Supply Chain -- Sage

Supply Chain

This sage notebook is here to understand what the "right" values for supply chain and competition are. There is a fixed daily demand for food:

q F = 101 - p F

$q_F=101-p_F$

The food (output) is produced by one or more food producers by hiring workers. Each worker every day turns one unit of beef into one unit of food. The labor supply curve is

w F = 5 7 q F

$w_F=\frac 5 7 q_F$

The beef (input) is produced by one or more beef producers by hiring workers. Each worker every day creates one unit of beef (no input). The labor supply curve is

w B = 5 7 q B

$w_B=\frac 5 7 q_B$ The two labor supplies are independent.

Perfect Competition

In perfect competition there are many food and beef producers. Each of them should act as a price taker.

The profit function of a competitive food producer is:

Π F = p F * q F - 5 7 q 2 F - p B q F

$\Pi_F = p_F * q_F - \frac 5 7 q^2_F - p_B q_F$

The profit function of a competitive beef producer is:

Π B = p B * q B - 5 7 q 2 B

$\Pi_B = p_B * q_B - \frac 5 7 q^2_B$

We also know that in equilibrium the amount of beef produced will be equal to the quantity consumed

q B = q F = q

$q_B = q_F = q$

When we add the demand curve we have 3 equations in 3 unknown

⎧ ⎩ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ \partial Π F \partial q F \partial Π B \partial q B q F = 0 = 0 = 101 - p F

$\begin{cases} \frac{\partial \Pi_F}{\partial q_F} &=0 \\ \frac{\partial \Pi_B}{\partial q_B} &=0 \\q_F &=101-p_F \end{cases}$

q, f,b = var('q,f,b') s=solve([ diff(f*q-q*q*5/7 - b*q,q)==0, diff(b*q-5/7*q^2,q)==0, q==101-f] ,f,b,q) s

[[f == (2020/27), b == (1010/27), q == (707/27)]]

Then

⎧ ⎩ ⎨ q p F p B \approx 26 \approx 75 \approx 37

$\begin{cases} q &\approx 26 \\ p_F &\approx 75 \\p_B &\approx 37 \end{cases}$

Food Monopolist, Beef Competitive

Now we have a lot of beef producers but only one food seller which will act as a monopolist.

The beef producers profit function is the same:

Π B = p B * q B - 5 7 q 2 B

$\Pi_B = p_B * q_B - \frac 5 7 q^2_B$

The food producer instead will consider both prices as a function of its production:

Π F = p F (q F) * q F - 5 7 q 2 F - p B (q F) q F

$\Pi_F = p_F(q_F) * q_F - \frac 5 7 q^2_F - p_B(q_F) q_F$ Where the solution of the profit function of the beef producer will be the supply function

pB(qF) $p_B(q_F)$ since

q F = q B = q

$q_F = q_B = q$ to insert in the profit function of the food monopolist.

supplyBeef=solve(diff(b*q-5/7*q^2,q)==0,b) supplyBeef

[b == 10/7*q]

solve(diff((101-q)*q-q*q*5/7 - 10/7*q*q,q)==0,q)

[q == (707/44)]

⎧ ⎩ ⎨ q p F p B \approx 16 \approx 85 \approx 23

$\begin{cases} q &\approx 16 \\ p_F &\approx 85 \\p_B &\approx 23 \end{cases}$

Food Competitive, Beef Monopolist

This is the inverse of the previous function. Now the beef producer takes into consideration how its production affect prices:

Π B = p B (q B) * q B - 5 7 q 2 B

$\Pi_B = p_B(q_B) * q_B - \frac 5 7 q^2_B$

The food producers instead take all prices as given:

Π F = p F * q F - 5 7 q 2 F - p B q F

$\Pi_F = p_F * q_F - \frac 5 7 q^2_F - p_B q_F$ So we need to build beef demand first to solve the monopolist problem

diff(f*q-q*q*5/7 - b*q,q)==0

-b + f - 10/7*q == 0

solve(-b + 101-q - 10/7*q == 0,b)

[b == -17/7*q + 101]

That should be the demand of beef

solve(diff((-17/7*q + 101)*q-5/7*q^2,q)==0,q)

[q == (707/44)]

⎧ ⎩ ⎨ q p F p B \approx 16 \approx 85 \approx 62

$\begin{cases} q &\approx 16 \\ p_F &\approx 85 \\p_B &\approx 62 \end{cases}$