The other ingredient you are leaving out is OIL. You can't make CP/HP without oil, so it's not just the lye that is the cost difference between those and MP. The oil and lye are already in the MP soap.
Costwise, it really really depends on the vendors you choose and the oils you choose. For example, do you pay $12.00 for 2 pounds of MP at a local Michael's store, or do you order online and pay $6.00 for 2 pounds (plus shipping fees)? Another example is for lye: Do you pay $14.00 for 2 pounds of lye at your local hardware store (some say they get it for a lot less, but that is totally dependent on where you live, as around here it's $14 for 2 pounds) or do you order online and pay $7.58 for two pounds (plus shipping if applicable)? Oils in your grocery store can be costly or not, depending on if you buy 'specialty' oils or lard, for example. Oils online usually go along with a shipping fee, except in cases where the vendor is running a 'no shipping costs' special.
So you can see how the cost can be variable depending on your product choices and the vendors, but always take into account shipping and/or the cost of gas if you have to go out of your way to find the product.
edited to correct numbers (lye online for me is about 3.79 per pound free shipping so I doubled for 2 pounds)
Here is a potential cost comparison for a soap recipe using ingredients from local sources. This is NOT a recipe I use, but I made it up based on ingredients I can find in the small town where I live (except the lye, I have to travel for that).
2 pounds CP soap using 20.8 ounces of oils (I normally use grams, but for comparison to MP, which here is sold in pounds...)
Castor Oil sold at Walmart in 2 ounce bottles as a laxitive: $2.64 per bottle (using about half) so $1.32
Crisco new w/palm: $5.48 for 48 ounces (using 10.4 ounces) so $1.21
Olive oil: $2.56 for 17 ounces (using 10.4 ounces) so $1.57
Coconut Oil: $9.98 for 60 ounces (using 3 ounces) so $0.49
Lye: $14 for lye at TSC (gasoline cost to get there: $14.00) (using 2.83 ounces) so $2.47
Distilled water (required to make lye solution) comes to $0.05 for this batch
Total cost of oils for 2 pounds of this CP soap: $4.59
Lye solution: $2.47 (includes the cost of the gasoline to go buy it)
Total cost for this 2 pounds of soap using local stores as the source for oil and lye: $7.11
Michaels MP base $12.00 for 2 pounds. (That seems to be their standard price for different MP bases)
For me this would require a 60 mile round trip drive, so the price goes up based on one's gas mileage.
So there you have it, if I go to my local and local-ish stores, some of which require a 100 mile round trip, the CP soap is till cheaper than the MP. BUT that doesn't include some of the nicer oils that I like to use. What if I want to use shea butter or avocado oil, or cocoa butter, etc? The cost of the CP/HP soap goes up with those ingredients. And the recipe I used in this example doesn't include any luxury oils. And it's a lot like comparing apples to oranges because the ingredients list on any given MP base is not going to be the same as this particular soap recipe in this example.
Oh, and I forgot to include sales tax! That skews the numbers even more. Sales tax for me on food is 1%. I forgot to add that in for the oils. Sales tax for the lye would be 7.75% and for the MP it would be 7%, as different states and counties are involved in these purchases for me.