Question

You are the manager of Taurus Technologies, and your sole competitor is Spyder Technologies. The two firms’ products are viewed as identical by most consumers. The relevant cost functions are C(Qi )= Qi , and the inverse market demand curve for this unique product is given by$\mathbf{P}=\mathbf{160}-\mathbf{2}\mathbf{Q}.$ Currently, you and your rival simultaneously (but independently) make production decisions, and the price you fetch for the product depends on the total amount produced by each firm. However, by making an unrecoverable fixed investment of \($\mathbf{200}$ , Taurus Technologies can bring its product to market before Spyder finalizes production plans. Should you invest the \) $\mathbf{200}$? Explain

Short answer

It is not easy to invest the $200.

Step-by-step solution

Define oligopoly

A market structure known as an oligopoly is one that has few competitors and high entrance barriers.

Given by 

In the first scenario, the two firms are in a Cournot oligopoly model with simultaneous production decisions and cost functions. In the other scenario where Taurus invests of$200 in improvements, it makes it have a leadership position, and its rival Spyder becomes a follower.

The inverse demand and cost functions for both firms are as follows:

Inverse Demand Function: $\text{P=160-2Q}$

Where Q $\text{=}\left({\text{Q}}_{\text{-}}{\text{T+Q}}_{\text{-}}\text{S}\right)$

Taurus Cost Function: $\text{C}\left({\text{Q}}_{\text{-}}\text{T}\right){\text{=4Q}}_{\text{-}}\text{T}$

Spyder Cost Function: $\text{C}\left({\text{Q}}_{\text{-}}\text{S}\right){\text{=4Q}}_{\text{-}}$

To find the reaction function

Given that in the Cournot Model the condition that the marginal revenues are equal to the marginal costs is fulfilled, we can derive the total revenues and total costs to obtain the reaction function of both firms. First we find the reaction function of Taurus.

$\begin{array}{l}\text{TotalRevenue=PricexQuantity}\\ =(160-2{\mathrm{Q}}_{\mathrm{T}}-2{\mathrm{Q}}_{\mathrm{S}}){\mathrm{Q}}_{\mathrm{T}}\\ =160{\mathrm{Q}}_{\mathrm{T}}-2{\mathrm{Q}}_{\mathrm{S}}{\mathrm{Q}}_{\mathrm{T}}-2{\mathrm{Q}}_{\mathrm{T}}^2\end{array}$

To obtain the marginal revenue:

$\text{MR=}\frac{{\text{dR}}_{\text{T}}}{{\text{dQ}}_{\text{T}}}{\text{=160-2Q}}_{\text{S}}{\text{-4Q}}_{\text{T}}$

And the marginal cost:

$\text{MC=}\frac{{\text{dC}}_{\text{T}}}{{\text{dQ}}_{\text{T}}}\text{=4}$

To Equate the marginal revenue and marginal cost

Equating marginal revenue and marginal cost and solving for ${\text{Q}}_{\text{T}}$:

$\begin{array}{c}160-2{\mathrm{Q}}_{\mathrm{S}}-4{\mathrm{Q}}_{\mathrm{T}}=4{\mathrm{Q}}_{\mathrm{T}}\\ =\frac{160-4-2{\mathrm{Q}}_{\mathrm{S}}}{4}{\mathrm{Q}}_{\mathrm{T}}\\ =39-\frac{1}{2}{\mathrm{Q}}_{\mathrm{S}}\end{array}$

Equating marginal revenue and marginal cost and solving for${\text{Q}}_{\text{S}}$:

$\begin{array}{c}160-2{\mathrm{Q}}_{\mathrm{T}}-4{\mathrm{Q}}_{\mathrm{S}}=4{\mathrm{Q}}_{\mathrm{S}}\\ =\frac{160-4-2{\mathrm{Q}}_{\mathrm{T}}}{4}{\mathrm{Q}}_{\mathrm{S}}\\ =39-\frac{1}{2}{\mathrm{Q}}_{\mathrm{T}}\end{array}$

To obtain the equilibrium output level

Having the reaction functions we can substitute the Spyder function in the Taurus function and solve for ${\text{Q}}_{\text{T}}$ to obtain the equilibrium output level.

$\begin{array}{c}{\mathrm{Q}}_{\mathrm{T}}=39-\frac{1}{2}\left(39-\frac{1}{2}{\mathrm{Q}}_{\mathrm{T}}\right){\mathrm{Q}}_{\mathrm{T}}\\ =39-19.5+\frac{1}{4}{\mathrm{Q}}_{\mathrm{T}}-{\mathrm{Q}}_{\mathrm{T}}\\ =26\end{array}$

With the equilibrium output value of the Taurus firm we can substitute it in the reaction function of the Spyder firm and obtain the equilibrium output.

