This is a fun little task I need help completing. Im creating a game-theoretical model for the utility a government get from labour rights (R) and FDI (F), and I need a utility function to have certain properties to be a good model of the theory Im using. An important factor is the average level of labour rights (C) in other countries. The point of the theory is that states cuts competitively in labour rights to attract more FDI. So, Im hoping someone on this forum could pick up the challenge of making a Utility function for me that I can use.
Here are the Rules. P is a function of R and F (Higher R and F gives higher P), F is a function of R and C, higher R relative to C gives lower F. There has to be a parameter alfa(a) that gives the "sensitivity" of a state to P and beta (b) that gives the sensitivty to F. a+b=1 Higher a means that the government gets more utility per increase in R. Higher b means that the government gets more utility per increase in F.
The optimal choice of R for a government should be a function of a,b and C in which the entire idea of this is that the more a government prefers FDI instead of labour rights, the lower R should be set relative to C.
I need the function itself, its first derivative, and the value of R at which the first derivative is zero.
I hope this makes sense and that someone will find this challenge to be fun.
Best regards Magnus