# DYNAMIC PRICING FOR MOVIE TICKETS

In our everyday life, we come across various instances where we see that the prices of a product change according to changing conditions. Let’s just take an example of a concert. In a concert, the earlier you book the ticket, the cheaper it will be. The further away the arena is from the main stage, the cheaper it is. Even the number of people demanding a ticket to that concert, (which may depend on who the celebrity is) the ticket price may increase or decrease accordingly. Thus, we can see that the prices for the desired products are changing dynamically as the factors change. This concept is called Dynamic Pricing. Simply said, dynamic pricing is a variable pricing strategy that takes into account a range of elements such as market demand, price boundaries, and seasonality. You can reprice fast and at scale with a good dynamic pricing strategy while also analyzing the impacts of your adjustments. This is a brilliant concept to increase sales without changing our timetables, nor lowering our average price per seat, by applying very basic principles of microeconomics. Thus we shall take a look at various microeconomics terms that apply here. Bear with me, it will all make sense towards the end!

## Dynamic Pricing factors

The dynamic pricing approach applies to any business that sells a service that has some similarities to concert seats, i.e., the cost of selling one additional unit, or marginal cost, is near to $0.

If a unit isn't sold by the time the service is given (for example, when the concert is over), it can't be held in stock and its potential worth is gone permanently.

This would also be the case of hotel rooms, airlines, long-haul bus tickets, cinema, theater, railway tickets, zoos, cruises, sporting events, etc.

This in order to build a model, we need a brief idea about the following:

How much do customers buy of a specific product?

How much are they willing to pay for that product?

When do they want to buy?

We’ll get a deeper insight once we have grasped the above concepts. We will be considering a cinema ticket system.

**How much do customers buy of a specific product?**

This above graph is a typical demand curve. To explain it vaguely, as the price of a product increases, it’s demand decreases and vice versa, keeping all other factors constant. Thus if you want to increase the sales of your theatre, you need to decrease the prices and go backwards on the curve. The lesser the price, the more in demand the ticket will be, keeping rest factors constant.

**How much are they willing to pay for that product?**

Willingness to pay plays an important part in pricing your tickets for the movie accordingly. An economist (here, the theatre owner/manager) needs to ask himself, how much are the customers willing to pay for the ticket at our theatre, for a specific movie, for a specific seat, or at a specific screening time.

To make things easier, let's pretend that a customer can only purchase one unit of the service (in this case a movie ticket). If you price the ticket at €9, you'll find that 30 people are interested in purchasing it. If you price it at €3, you'll get 90 people to buy a ticket (30 people who would purchase at €9 + the 60 people who would purchase at €3).

**Maximizing Revenue**

Revenue is price times quantity. So far, we’ve been charging a constant price for a product for everyone, and thus revenue was a single variable function. However, let’s now think a little differently. How can we change this constant revenue? As the number of tickers depends only on the price, according to the demand function, we can only vary price.

But this varying will be done discriminatorily, i.e. charge different prices for different customers. This may seem unfair but it’s not how you think it is.

In a theatre, if it can only charge one and the same amount to all customers, and because revenue equals price times quantity, if they charge 6 Euros per ticket, it will sell 60 tickets, generating €360 in income.

But now let’s discriminate between the prices of an adult and a child. Then they could potentially increase revenues from a maximum of €360 they could reach with a single price, to €440 (40 tickets at €8 per adult, plus 20 tickets at €6 per child).

Let’s take a look at the demand graphs for both the cases.

The graphs are self-explanatory now.

In our case, we charged a different amount for two subgroups. However to get the maximum area under the graph, (which is the revenue) we need to divide the revenue between several subgroups. This is the method of maximizing revenue.

**When do they want to buy?**

We always notice a similar trend in every customer, irrespective of the product. A person will keep postponing buying a product as long as possible, until either the product’s timeline is about to end ( in case of tickets), or till the price decreases.

Thus, we can safely say that once a movie is declared, very few people will buy tickets to the screening a month before the date. More people will buy tickets during that month and the maximum will choose to buy it the week of the screening. And we also notice the price of tickets starting off low, but as the screening approaches, the prices increase. Thus, we have figured out another factor of price discriminations which is time.

**What happens if we put it all together?**

Now we need to figure out what price we should charge at what time, considering that we only have a limited amount of tickets to sell.

We can bring it all down to two equations that share two variables and can be solved concurrently:

The Price Demand Function where

*Price***(P)**is a function**(g)**of*Quantity***(Q).**

And the Number of Tickets left:

where

*Quantity (Q)*is a function (h) of*Time (T).*

This gives us a new function, where Price is a function of Time.

Using this function, we can also calculate the expected price P at any given value of time T and quantity Q to populate a matrix, and show this function in tabular form.

As you can see, we have only filled the diagonal elements. That is the expected results that we have achieved. However, what happens when things don’t go as planned?

That’s where the rest of the elements come into play.

If the number of items sold at any particular time (T) and price (P) was more than predicted, it suggests that more individuals were willing to buy a ticket at the current time and price. So accordingly, we need to increase the prices (in other cases, we may need to decrease).

We can represent this variance as an upwards shift in the demand function. When there is an upward shift in demand curve, people are willing to pay more for the same product as it’s more in demand. Thus to make sure we have a higher revenue, we increase prices. A side effect of this is that the demand decreases and thus we have tickets left to sell at a later time.

On the other hand, if the sales we are making are lesser than expected, the demand function shifts downwards. Thus we need to make the prices lower to sell the same quantity as we did before.

Calculating all of the values that these upward and downward shifted demand functions give us all of the Prices P, given combinations of Times T and Quantities Q, that we need to fill the full Price Matrix for our hypothetical movie screening, showing us the prices that we need to charge at any given combination of time to screening and seats left to sell to maximize revenue for that movie screening.

All of this looks good on paper. But how do we go about implementing it?

I’m pretty sure you’re also asking if we reduce the prices, won’t the cinema owner face a loss? Why give such a discount based on time afterall?

So let’s see how to implement such a model.

**Implementation**

All your questions were valid. We have to set a few policies on how to transition the prices effectively, without incurring losses. We didn't make too many changes to the original pricing scheme, but we did end up putting the price matrix on top of it. This price matrix can now be converted as a percentage matrix of the original price.

As you can see, we never reduce the price to less than 20% of the original price. This way we can ensure that the firm incurs no loss in the long run. We also do not charge more than the actual price in order to be fair to the consumers.

**Conclusions**

Dynamic pricing is a very intuitive idea where markets can benefit by dynamically changing their prices and making negligible changes to their management and economy. As you saw with the above example, taking time and quantity as factors, we produced a matrix for the price change. Accordingly, various other factors can be accounted for and a matrix for each can be made and correlated.

In practice, the concept of dynamic pricing is a lot more complex than what we explored in this blog. However, the basic intuition remains the same. Change prices according to the situation of the market to benefit both parties in the long run.

**References**

**Pictures and examples: **

https://towardsdatascience.com/how-to-build-a-dynamic-pricing-model-5544151e8bce

https://blog.wiser.com/dynamic-pricing-what-why-how/