$\backslash$#$1.$ Consider a simple duopoly problem with two firms that are identical except in one important sense. The two produce an undifferentiated good, the demand relationship is summarized by the inverse demand curve $\backslash ($ P$=1-$X $\backslash ),$ where $\backslash ($ P $\backslash )$ is the price and $\backslash ($ X $\backslash )$ is the total quantity produced by the two firms $\backslash (\backslash$left$($X$=$x$\_\{1\}+$x$\_\{2\}\backslash$right$)\backslash ).$ The two firms compete as Cournot quantity competitors $($each assuming the other firm's supply is fixed$)$; each names a quantity supplied, simultaneously and independently of the other, and the market price is set so that demand equals the sum of the two supplies. Both firms have zero marginal costs of production. One firm is owner managed. The quantity decision for this firm is made by the owner, who retains any profits, $\backslash (\backslash$pi$\_\{1\}\backslash ).$ This owner is risk neutral.

The second firm is not managed by its owner. The owner of the second firm has hired a manager and has given this manager a contract wherein the compensation paid to the manager is a linear function of the profits of the firm, $\backslash (\backslash$pi$\_\{2\}\backslash )$ and the quantity that the firm sells, $\backslash ($ x$\_\{2\}\backslash ).$ Specifically, this contract calls for the manager to receive a payment equal to $\backslash (\backslash$alpha $\backslash$pi$\_\{1\}+\backslash$beta x$\_\{2\}\backslash ),$ where $\backslash (\backslash$alpha $\backslash )$ and $\backslash (\backslash$beta $\backslash )$ are given constants. The owner retains any residual profits $($profits net of the payment to the manager$)$ made by the firm. Note well that the manager makes the quantity decision and not the owner, and the manager does so to maximize her compensation and not the net profits of the owner. What, in this case $($for given $\backslash (\backslash$alpha $\backslash )$ and $\backslash (\backslash$beta $\backslash ))$ is the Cournot equilibrium in the market in terms of price and quantities supplied by the two firms? What $($as a function of $\backslash (\backslash$alpha $\backslash )$ and $\backslash (\backslash$beta $\backslash ))$ are the profits of firm one, the manager's compensation and the net profits $($to the owner$)$ of firm two?