Question

As the manager of Smith Construction, you need to make a decision on the number of homes to build in a new residential area where you are the only builder. Unfortunately, you must build the homes before you learn how strong demand is for homes in this large neighbourhood. There is a $\mathbf{60}$percent chance of low demand and a $\mathbf{40}$percent chance of high demand. The corresponding (inverse) demand functions for these two scenarios are $\mathit{P}\mathbf{=}\mathbf{300}\mathbf{,}\mathbf{000}\mathbf{‐}\mathbf{400}\mathit{Q}$and $\mathit{P}\mathbf{=}\mathbf{500}\mathbf{,}\mathbf{000}\mathbf{‐}\mathbf{275}\mathit{Q}$, respectively. Your cost function is $\mathit{C}\left(Q\right)\mathbf{=}\mathbf{140}\mathbf{,}\mathbf{000}\mathbf{+}\mathbf{240}\mathbf{,}\mathbf{000}\mathit{Q}$. How many new homes should you build, and what profits can you expect?

Short answer

The profit that Smith Construction's manager may predict is: $\cancel{S}13,860,000$.

The manager should build $200$new houses.

Step-by-step solution

Define Asymmetric Information

The term asymmetric information implies that in a deal between two groups, one group cheats another because it has more information. This phenomenon is common in insurance deals.

Evaluating the grouping data

Smith Construction's management has two demand functions, each with a distinct probability of high or low demand for dwellings in the new development. Similarly, it has the same cost function for each of these requests, allowing us to determine the best number of houses to build and the projected profits.

The probability of occurrence and demand functions can be replaced using the anticipated value given below:

$$\begin{gathered} E\left(x\right)=q_1{x}_1+q_2{x}_2+......+q_n{x}_n \\ E\left(x\right)=60\%\left(300,000-400Q\right)+40\%\left(500,000-275Q\right) \\ E\left(x\right)=\left(180,000-240Q\right)+\left(200,000-110Q\right) \end{gathered}$$

The grouping data gives the following:

$E\left(x\right)=380,000-350Q.........\left(1\right)$

Evaluating the cost

Assuming that profits can be computed as (revenue - cost), the following demand function with the cost function is matched:

$Profit=\left(380,000-350Q\right)Q-\left(140,000+240,000Q\right)$

When it comes to profit maximization, the predicted marginal revenue may be calculated by taking the derivative of the prior equation:

$$\begin{gathered} \frac{dprofit}{dq}=\left(380,000-350Q^2\right)-240,000 \\ \frac{dprofit}{dp}=\left(380,000-700Q\right)-240,000 \end{gathered}$$

As a result, the marginal revenue will be calculated using the following formula: $380,000-700Q$, with a cost of $240,000$.

Evaluating the number of houses that manger should built

To find the new residences that the management should build, set the equation to zero and solve for Q from the previous equation:

$$\begin{gathered} \left(380,000-700Q\right)-240,000=0 \\ 380,000-240,000=700Q \\ 140,000=700Q \\ Q=\frac{140,000}{700} \\ =200 \end{gathered}$$

The manager should build $200$new houses.

Evaluating the profit

Q can be used to replace the predicted value in equation one to get the total amount of the incomes:

$$\begin{gathered} E\left(p\right)=380,000-350\left(200\right) \\ =380,000-70,000 \\ =310,000 \end{gathered}$$

We get the value of the expected benefit by subtracting the whole value of the income from the cost equation and inserting Q:

$$\begin{gathered} E\left(p\right)=\left(310,000-240,000\right)\left(200\right)-140,000 \\ =70,000\times \left(200\right)-140,000 \\ =13,860,000 \end{gathered}$$

Therefore, the profit will be: $\cancel{S}13,860,000$.