<p>Below is an example based on the general formulation in your linked article. <em>This will very likely need a lot more fine-tuning.</em> 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. </p>

<p>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</p>

<ul>
<li>$d=$ down payment; $q=$ car quality; $\theta=$ creditworthiness; $L=$ loan amount</li>
<li>car with quality $q$ is priced at $p(q)=a+bq$, where $a,b$ are parameters to be estimated/calibrated</li>
<li>loan amount: $L=p(q)-d=a+bq-d$</li>
<li>probability of purchase $G(d,q,\theta)$ is modeled by a <a href="https://en.wikipedia.org/wiki/Logistic_distribution" rel="nofollow noreferrer">logistic distribution</a>, which is parameterized by the variables of interest, and the $\alpha_i$'s are parameters to be estimated/calculated </li>
<li>$\theta T$ is the term of the loan, which is increasing in $\theta$</li>
<li>assuming repayment in each period is the same, i.e. $L/\theta T$, the <a href="https://en.wikipedia.org/wiki/Present_value#Present_value_of_an_annuity" rel="nofollow noreferrer">PV of loan payment</a> is $M(L,q,\theta)$, where $r$ is the periodic interest rate</li>
<li>cost of producing a car with quality $q$ is modeled by $c(q)=q^2$</li>
</ul>

CC BY-SA 3.0
2018-03-07T04:48:47.910Z
this surely gives an idea on how to go along formulating the utility function. Much thanks
Owadgi Akinyi
CC BY-SA 3.0
2018-03-06T20:53:20.360Z
2018-03-06T20:53:20.360Z
Herr K.
<p>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 <a href="https://pdfs.semanticscholar.org/3207/55db4a277766043b5a1cb73f3b84df9cb613.pdf" rel="nofollow noreferrer">https://pdfs.semanticscholar.org/3207/55db4a277766043b5a1cb73f3b84df9cb613.pdf</a></p>

CC BY-SA 3.0
2018-03-05T17:32:02.210Z
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?
Herr K.
CC BY-SA 3.0
2018-03-06T04:09:04.207Z
@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.
Owadgi Akinyi
<microeconomics><utility><consumer-credit>
Utility Function in Consumer Credit
Owadgi Akinyi
# Utility Function in Consumer Credit

(Asked by: username_0 on 2018-03-05)

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>

**Tags:** microeconomics, utility, consumer-credit

> **Comments:**
> **username_1**: 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?  
> **username_0**: @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.  

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## 1 answers

### Answer by username_1 on 2018-03-06. Score: 0 | Accepted answer

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](https://en.wikipedia.org/wiki/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](https://en.wikipedia.org/wiki/Present_value#Present_value_of_an_annuity) is $M(L,q,\theta)$, where $r$ is the periodic interest rate
* cost of producing a car with quality $q$ is modeled by $c(q)=q^2$

> **Comments:**
> **username_0**: this surely gives an idea on how to go along formulating the utility function. Much thanks  

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