Question

According to the American Metal Markets Magazine, the spot market price of U.S. hot rolled steel recently reached \(600 per ton. Less than a year ago this same ton of steel was only \)300. A number of factors are cited to explain the large price increase. The combination of China’s increased demand forraw steel—due to expansion of its manufacturing base and infrastructure changes when preparing for the 2008 Beijing Olympics—and the weakening U.S. dollar against the euro and yuan partially explain the upward spiral in raw steel prices. Supply-side changes have also dramatically affected the price of raw steel. In the last 20 years there has been a rapid movement away from large integrated steel mills to mini-mills. The mini-mill production process replaces raw iron ore as its primary raw input with scrap steel. Today, mini-mills account for approximately 52 percent of all U.S. steel production. However, the worldwide movement to the mini-mill production model has bid up the price of scrap steel. In December, the per-ton price of scrap was around \(140, and it soared to \)285 just two months later. Suppose that, as a result of this increase in the price of scrap, the supply of raw steel changed fromQ^s_raw=4,400 +4P to Q^s_raw=800+4P. Assuming the market for raw steel is competitive and that the current worldwide demand for steel is Q^d_raw=4,400+ 8P, compute the equilibrium price and quantity when the per-ton price of scrap steel was \(140, and the equilibrium price–quantity combination when the price of scrap steel reached \)285 per ton. Suppose the cost function of a representative mini-mill producer is C(Q)=1,200+15Q². Compare the change in the quantity of raw steel exchanged at the market level with the change in raw steel produced by a representative firm. How do you explain this difference?

Short answer

After the change, the required quantity would be 10 units, which was initially 0.

Step-by-step solution

Given 

To compute the equilibrium price and quantity of the scrap steel when the price is $140 and then increases to $285, we must match the demand and supply functions as appropriate to each price. Therefore, it is important to bear in mind that the supply function for each price will be the following, and the demand function will remain constant for both prices.

The supply function at a price of $140 is:$Q_{raw}^S=4,400+4P$

The supply function at a price of $285 is:$Q_{raw}^S=800+4P$

The supply function for both price is:$Q_{raw}^D=4,400-8P$

Part 1

After matching supply and demand at a price of $140 and solving P to obtain the equilibrium price of the scrap steel market, we get

$\begin{array}{c}4,400+4\text{P}=4,400-8\text{P}\\ 4,400-4,400=12\text{P}\\ \text{P}=0\end{array}$

Thus, the equilibrium price of per ton raw steel when the scrap metal price is $140 is $0. Therefore, the quantities that maximize profit will also be 0.

By matching supply and demand at a price of $285 and solve P to obtain the equilibrium price of the scrap steel market:

$\begin{array}{c}800+4\text{P}=4,400-8\text{P}\\ 3600=12\text{P}\\ \text{P}=300\end{array}$

Thus, the equilibrium price of per ton raw steel when the scrap metal price is $285 is $300.

By substituting this price that is greater than zero in the demand function, the equilibrium quantities of raw steel can be obtained when the price of scrap metal is $285.

Thus,

$\begin{array}{l}{Q}_{raw}=4,400-8\left(300\right)\\ Q_{raw}=2,000\end{array}$

Thus, the equilibrium quantity of raw steel is 2000 units.

Second Part:

Now having the following cost function, the change in the quantity of raw steel exchanged at the market level with the change in raw steel must be determined.

$C\left(Q\right)=1,200+15Q^2$

Deriving the cost function and equating it to the price we can obtain the supply of raw steel at each price level calculated in the first step:

$P=\frac{\text{dC}}{\text{dQ}}30Q$

By substituting the prices $0 and $300 obtained in the first step and solving for Q, we get-

$\begin{array}{c}0=30\text{Q}\\ Q=0\\ \\ 300=30\text{Q}\\ \text{Q}=10\end{array}$

Due to the increase in the price of scrap metal from $140 to $285, the mini mill will be able to increase its raw steel production and be profitable with a price greater than zero and with which they will be able to obtain a profit.