park. When they first opened they were charging $18 per day and the place was empty for weeks. No one wanted to use it.

Eventually they dropped the price to $7 per day and within a couple of weeks the car park was full every day.

So now they are slowly putting up the prices, it went from $7 to $7.50 and now it is $8.00. I now see a few spaces in the car park that are there all day so the price increase was affecting take-up which would be expected.

I was wondering what the math would look like that works out the sweet spot, i.e. what is the price to park which will earn the most amount of money per day for the car park. To do this I have to make an assumption regarding how many people stop using the car park for every 50cent price increase and I think that function is nonlinear. So lets assume it is nx2. So if the price goes up my 50c then there is 1 new vacant space every day, increase of $1 is 2 vacant spaces, $1.50 is 4 spaces etc.

If we assume there are 200 spaces to begin with and $7 means it is always full and also assume that there is no initial lag such that the first initial increases in price has an immediate effect (i.e. the people who couldn't get a space when it was $7 don't turn up and take the extra spaces when it goes up to $8 due to new availability).

I can imagine a bell curve showing an increase in revenue as the price is increased because the lost revenue from the number of extra vacant spaces does not exceed the money they gain from putting up the price, but then at some point the loss of income from the number of vacant spaces will be greater than the extra amount they get from putting up the price. So I would need to work out the peak of the bell curve, the point just before revenue starts going down with another price hike.

How would I approach this mathematically (BTW, this is not a course work problem, I was just trying to figure it out on my way home the other night after noticing the extra spaces in the car park after the last price hike, but my math is rusty and I was never that good anyway).

Any takers?

 

[Horatius]

This is a price optimization problem.

http://www.warrantyweek.com/archive/ww20091029.html


Basically, graph one function for their revenue (price) per space, another function for the number of spaces sold per price, and multiply those two functions. Take the maximum of that resultant function, and that's the price point to maximize profits.

 

[Foolmewunz]

This is a price optimization problem.

http://www.warrantyweek.com/archive/ww20091029.html


Basically, graph one function for their revenue (price) per space, another function for the number of spaces sold per price, and multiply those two functions. Take the maximum of that resultant function, and that's the price point to maximize profits.

Yeah, you can do it informally, but that's the intended result and underlying methodology.

It's essentially the same as a "price point" in retail. But retail has all sorts of consumer/shopper surveys in addition to just watching the results.** If it is determined the Mrs. Consumer just won't go above $9.95 for a pair of kid's peejays, you can see a drop-off of 20% if you come in at $10.49 If you sell 10,000 pairs, your new income is 8000 pr @ 10.49 versus the original $99,500. Which is higher? 20% of units is obviously too large a loss to bear (you'd be down by about $16K), but if the number is only 5% loss of sales, it would be slightly better at the new price, and more importantly, you'd have broken the psychological 9.99 barrier. You'll probably get that missing 5% back.

For the parking lot. Just count the empty spaces. At a certain dollar value X number of empty spaces loses you more than Y number of dollars gains you.

The question with the parking lot, though, is whether those empty spaces will be filled after the fifteen or twenty petulant drivers* return. Do the empty slots show up when the price goes up but then re-fill in a few weeks to a month? Or are they permanently empty. Is there any growth in commerce or industry in the area?

*Where are they going? Competing lots? Or going to circle the block fifteen times waiting for a street parking spot to open? Taking a chance on nursing a meter all day? Whatever! In short, there are other elements.

**Because in retail, you've laid money out for the goods and are stuck with them. You have to sell them at a loss - not just loss of revenue but actual loss - in some cases, so knowing the price point ahead of time is crucial.

 

[Horatius]

Another factor is casual users vs. regular users. If you can sell one hour of parking for $4, vs. $7 for the day, then it's better to have a few open spaces available.

I just did that today. Went to work for a meeting, parked in my usual lot, paid $12 for about 2 hours. The maximum daily rate is $13, so if they could count on re-selling the spot I vacated, they'd almost double their revenue.

Actually, thinking about it, I snagged that spot just as someone else was leaving, so they probably already did that.

 

[Giordano]

Another factor is casual users vs. regular users. If you can sell one hour of parking for $4, vs. $7 for the day, then it's better to have a few open spaces available.

I just did that today. Went to work for a meeting, parked in my usual lot, paid $12 for about 2 hours. The maximum daily rate is $13, so if they could count on re-selling the spot I vacated, they'd almost double their revenue.

