Question $4(25$ points$)$

Suppose a firm is composed of two divisions: an upstream division and a

downstream division, with no external market.

Upstream division:

The upstream division produces Good Y as an intermediate good used in

producing Good X$.$

$$ The total cost function of producing Good Y is $=++.$

$$

$,$

where QU is the quantity of Good Y produced.

Downstream division:

The downstream division produces Good X as a final good and sells it to

customers.

$$ The market demand function is Q $=500\mathrm{.}5$P$,$ where Q is the quantity

of Good X produced, and P is the price per unit of Good X$.$

$$ The total cost function of producing, assembling, and selling Good X to

customers is TCD $=100+20$Q$\mathrm{.}11$

$4\mathrm{.}1(7$ points$)$ Write down the profit functions.

Downstream profit function, i$.$e$.,$ D $=$ f$($QU$,$ Q$)$:

$$ D $=$

$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

Upstream profit function, i$.$e$.,$ U $=$ f$($QU$)$:

$$ U $=$

$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

Total profit function, i$.$e$.,$ $=$ f$($QU$,$ Q$)$:

$$ $=$

$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

Total profit function with no external market, i$.$e$.,$ $=$ f$($Q$),$ where QU $=$ Q:

$$ $=$

$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

$4\mathrm{.}2(7$ points$)$ What are the quantity of Good X $($Q$*),$ Good Y $($QU

$),$ and the price

per unit of Good X $($P$*)$ that maximize the firm$$s total profit?

$$ Q$*=\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$ units.

$$ QU

$=\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$ units.

$$ P$*=$ $ $\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_12$

$4\mathrm{.}3(4$ points$)$ What is the optimal transfer price $($P U

$),$ i$.$e$.,$ the price per unit of

Good Y that maximizes the firm$$s total profit?

$$ P U

$=$ $ $\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

$4\mathrm{.}4(7$ points$)$ Using the information from Question $4,$ what are the profits of

the downstream division $($D$),$ the upstream division $($U$),$ and the total profit

of the firm $($$)?$

$$ D $=$ $ $\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

$$ U $=$ $ $\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

$$ $=$ $ $\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$