Consider a stylized model describing how John Deere $($JD$)$ and Caterpillar $($C$)$ produce tractors for sale. Both JD and $\backslash (\backslash$mathbf$\{$C$\}\backslash )$ first build plants of preferred capacity and then set their prices. For simplicity, assume that the customers make no distinction between JD and $\backslash (\backslash$mathbf$\{$C$\}\backslash )$ and simply buy the tractors at whatever the lowest price might be$.$

In South America, the inverse demand for tractors is given by

$\backslash [$

P$=80-20$ Q

$\backslash ]$

where $\backslash ($ P $\backslash )$ is measured in thousands of USD and $\backslash ($ Q $\backslash )$ is measured in hundreds of thousands. Once the plants are built, the marginal costs of the two firms are given by

$\backslash [$

M C$\_\{$J D$\}=20\backslash$text $\{$ and $\}$ M C$\_\{$C$\}=30$

$\backslash ]$

$($a$)$ Suppose JD and $\backslash (\backslash$mathbf$\{$C$\}\backslash )$ already have pre$-$existing large production plants so that they effectively have no capacity constraints. At what price will the tractors be sold?

$($b$)$ Suppose the firms are building their plants simultaneously and choose quantities $\backslash ($ Q$\_\{$J D$\}\backslash )$ and $\backslash ($ Q$\_\{$C$\}\backslash ).$ Find best responses of the firms. What price will they be charging in the market?

$($c$)$ If JD could instead be the first to build their plant, how large would its plant be$?$ How does this choice compare to its capacity in $($b$)?$

$($d$)$ Suppose JD had a choice: either to compete against against $\backslash (\backslash$mathbf$\{$C$\}\backslash )$ la Cournot or to pay $\backslash (\backslash$mathbf$\{$C$\}\backslash )$ never to enter the market in the first place. What is the maximum amount $\backslash ($ X $\backslash )$ it would be willing to pay assuming $\backslash (\backslash$mathbf$\{$C$\}\backslash )$ will honor the deal?