Actually, thinking about it, I snagged that spot just as someone else was leaving, so they probably already did that.

A related phenomenon exists in airline flight in terms of first/business class vs. coach. If there are any empty seats in the premium classes and none in the coach, it pays to upgrade a coach passenger to premium and try to re-sell the coach seat. The airline is much more likely to be able to sell the now-empty coach seat than the previously empty premium class seat (it is guaranteed if there are already standby passengers). The extra cost of flying a coach passenger first class or business is very little (just the cost of a meal and some free drinks). The airline has collected twice for the same coach seat. And with all these benefits to the airline, they still have the nerve to charge the upgraded passenger a bundle of dollars and frequent flyer miles for the privilege!

 

[Crossbow]

I think what needs to be done is to survey the parking that is available near the new parking lot.

If the existing parking is both affordable and available, then the new parking lot will have to charge a low price in order to entice parking customers to use the new parking lot.

On the other hand, if the existing parking around the new parking lot is of poor quality and/or is expensive, then the new parking lot can charge more and still attract parking customers.

Generally speaking, when starting a new service it is better to charge a low fee to attract new customers, and after that is established, then start slowly raising prices.

 

[Dr. Keith]

I've been having similar conversations with a client who is selling items online. When they run an enticing ad, say buy three items get one free, they see their sales increase by somewhere around 3 fold. This is true even though the advertised deal is only used in about 20% of the recorded transactions. The advertised deal drives shoppers to the site, but then some decide that they don't really need that much and they end up buying something less.

It is really fascinating how discounts impact shopping.

 

[aleCcowaN]

The OP is departing from a fixed-cost structure, otherwise it'd make no sense mixing up profit with income.

Some comments:

-with the elements in the OP, no bell curve but a parabola (an "inverted" one, if you like).

-from what is described in the OP, unless this parking lot is just working with the public of some sport stadium with no any additional users, the same space may have several users paying for a whole "day" during the same day (unless there's some element in the Usaian/local legislation and/or commercial uses I ignore, something like "if you pay for the day you may take your car and later park it again within the day"). At 7$ there may be customers using the place just for 2 hours, what may not be the case with $18.

So, there's an answer to the oversimplified question in the OP (implied fixed costs, a parabola that maximizes income/profit for X= -b/2a with a and b being the coefficients of the quadratic equation of income).

And there's an answer for the real business (strategy of products -day and hour-, real structure of costs -typically quasi-fixed-, marginal costs that decide the hours the parking is operating (unless there're conditions in the contract). That would make this problem a bit more entertaining.

 

[psionl0]

I was wondering what the math would look like that works out the sweet spot, i.e. what is the price to park which will earn the most amount of money per day for the car park. To do this I have to make an assumption regarding how many people stop using the car park for every 50cent price increase and I think that function is nonlinear. So lets assume it is nx2. So if the price goes up my 50c then there is 1 new vacant space every day, increase of $1 is 2 vacant spaces, $1.50 is 4 spaces etc.

That is not a square law variation, it is an exponential variation.

Let T be the total number of spaces available
N the number of cars parked for a certain price
x be the number of 50c increments
and R be the total revenue collected.

We can demonstrate that for x >= 1, N = T - 2^x-1Also R = (7 + 0.5x) N = (7 + 0.5x)(T - 2^x-1)

We could use calculus to find the optimum value for x but in this case it seems a bit too messy for my liking. I input the formula into my spreadsheet (using T=200 as you suggested) and found that you could get a maximum revenue of $1,749 when you have 4.8 50c price increases. (Without any price increases, the revenue would be $1,400).

ie If the car park charged $9.40 then it would get a maximum revenue of $1,749.

 

[BadBoy]

This is a price optimization problem.

http://www.warrantyweek.com/archive/ww20091029.html


Basically, graph one function for their revenue (price) per space, another function for the number of spaces sold per price, and multiply those two functions. Take the maximum of that resultant function, and that's the price point to maximize profits.

Interesting link. I come from a technical background and have zero sales and marketing skills that I'm aware of. I did imagine two curves actually at one point, thinking about where they cross is the sweet spot. Thanks

 

[BadBoy]

Yeah, you can do it informally, but that's the intended result and underlying methodology.

