# Profit maximization under stochastically independence

Consider a risk-neutral profit-maximizing firm that produces output y from input x using the technology y = f(x), f’(x) > 0 and f’’(x) < 0. It is possible to ob-tain x from two sources: either directly from a primary supplier or from a secondary supplier (retailer) who purchases it from various primary suppliers and resells it. Pri- mary suppliers charge a lower price, p, but are prone to supply disruptions, whereas secondary suppliers charge a higher price, r, but guarantee delivery, where r > p > 0. The price of output is normalized to 1. Let a$\in$ (0; 1) denote the probability that a primary supplier will deliver the input. The contract between a firm and a primary supplier stipulates that the firm orders and pays for the input in advance even if a supply disruption occurs and the input cannot be delivered. It is verifiable whether a disruption occurs, and, if not, then the supplier is obligated to deliver the input. In what follows, let xp and xr denote the quantities of x bought from primary and secondary sellers (retailers) respectively.

As stated above, a secondary seller, unlike the firm, can purchase the input from many suppliers. But like the firm, he is subject to the same contract restrictions, namely, it must prepay for inputs even if they are not delivered. Assuming that supply disruptions among primary sellers are stochastically independent and there is a large number of primary suppliers in the industry, justify (using mathematics) why secondary suppliers can (almost) guarantee supply of the input.

Question is as above.

I have done the following

Step 2 firm’s problem

$$max(f(x_r)-rx_r)$$

By FOC

$$f’(x_r)-r=0$$

Step 1 retailers problem

$$max (rx_r-(p_1x_{p_1}+...+p_nx_{p_n})$$

$$max (r\sum_i^na_ix_{p_i}-\sum^n_ip_ix_{p_i}))$$

FOC

For i=1, $r a_1-p_1=0$

For i=2, $r a_2-p_2=0$

For i=n, $r a_n-p_n=0$

$\sum_i^n r*a_i=\sum_i^np_i$

$r\sum_i^n a_i=\sum_i^np_i$

$\sum_i^n a_i={\sum_i^np_i\over r}$

Since $p_i<r$, then $\sum_i^n a_i=\frac{p_1}{r}+…+\frac{p_n}{r}$ converges to one.

So $\sum_i^n a_i$ converges to 1.

But after that, I don’t know how to apply stochastically independence in order to deal with this maximization problem.