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$2.$ Cournot competition

Consider a town in which only two residents, Gilberto and Juanita, own wells that produce water safe for drinking. Gilberto and Juanita can pump and sell as much water as they want at no cost$.$ Assume that outside water cannot be transported into the town for sale. The following questions will walk you through how to compute the Cournot quantity competition outcome for these duopolists.

Consider the market demand curve for water and the marginal cost for collecting water on the following graph. Assume Gilberto believes that Juanita is going to collect $12$ gallons of water to sell.

On the graph, use the purple points $($diamond symbols$)$ to plot the demand curve $($D$1$D$1)$ Gilberto faces; then use the grey points $($star symbol$)$ to plot the marginal revenue curve $($MR$1$MR$1)$ Gilberto faces. Finally, use the black point $($plus symbol$)$ to indicate the profit$-$maximizing price and quantity $($Profit Max $1)$ in this case.

Note: Dashed drop lines will automatically extend to both axes.

D$1$MR$1$Profit Max $10246810121416182022246\mathrm{.}005\mathrm{.}505\mathrm{.}004\mathrm{.}504\mathrm{.}003\mathrm{.}503\mathrm{.}002\mathrm{.}502\mathrm{.}001\mathrm{.}501\mathrm{.}000\mathrm{.}500$PRICE $($Dollars per gallon$)$QUANTITY $($Gallons of water$)$Market DemandMC$4,4$

Instead, now assume Gilberto believes that Juanita is going to collect $16$ gallons of water to sell, rather than $12.$

On the following graph, use the purple points $($diamond symbol$)$ to plot the demand curve $($D$2$D$2)$ Gilberto faces; then use the grey points $($star symbol$)$ to plot the marginal revenue curve $($MR$2$MR$2)$ Gilberto faces. Finally, use the black point $($plus symbol$)$ to indicate the profit$-$maximizing price and quantity $($Profit Max $2)$ in this case.

D$2$MR$2$Profit Max $20246810121416182022246\mathrm{.}005\mathrm{.}505\mathrm{.}004\mathrm{.}504\mathrm{.}003\mathrm{.}503\mathrm{.}002\mathrm{.}502\mathrm{.}001\mathrm{.}501\mathrm{.}000\mathrm{.}500$PRICE $($Dollars per gallon$)$QUANTITY $($Gallons of water$)$Market DemandMC

Fill in the following table with the quantity of water Gilberto produces, given various production choices by Juanita.

Juanita's Water Production

Gilberto's Water Production

$($Gallons of water$)$

$($Gallons of water$)$

$12$

$16$

Given the information in this table, use the green points $($triangle symbol$)$ to plot Gilberto's best$-$response function $($BRF$)$ on the following graph. Since Gilberto and Juanita face the same costs for producing water, Juanita's best$-$response function is simply the reverse of Gilberto's; that is$,$ the curve has the same shape, but the horizontal and vertical intercept values are switched. Therefore, you can derive Juanita's best$-$response function by following the same analysis as in the previous question, but from Juanita's perspective. Use the purple points $($diamond symbol$)$ to plot her best$-$response function on the graph. Finally, use the black point $($plus symbol$)$ to indicate the unique Nash equilibrium under Cournot quantity competition.

Gilberto's BRFJuanita's BRFProfit Max $2024681012141618202224242220181614121086420$JUANITA'S QUANTITY $($Gallons of water$)$GILBERTO$\text{'}$S QUANTITY $($Gallons of water$)$

True or False: According to her best$-$response function, Juanita will always want to decrease her output as Gilberto decreases his.

True

False

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Cournot competition

Consider a town in which only two residents, Gilberto and Juanita, own wells that produce water safe for drinking. Gilberto and Juanita can pump and

sell as much water as they want at no cost$.$ Assume that outside water cannot be transported into the town for sale. The following questions will walk

you through how to compute the Cournot quantity competition outcome for these duopolists.

Consider the market demand curve for water and the marginal cost for collecting water on the following graph. Assume Gilberto believes that Juanita

is going to collect $12$ gallons of water to sell.

On the graph, use the purple points $($diamond symbols$)$ to plot the demand curve $($ D$\_(1))$ Gilberto faces; then use the grey points $($star symbol$)$ to plot

the marginal revenue curve $($MR $)$ Gilberto faces. Finally, use the black point $($plus symbol$)$ to indicate the profit$-$maximizing price and quantity $($Profit

Max $1)$ in this case.

Note: Dashed drop lines will automatically extend to both axes.

Profit Max $1$

Instead, now assume Gilberto believes that Juanita is going to collect $16$ gallons of water to sell, rather than $12.$

On the following graph, use the purple points $($diamond symbol$)$ to plot the demand curve $($ D$\_(2))$ Gilberto faces; then use the grey points $($star symbol$)$

to plot the marginal revenue curve $($MR$2)$ Gilberto faces. Finally, use the black point $($plus symbol$)$ to indicate the profit$-$maximizing price and quantity

$($Profit Max $2)$ in this case.

Fill in the following table with the quantity of water Gilberto produces, given various production choices by Juanita.

Juanita's Water Production

$($Gallons of water$)$

Gilberto's Water Production

$($Gallons of water$)$