# Discount function

Company A needs to provide service $p$ to company B. However if the company B will reserve the $(1-\alpha)$ fraction of resource for the serving customers of company A, company A will provide a discount proportional to ($1-\alpha$) to company B.

The utility function of company A is composed of benefit obtained from company B given as

$$U(A)=\alpha*p*c*f(1-\alpha)$$

c here is the unit cost of resource $p$ and this function is maximized in c.

The utility of company B is based on benefit-cost function and given as :

$$U(B)=log(1+\alpha p)-\alpha *p * c * f(1-\alpha)$$

Which is maximized in $\alpha$

I need a function$f(1-\alpha)$ which reduces the price $p*c$ .

I have considered the following functions

$$exp(-(1-\alpha))$$ But I could not find the value of $\alpha$ that would maximize the utility of company B

$$\frac{\alpha p}{(1+(1-\alpha)^2)}$$ But this function also does not works.

Any ideas for function $f(1-\alpha)$

• Have you tried $f(1-\alpha)=-(1-\alpha)^2$? Jun 21 '18 at 12:53
• yes this gives a negative value of $p$ Jun 21 '18 at 14:32
• I don't understand the second sentence Jun 23 '18 at 17:23

Note: The story behind the problem is still confusing, but I will merely focus on your utility functions.

Edit

In the problem, it seems that both are minimizing costs: $A$ is doing so via $c$, and $B$ via $\alpha$. I assume you want a functional form that yields a closed-form solution (and not corner solutions). In that case, you may want $U$'s to be quasi-concave (i.e. to have a unique maximum). Namely, you want \begin{eqnarray*} \frac{\partial U(A)}{\partial c} >0,&& \frac{\partial^2 U(A)}{\partial c^2} <0, \\ \frac{\partial U(B)}{\partial \alpha} >0,&& \frac{\partial^2 U(B)}{\partial \alpha^2} <0. \end{eqnarray*}

It seems that $f(1-\alpha) = - \ln(1-\alpha)$ satisfies those inequalities.

• Exactly You are right until "the problem boils down to finding response functions of each player." and I missed a point here that instead of $log(1+P)$ it is $log(1+\alpha P)$,because it is using only an $\alpha$ share resource of P. Now the best response is not independent and similarly the best response of company A is also not independent of $\alpha$. Jun 25 '18 at 13:18
• @user7341333 Please, make corresponding changes in your OP Jun 25 '18 at 15:32
• using $-ln(1-\alpha)$ gives me the following equation : $\frac{p}{ln(2)(1+\alpha p)}+p*c*ln(1-\alpha)-p*c\frac{\alpha}{1-\alpha}=0$ How can I solve it for $\alpha$, even if I am able to solve it, for the value of $c$using backward induction in utility of Company B, if I replace the value of $\alpha$ it will become too complex to solve. Jun 27 '18 at 5:39
• @user7341333 You cant get an explicit form solution, but you can show that a solution exists Jun 27 '18 at 9:06
• also, your derivative is wrong: the first term should be $\frac{p}{1+\alpha p}$ Jun 27 '18 at 9:33