How do we compute surplus and who got them here?

A billionaire buys a yacht from a yacht wizard.

the wizard cast spells produce a yacht that he sells at $1 billion.

the billionaire values the yacht at $1.1 billion and hence buys the yacht.

so total economic surplus is $1.1 billion. billionaire got $1.1 billion surplus but pay $1 billion. he is better off by $100 million.

want another yacht? asks the wizard. let me think. says the billionaire. I have diminishing marginal utility and the second one may only worth $1.0001 billion.

a pretty woman comes to the billionaire. how much would you value an heir you can pass on a billion dollar?

the billionaire think. $1.5 billion dollar.

that means less yacht for me. but I think I will be happier having another biological heir.

well. I am like that wizard. I can create heirs to you. let me play with some wand and wait 9 months to cast a spell and voila you got an heir that will of course pass paternity tests. I want $1 million. I want $1 billion for our children. That $1 billion is worth around $500 million to me.

so both of our utility function is something like

U = log(c) + β × n × log(w_child) which is a Barro Becker model. it's a function of consumption + passing on wealth to children. here β reflects how much we love our children. the higher the β the higher the marginal cost we get by passing more wealth to our children.

so mom got surplus $500. the billionaire got surplus of $1.5 billion but he pays $1 billion as inheritance to son. so he got a $500 million surplus. paying $1 billion inheritance cost the billionaiere money. that's money he can't use to buy more yachts. but he prefers that anyway because having genetic offspring makes him happier than another yacht.

let's say the billionaire keep having children till marginal happiness of having one additional children meet the marginal cost. but let's talk about one child at a time here.

Mom is also better off by $500 million. she prefers her children to be rich and the arrangements make her child rich.

So the billionaire value having a $1 billion son at $1.5 billion. mom value having a $1 billion son at $500million. there is a $2 billion surplus at least. not to mention the son himself got $1 billion that worth $1 billion

in wizard case the total surplus of $1.1 billion is split. wizard got $1 billion and billionaire got $100 million.

here we have surplus of at least $2 billion. yet billionaire got $500 million and mom got $500 million. numbers don't match.

why?

double counting?

I mean surplus should be $3 billion if we count the child to literally get $1 billion. but ai says we shouldn't double count. I mean the child don't agree to be born. so we should only look at expected improvement to utility of mom and dad.

which is only $500 million each.

where is the other $1 billion surplus?