Problem $4.(20$ points$)$ Suppose that the inverse demand function $is$ described $byP=$

$120-Q,$ where $q_{1}is$ the output $of$ the incumbent firm and $q_{2}is$ the output $of$ the entrant.

Let the labor cost and capital cost per unit $be20$ and $20,i.e.,w=r=20.In$ addition, let

each firm have a fixed cost $of{F}_1=F_2=200.$

Suppose that $in$ stage one, the incumbent invests $in$ capacity $\frac{?}{\text{b}\text{a}\text{r}}(K{)}_1.$ Calculate the $in-$

cumbent's best response function $in$ stage two for the cases $q_1\frac{?}{b}ar(K{)}_1$ and $q_1\frac{?}{b}>ar(K{)}_1.$

Additionally, calculate the entrant's best response function $in$ stage two. $(5$ points$)$

Determine the profit$-$maximizing output for the Stackelberg leader and specify the pro$-$

duction capacity the incumbent should establish $to$ maximize profits. $If$ the incumbent

commits $to$ a production capacity $of\frac{?}{\text{b}\text{a}\text{r}}(K{)}_1,$ calculate the entrant's optimal output $in$

stage two. Finally, compute the profits for both the incumbent and the entrant and

determine which firm gains a greater advantage. $(5$ points$)$

Now show that $if$ the incumbent chooses a capacity $\frac{?}{\text{b}\text{a}\text{r}}(K{)}_1=32.$ How does the entrant's

behavior change? What would the incumbent's profit $bein$ this case? Compare $it$ with

the results obtained previously. $(5$ points$)$

$Is$ choosing less than $30$ and calculate