Chapter 12, Problem 12.1P

a)

To determine

To find:The short − run supply curve when q is the function of market price.

Explanation of Solution

First find MC from TC function and then equate MC with P.

$\begin{array}{l}\text{C(q) = }\frac{1}{300}{q}^3+0.2q^2+4q+10\\ MC=\frac{\Delta C(q)}{\Delta q}\\ MC=0.01q^2+0.4q+4\end{array}$

Now equate MC = P.

$\begin{array}{l}MC=0.01q^2+0.4q+4\\ P=0.01q^2+0.4q+4\\ \text{Thus supply curve, }P=0.01q^2+0.4q+4\\ \text{Rewrite in q form}\text{.}\\ \text{q = 10}\sqrt{P}-20\end{array}$

b)

To determine

The short-run industry supply curve need to be computed.

Explanation of Solution

Multiply 100 to find the industry supply curve.

$\begin{array}{l}\text{q = 10}\sqrt{P}-20\\ q=1000\sqrt{P}-2000\\ \text{Thus, industry}\text{supply curve:}\\ q=1000\sqrt{P}-2000\end{array}$

c)

To determine

To find:The combination of short run equilibrium price and quantity.

Explanation of Solution

Given market demand:

$\begin{array}{l}Q=-200P+8000\\ \text{Now, Qs = Qd}\\ \text{1000}\sqrt{P}-2000=-200P+8000\\ \text{1000}\sqrt{P}+200P=8000+2000\\ \text{1000}\sqrt{P}+200P=10000\\ 5\sqrt{P}+P=50\\ {\left(5\sqrt{P}\right)}^2={\left(50-P\right)}^2\\ 25P=2500+P^2-100P\\ P^2-125P+2500=0\\ \text{Now solve for the P:}\\ \text{P = 25 and P =100 (ignore this )}\\ \text{Thus, Price = \$25}\\ \text{Now find Q:}\\ \text{Q = -200(25) + 8000}\\ \text{Q = -5000+8000}\\ \text{Q = 3000}\end{array}$

Thus, equilibrium price and quantity is $25 and 3000 units, respectively.