It's essentially the same as a "price point" in retail. But retail has all sorts of consumer/shopper surveys in addition to just watching the results.** If it is determined the Mrs. Consumer just won't go above $9.95 for a pair of kid's peejays, you can see a drop-off of 20% if you come in at $10.49 If you sell 10,000 pairs, your new income is 8000 pr @ 10.49 versus the original $99,500. Which is higher? 20% of units is obviously too large a loss to bear (you'd be down by about $16K), but if the number is only 5% loss of sales, it would be slightly better at the new price, and more importantly, you'd have broken the psychological 9.99 barrier. You'll probably get that missing 5% back.

For the parking lot. Just count the empty spaces. At a certain dollar value X number of empty spaces loses you more than Y number of dollars gains you.

The question with the parking lot, though, is whether those empty spaces will be filled after the fifteen or twenty petulant drivers* return. Do the empty slots show up when the price goes up but then re-fill in a few weeks to a month? Or are they permanently empty. Is there any growth in commerce or industry in the area?

*Where are they going? Competing lots? Or going to circle the block fifteen times waiting for a street parking spot to open? Taking a chance on nursing a meter all day? Whatever! In short, there are other elements.

**Because in retail, you've laid money out for the goods and are stuck with them. You have to sell them at a loss - not just loss of revenue but actual loss - in some cases, so knowing the price point ahead of time is crucial.

Where I work is sort of on the edge of the city. It's mainly residential actually, and other businesses tend to be small places to grab something to eat and a coffee (non chain type establishments). However on the site my new office block is on (5 stories built 4 years ago) I think was some very old industrial units. I say that because next door was similar which they cleared and put up some residential flats (with a shared swimming pool !) and on the other side was the same which they cleared and is now a temporary parking lot. I am guessing the owners are waiting for planning permission or something similar before they can start building there.

The only other public parking is roadside meter parking and the whole day cost around $18 I think.

There is a cheaper price for parking for a shorter period but based on where it is I don't think it would be used much for that. The city is just about walkable and I recon most people use it who work all day in my office block which is the only one around for miles, or they park cheap and walk into the city.

 

[BadBoy]

The OP is departing from a fixed-cost structure, otherwise it'd make no sense mixing up profit with income.

Some comments:

-with the elements in the OP, no bell curve but a parabola (an "inverted" one, if you like).

-from what is described in the OP, unless this parking lot is just working with the public of some sport stadium with no any additional users, the same space may have several users paying for a whole "day" during the same day (unless there's some element in the Usaian/local legislation and/or commercial uses I ignore, something like "if you pay for the day you may take your car and later park it again within the day"). At 7$ there may be customers using the place just for 2 hours, what may not be the case with $18.

So, there's an answer to the oversimplified question in the OP (implied fixed costs, a parabola that maximizes income/profit for X= -b/2a with a and b being the coefficients of the quadratic equation of income).

And there's an answer for the real business (strategy of products -day and hour-, real structure of costs -typically quasi-fixed-, marginal costs that decide the hours the parking is operating (unless there're conditions in the contract). That would make this problem a bit more entertaining.

I wont pretend I understand most of that Sounds like you've done a bit of business and marketing? But I agree that the OP was simplified somewhat.

And yes of course, it would be a parabola, rate of increase in vacant spaces not going to drop as the price reaches a maximum, that would be daft.

Cheers

 

[BadBoy]

That is not a square law variation, it is an exponential variation.

Let T be the total number of spaces available
N the number of cars parked for a certain price
x be the number of 50c increments
and R be the total revenue collected.

We can demonstrate that for x >= 1, N = T - 2^x-1Also R = (7 + 0.5x) N = (7 + 0.5x)(T - 2^x-1)

We could use calculus to find the optimum value for x but in this case it seems a bit too messy for my liking. I input the formula into my spreadsheet (using T=200 as you suggested) and found that you could get a maximum revenue of $1,749 when you have 4.8 50c price increases. (Without any price increases, the revenue would be $1,400).

ie If the car park charged $9.40 then it would get a maximum revenue of $1,749.

Excellent, that's what I was looking for, and of course the vacancy rate would be 2^x which is what I was trying to get to (and did later, I'm a software developer so I know 2^x function very well of course)

So working out where the maximum is would require a bit of calculus, interesting. I had to learn some Calculus for my first degree, but that was a long time ago.

(As an aside, I purchased a book called A Tour of the Calculus by David Berlinski years ago which I quite enjoyed. Later on I found out the author is a religious nutcase and an evolution denier! Crap! It just goes to show how a people can be very bright and still think Evolution is bunk - I guess that's where the religious dogma has messed with his ability to see clearly on certain topics)

 

[psionl0]

Excellent, that's what I was looking for, and of course the vacancy rate would be 2^x which is what I was trying to get to (and did later, I'm a software developer so I know 2^x function very well of course)

It's good to see that you now know that the solution is not "a parabola that maximizes income/profit for X= -b/2a with a and b being the coefficients of the quadratic equation of income".

