How does the companies set up utility function for its own purposes?
In another word, what is the types statistics data that is usually consider and statistical method being used to set up these functions? Just name the types of methods and types of data out.
I heard that the function itselves is always to be a secret of a company but why are they always different between different companies?
Most agents, including firms, do not actively set-up utility functions in their heads. A firm might however might have a worker create a profit function. A course related to labor economics will usually give you an idea of what sorts of things a firm considers.
A generic macroeconomic profit maximization problem that you'll commonly see is
$$\max_{K, L} \ \Pi = A[\delta K^{-\rho} + (1- \delta)L^{-\rho}]^{-1/\rho} - (rK + wL)$$
Where you have a constant elasticity of substitution ($\rho$ is the substitution parameter) with some level of technology ($A$) and weight of distribution ($\delta$) of inputs between capital and labor ($K, L$) which have costs per unit in the form of the rental rate of capital and wages ($r, w$). Maximize profit with respect to capital and labor.
In labor, you might see a very basic profit maximization setup where the worker maximizes
$$\max_{e} \ U = \mathbb{E}(\alpha + \beta (e + v) - C(e))$$
where $e$ is some level of effort, $\alpha + \beta q$ is pay given $q$ output and $q = e + v$, where $v$ is an i.i.d. random variable with an expected value of zero in this case.
The first order condition is
$$\frac{\partial}{\partial e} : \beta - c'(e) = 0$$
which will create the labor supply function.
The constraint is that the worker is individually rational, so $\alpha + \beta e \geq C(e)$, which can be simplified to just equality since the firm will maximize profit by giving just enough pay to elicit the effort they are looking for.
So the firm maximizes
$$\max_{\alpha, \beta} \ E(q) - (\alpha + \beta e) = e - C(e)$$
FOC:
$$\frac{\partial}{\partial \beta} : [1 - c'(e)] \frac{\partial e}{\partial \beta} = 0$$
$$\implies \beta = 1 \ (= c'(e))$$
In the above, we see that the firm makes sure that they get the full pay for their value of effort, but in real life, that may not mean the worker gets a piece rate part of their wage equal to 1; a worker like a car salesman may only make a commission of 8-20% of the value of the car because there are other labor and capital inputs that go into selling the car, so the salesperson may be contributing 8-20% of the effort needed to sell the car.
There are other things the firm needs to consider like screening costs, heterogeneous labor markets (differentiated skill levels), signalling versus human capital or firm specific capital, etc. I'll provide one more economic setup a firm might look at when seeing how their workers invest in human capital.
This setup comes from the Heckman and Killingsworth paper (1986 I want to say) on women in the labor force.
We have a dynamic labor supply with endogenous wages.
$$E(t) = E[H(t), K(t)]$$ where $E(t)$ is earnings at time $t$. $H(t)$ is hours of work for some period of time at time period $t$, and $K(t)$ is the stock of human capital.
$$\frac{\partial K(t)}{\partial t} \equiv \dot{K} = i[I(t), G(t), K(t)] - qK(t)$$
where $q$ is the depreciation of human capital, $I(t), G(t)$ are investment and goods meant to increase human capital (note that human capital is an input into its own production.)
The worker then has some utility to maximize:
$$U = \int_0^D e^{-st} u[c(t), m(t)L(t), K(t)] \ dt$$
where $c(t)$ is consumption at time $t$, $L(t)$ is leisure, and $m$ is a multiplicative factor, an underlying preference for leisure over time that changes involvement in the labor force.
Constraints:
$T = I(t) + H(t) + L(t)$
$W(t) = E(t)/(I(t) + H(t))$
$A(D) = A(0) + \int_0^D e^{-rt} [E(t) - P(t)C(t)] \ dt \geq 0$
(the last condition is just a transversality condition; cannot die in debt)
Heckman neutrality:
$E(t) = k H(t)K(t)$
$\dot{K} = i[I(t)K(t)] - qK(t)$
$U = \int_0^D e^{-st} u[c(t), m(t)L(t)K(t)] \ dt$
(the third condition is different from the above actually)
$u(t)$ will be the shadow value of cash, and $w(t)$ will be the shadow value of human capital.
Solving from the first order conditions will get you a time path for $w(t)$
Other recommended reading: "Personnel Economics in Practice", 2009; Lazear, Edward P. and Gibbs, Michael
