I have been studying the area of unsecured consumer credit (consumer loans and credit cards) and credit scoring. My question is: can we have a utility function (either a lender's or borrower's utility) that can take both the borrower's credit score (probability of repayment) and loan limit (amount) as arguments in an explicit manner? What I have managed to get on the web are implicit functions of the form $f(x,y)$ e.g. on equation (1) in the article https://pdfs.semanticscholar.org/3207/55db4a277766043b5a1cb73f3b84df9cb613.pdf

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    $\begingroup$ Short answer is yes. But you have to be more explicit about the properties that you want the utility function to possess. For example, should the function be increasing/decreasing in either creditworthiness or loan amount, and is there any interaction effects between the two variables? $\endgroup$ – Herr K. Mar 5 '18 at 17:32
  • $\begingroup$ @HerrK. the utility function is to be a non-decreasing function of credit worthiness. As for the interaction between the two variables, we can consider a case whereby the loan amount is influenced (implicitly) by the credit worthiness variable. Further, the loan amount can be considered as a control variable, as in stochastic optimization. $\endgroup$ – Owadgi Akinyi Mar 6 '18 at 4:09

Below is an example based on the general formulation in your linked article. This will very likely need a lot more fine-tuning. Moreover, the following example does not necessarily satisfy the theoretical properties laid out in footnote 16 of the paper. (You should check this.) Nevertheless I hope this could at least give you an idea of how to think about writing down an appropriate utility function that suits your purpose.

Consider \begin{equation} \Pi(d,q,\theta)= \underbrace{\left[1+\mathrm e^{-(\alpha_0+\alpha_1 d+\alpha_2q+\alpha_3\theta)}\right]^{-1}}_{G(d,q,\theta)} \biggl[d+\underbrace{\frac{a+bq-d}{\theta T}\left(\frac{1-(1+r)^{-\theta T}}{r}\right)^{\!\!-1}}_{M(L,q,\theta)}-q^2\biggr]\,, \end{equation} where

  • $d=$ down payment; $q=$ car quality; $\theta=$ creditworthiness; $L=$ loan amount
  • car with quality $q$ is priced at $p(q)=a+bq$, where $a,b$ are parameters to be estimated/calibrated
  • loan amount: $L=p(q)-d=a+bq-d$
  • probability of purchase $G(d,q,\theta)$ is modeled by a logistic distribution, which is parameterized by the variables of interest, and the $\alpha_i$'s are parameters to be estimated/calculated
  • $\theta T$ is the term of the loan, which is increasing in $\theta$
  • assuming repayment in each period is the same, i.e. $L/\theta T$, the PV of loan payment is $M(L,q,\theta)$, where $r$ is the periodic interest rate
  • cost of producing a car with quality $q$ is modeled by $c(q)=q^2$
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  • $\begingroup$ this surely gives an idea on how to go along formulating the utility function. Much thanks $\endgroup$ – Owadgi Akinyi Mar 7 '18 at 4:48

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