BTW I did not round off the number of cars in the car park which would change the optimum price slightly. (At $9.35 you would get 187 whole cars in the car park for a peak revenue of $1,748.45).

So working out where the maximum is would require a bit of calculus, interesting. I had to learn some Calculus for my first degree, but that was a long time ago.

(As an aside, I purchased a book called A Tour of the Calculus by David Berlinski years ago which I quite enjoyed. Later on I found out the author is a religious nutcase and an evolution denier! Crap! It just goes to show how a people can be very bright and still think Evolution is bunk - I guess that's where the religious dogma has messed with his ability to see clearly on certain topics)

It is only when you are not within the confines of the real world of mathematics that quackery becomes possible.

 

[aleCcowaN]

It's good to see that you now know that the solution is not "a parabola that maximizes income/profit for X= -b/2a with a and b being the coefficients of the quadratic equation of income".

Look, algebra teacher, I'm sorry to say that nx2 (see OP), if taken as a continuous function, is linear. I simply ignored the contradicting text, which supposed the demand to be a rarely to be seen concave function, price elasticity of the demand quickly tending to infinite, and the parking business to have awfully negative externalities. Not real unless there's another parking lot very near and a price war has started. Or abundance of free shady parking space in an area without car theft.

I congratulate you on your innate ability to make math to become increasingly useless as you go more precise with it.

BadBoy didn't have to know how the system works. Yours is inexcusable. You should know better as you always dive in every discussion about economics, finances and business -with variable luck-, and you obviously have a strong formation centred around algebra.

@BadBoy: if you haven't studied the actual demand, you can't make the kind of assumption you did with its function. You know that the demand is 200 at a price of 7$ and 0 at a price of 18$. You should assume a linear function between those 2 points and confirm if your observations for a price of $8 reasonably match the function (one each 11 spaces being empty would be a "perfect" match, but any similar proportion would work). Then you have your parabola and you have your solution: a price of $9 or $9,50.

The reason of you assuming such linear function is that you don't have any additional information about the demand, nor you know about all the variables in that parking business. Then, it makes no sense that you assume demand to be a complex and unnatural function, nor it has real value the analysis provided by psionl0 following your verbal specifications.

In a problem with multiple factors, the quality of the global solution follows the quality of the weakest component. In this case, all the pertinent elements in the parking business you simply left out.

 

[WhatRoughBeast]

Look, algebra teacher, I'm sorry to say that nx2 (see OP), if taken as a continuous function, is linear. I simply ignored the contradicting text, which supposed the demand to be a rarely to be seen concave function, price elasticity of the demand quickly tending to infinite, and the parking business to have awfully negative externalities. Not real unless there's another parking lot very near and a price war has started. Or abundance of free shady parking space in an area without car theft.

But you had no good cause to do ignore the contradictory text. The question asked

I was wondering what the math would look like that works out the sweet spot

BadBoy supplied enough information to make it clear that the proposed demand function is not linear, and psion10 quite accurately described the math.

@BadBoy: if you haven't studied the actual demand, you can't make the kind of assumption you did with its function. You know that the demand is 200 at a price of 7$ and 0 at a price of 18$. You should assume a linear function between those 2 points and confirm if your observations for a price of $8 reasonably match the function (one each 11 spaces being empty would be a "perfect" match, but any similar proportion would work). Then you have your parabola and you have your solution: a price of $9 or $9,50.

But your own assumption, while slightly less unrealistic than BadBoy's, is even less justifiable given your background. You have assumed a linear demand function over the price range of 7 to 18 dollars, based on the observation of zero demand at 18 dollars. Really? On the one hand, assumptions of perfectly linear demand are commonly used in economics due to the mathematical simplicity which results, but real-world businesses know better. At the very least, very high asking prices can produce non-zero demands due to perceived status effects. Worse, even assuming a linear demand model, there is simply no evidence that, for instance, a cost of 17 dollars would not have likewise produced zero demand. In other words, your assumption that 18 dollars/zero demand establishes a real end point to the linear demand function is one you pulled out of a hat, and you should know better.

The reason of you assuming such linear function is that you don't have any additional information about the demand, nor you know about all the variables in that parking business.

Neither do you.

