The market equilibrium point for a product is reached when 12000 units are produced and sold at $27 per unit. The manufacturer will not produce any units at the price of $6, and the customers will not buy any at the price of $69. Find the supply and demand equations, assuming they are linear.

The equations should express price p in terms of quantity q.

Supply equation: p=

Demand equation: p=

If the equilibrium is such that only 12000 units are sold for $27, then the total earnings from the given scenario is $324,000. The supply equation would then be,
supply: 324000 = 6p ; p = 324000/6 = 54000
demand: 324000 = 69p ; p = 324000/69 = 4695.65 ≈ 4696

Answer Link

MrRoyal MrRoyal · Answer 2 · 2022-02-15T10:46:11-05:00

The demand and the supply equations are $p = \frac{-24000 + 4000q }{7}$ and $p = \frac{138000 - 2000q }{7}$

How to determine the equations

At the equilibrium point, we have:

(p,q) = (12000, 27)

On the supply equation, we have:

(p,q) = (0, 6)

So, we start by calculating the slope:

$m = \frac{q_2 - q_1}{p_2 -p_1}$

This gives

$m = \frac{6 - 27}{0 - 12000}$

$m = \frac{21}{12000}$

Simplify

$m = \frac{7}{4000}$

The supply equation is then calculated as:

$q = m(p - p_1) + q_1$

So, we have:

$q = \frac{7}{4000}(p - 0) + 6$

$q = \frac{7}{4000}p + 6$

Subtract 6 from both sides

$-6 + q = \frac{7}{4000}p$

Multiply through by 4000

$-24000 + 4000q = 7p$

Rewrite as:

$7p = -24000 + 4000q$

Solve for p

$p = \frac{-24000 + 4000q }{7}$

On the demand equation, we have:

(p,q) = (0, 69)

So, we start by calculating the slope:

$m = \frac{q_2 - q_1}{p_2 -p_1}$

This gives

$m = \frac{69 - 27}{0 - 12000}$

$m = -\frac{42}{12000}$

Simplify

$m = -\frac{7}{2000}$

The demand equation is then calculated as:

$q = m(p - p_1) + q_1$

So, we have:

$q = -\frac{7}{2000}(p - 0) + 69$

$q = -\frac{7}{2000}p + 69$

Subtract 69 from both sides

$-69 + q = -\frac{7}{2000}p$

Multiply through by -2000

$138000 - 2000q = 7p$

Rewrite as:

$7p = 138000 - 2000q$

Solve for p

$p = \frac{138000 - 2000q }{7}$

Hence, the demand and the supply equations are $p = \frac{-24000 + 4000q }{7}$ and $p = \frac{138000 - 2000q }{7}$

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