Cost 

If we ask you whether you would spend more living your entire life (or even a year) in a hotel or a self-owned/rented house, the answer will clearly be the latter. This is the same case in the cloud world. 

The public cloud can be considered akin to a full service hotel. They will upgrade the hardware, take care of the maintenance of the servers, data center, network, and so on. They will manage the vendors and ensure the load gets taken off you. 

The CoLo (Co-Location) model is comparable to a rented apartment, where you pay for the space and the basic features, but you have to bring your own furniture (hardware). This is less expensive when compared to the public cloud, but our responsibility increases multi-fold. 

A private data center is similar to owning a house or an apartment. In this we are responsible for everything, and it gives us the maximum flexibility and freedom to make whatever changes we might desire. That puts an onus back on us to ensure that we know what we are doing, or hire experts temporarily (such as an architect) to help us make the changes. 

As far as costs are concerned, houses are cheaper in most cases. A study on statista has shown that in the US, the average daily rate of hotels is $122.64 (https://www.statista.com/statistics/208133/us-hotel-revenue-per-available-room-by-month/) and the median square foot price in buying a home is $140 (https://www.zillow.com/home-values/), and the average house costs us over $250000. 

Considering a hotel room is 100 sq-foot, the same size home (which is not easily possible) will take only 6 months to break even, and even if we use one room in the house, the hotel will break even with the home at approximately 6 years. 

In order to further understand the point, let's take a look at a study done. However, remember that it is a like to like service comparison; we are not comparing the fancy cognitive services that can be had the public cloud. 

Case in point, research by ServerPronto University and a survey conducted by 451 Research, point to the fact that a well designed (read, automated) private cloud is indeed several times cheaper (ServerPronto University claims 3X) than that of an equivalent public cloud. 

Again, the internet is divided on this, however this is the truth given a certain volume. If it were not true, then AWS/Azure/GCP and the like would never make significant profits at all. In order to evade unnecessary debates, look at the following graph on the costs: 

Now, let's take a look at the public cloud costs. It's a linear curve (with almost a 45 degrees slope), as we know it follows a pricing model per virtual machine (VM). As an example, if the monthly cost of a VM is $100, then 500 virtual machines a month will automatically be $50000. 

In the private cloud, the initial capital expenditure makes it expensive initially, but for the private cloud, the additional capacity is incremental, and so the slope of the curve is a lot less than 45 degrees. For this example, we can consider it 30-35 degrees. This will ensure that there is a break-even point at which the private cloud will become cheaper than the public cloud. 

Depending on what we pit the cost against (the options are number of virtual machines, % utilization of virtual machines, and virtual machines managed per engineer), the break-even point will move from the mid-hundreds to high-hundreds. The 451 Research also came to a similar conclusion. 

In my experience, at 400 VMs (medium- or large-sized) running at 100% utilization 24 x 7, the private cloud becomes cheaper. This number is easily achieved in any medium to large enterprise environment.