Then, it makes no sense that you assume demand to be a complex and unnatural function, nor it has real value the analysis provided by psionl0 following your verbal specifications.

In a problem with multiple factors, the quality of the global solution follows the quality of the weakest component. In this case, all the pertinent elements in the parking business you simply left out.

Exactly. While I applaud the larger meaning, it also applies to your own assumptions.

With that said, BadBoy should be aware that his exponential demand function rises so sharply that for any price greater than 11 dollars, demand becomes negative, so the model function he proposed doesn't seem terribly likely.

But psion10's analysis is correct for the demand function given.

 

[psionl0]

BadBoy should be aware that his exponential demand function rises so sharply that for any price greater than 11 dollars, demand becomes negative, so the model function he proposed doesn't seem terribly likely.

Correct. If the exponential model applied at all, it could only apply over a narrow range of prices. (Note that it couldn't apply for values of x <= 0. If the car park charged less than $7 then the demand would be a constant 200).

But as you pointed out, it was a case of question asked, question answered. Others have already pointed out that there are more realistic scenarios.

 

[WhatRoughBeast]

Oops. I meant, of course. that the demand function falls so rapidly.

But I think y'all know what I meant.

Dammed dyslexic fingers.

 

[aleCcowaN]

But you had no good cause to do ignore the contradictory text. The question asked

The procedure is "follow the equation, ignore the text". But I concede I could perfectly ask BadBoy for additional information.

BadBoy supplied enough information to make it clear that the proposed demand function is not linear, and psion10 quite accurately described the math.

psionl0 ignored the equation and accurately provided the math matching the verbal information

But your own assumption, while slightly less unrealistic than BadBoy's, is even less justifiable given your background. You have assumed a linear demand function over the price range of 7 to 18 dollars, based on the observation of zero demand at 18 dollars. Really? On the one hand, assumptions of perfectly linear demand are commonly used in economics due to the mathematical simplicity which results, but real-world businesses know better. At the very least, very high asking prices can produce non-zero demands due to perceived status effects. Worse, even assuming a linear demand model, there is simply no evidence that, for instance, a cost of 17 dollars would not have likewise produced zero demand. In other words, your assumption that 18 dollars/zero demand establishes a real end point to the linear demand function is one you pulled out of a hat, and you should know better.

I don't know what do you believe to be saying with "given your background".

The rest of you paragraph makes no sense. It basically says "you don't know how it really works, you can't make assumptions". You could've set you imagination free and provide here 10Mb of possible demand functions with loose arguments for each of them.

Still the fact remains there are just two precise observations. A linear function is the safe first assumption until you get more information, why? because it's the simplest. Your suggestion of demand vanishing -I knew someone will use it, but expected that to be psionl0- at 17$ generates a multipart function with a change of law at some value you don't have any backing observation.

Your patronizing "you should know better" falls by its own incongruence.

Neither do you.

Did you think I was talking to you? Oh, man! There are certain hours or situations you shouldn't be posting. Not exactly "dyslexic fingers"

There's a reason there are standard assumptions, like the linear function linking two observation. Or do you think microeconomics hasn't studied this and has nothing to propose about it?

The fact remains this is a problem where just using and refining microeconomics is overkilling. Some factor like two customers working in the same office building and getting their cars scratched to learn that the parking owner have self-insured such eventualities and has unsatisfactory dealt with the claims. That itself could make him lost one third of his clients in the term of a few weeks, just by word of mouth.

Exactly. While I applaud the larger meaning, it also applies to your own assumptions.

Did you think I didn't mean that also. That's why I kept it simple and that seems to simply have slipped your attention.

The only allowed multipart function is that having constant demand for every price below some value, once the parking is full.

With that said, BadBoy should be aware that his exponential demand function rises so sharply that for any price greater than 11 dollars, demand becomes negative, so the model function he proposed doesn't seem terribly likely.

And good luck finding in microeconomics and finances such exponential functions with exponents greater than one other than continuously compounded interest in time of hyperinflation and other few colourful herrings.

Even found them subtracted and causing the demand to fall sharply.

But psion10's analysis is correct for the demand function given.

Yes. He provided a mathematical solution to a mathematical problem. The relation of it with the business at hand is, at most, tenuous. That's why his use of the term "quackery" was out of place and misleading and deserved a reaction.

 

[Beerina]

An actual curve might not be smooth because there could be, say, other paid lots nearby, and if the price exceeds them, a sudden exodus might occur.