$\begin{array}{c}{\mathrm{Q}}_{\mathrm{s}}=39-\frac{1}{2}26\\ =26\end{array}$

To obtain equilibrium price

We can obtain the equilibrium price by substituting the equilibrium output levels of both firms in the inverse demand function:

$\begin{array}{l}\mathrm{P}=160-2({\mathrm{Q}}_{\mathrm{T}}+{\mathrm{Q}}_{\mathrm{S}})\\ \mathrm{P}=160-104\\ \mathrm{P}=56\end{array}$

Thus, the equilibrium price for both firm is $56.

To take the equilibrium levels of output and price

Taking the equilibrium levels of output and price of both firms, the profit of each firm can be obtained as follows:

$\begin{array}{l}\text{Π=TotalRevenue-TotalCost}\\ =(56\times 26)-(4\times 26)\\ =1352\end{array}$

Given that under the Cournot oligopoly model the two firms have the same level of output and price, both will obtain a profit of $\$1,352.$

To obtain the equilibrium output level of the leading firm

In this other scenario, the Stackelberg model, the Taurus company will be the leading firm while Spyder will be the follower company. Therefore, we can take the reaction function obtained for the follower company $\left({\text{Q}}_{\text{S}}\right)$and obtain the equilibrium output level of the leading firm as follows:

$\begin{array}{l}{\text{Π}}_{\text{T}}\text{=Total\hspace{0.17em}\hspace{0.17em}Revenue-Total\hspace{0.17em}\hspace{0.17em}Cost}\\ =\mathrm{P}\times {\mathrm{Q}}_{\mathrm{T}}-4{\mathrm{Q}}_{\mathrm{T}}\\ =[160-2({\mathrm{Q}}_{\mathrm{T}}+{\mathrm{Q}}_{\mathrm{S}})]{\mathrm{Q}}_{\mathrm{T}}-4{\mathrm{Q}}_{\mathrm{T}}\\ =[160-2({\mathrm{Q}}_{\mathrm{T}}+39-\frac{1}{2}{\mathrm{Q}}_{\mathrm{T}})]{\mathrm{Q}}_{\mathrm{T}}-4{\mathrm{Q}}_{\mathrm{T}}\\ =(160-2{\mathrm{Q}}_{\mathrm{T}}-78+{\mathrm{Q}}_{\mathrm{T}})({\mathrm{Q}}_{\mathrm{T}}-4{\mathrm{Q}}_{\mathrm{T}})\\ =82{\mathrm{Q}}_{\mathrm{T}}-{\mathrm{Q}}_{\mathrm{T}}^2-4{\mathrm{Q}}_{\mathrm{T}}\\ =78{\mathrm{Q}}_{\mathrm{T}}-{\mathrm{Q}}_{\mathrm{T}}^2\end{array}$

To derive the function

Deriving the above function and solving for${\text{Q}}_{\text{T}}$ :

$\begin{array}{c}\frac{{\mathrm{d\Pi }}_{\mathrm{T}}}{{\mathrm{dQ}}_{\mathrm{T}}}=78-2{\mathrm{Q}}_{\mathrm{T}}{\mathrm{Q}}_{\mathrm{T}}\\ =78/2\\ =39\end{array}$

Taking the equilibrium output for the leading company, we can obtain the output of the follower company by substituting the value of ${\text{Q}}_{\text{T}}$ in the reaction function of ${\text{Q}}_{\text{S}}$.

$\begin{array}{c}{\mathrm{Q}}_{\mathrm{S}}=39-\frac{1}{2}\left(39\right)\\ =19.5\end{array}$

Taking the equilibrium output levels for the leading and following firm, we can obtain the equilibrium price by substituting in the inverse demand function.

$\begin{array}{c}\mathrm{P}=160-2(39+19.5)\\ =160-117\\ =43\end{array}$

To obtain the profits of the leading and following firm

Taking the equilibrium price and output in the Stackelberg model, we can obtain the profits of the leading and following firm.

Leader Firm Proft:

$\begin{array}{l}{\text{Π=TotalRevenue}}_{\text{T}}{\text{-TotalCost}}_{\text{T}}\\ =\mathrm{P}\times \mathrm{Q}-4\mathrm{Q}\\ =(43\times 39)-(4\times 39)\\ =1,677-156\\ =1,521\end{array}$

Follower Firm Proft:

$\begin{array}{c}\text{Π}={\text{Total\hspace{0.17em}\hspace{0.17em}Revenue}}_{\text{S}}-{\text{Total\hspace{0.17em}\hspace{0.17em}Cost}}_{\text{S}}\\ =\mathrm{P}\times \mathrm{Q}-4\mathrm{Q}\\ =(43\times 19.5)-(4\times 19.5)\\ =838.5-78\\ =760.5\end{array}$

The profit of Taurus will be $\mathrm{\$}1,521$and the profit of Spyder will be $760.5.

As managers of the firm Taurus we must analyze the two profit results of our firm under the two models and taking into account the investment made of $200 to be a leading firm.

In the Cournot model we will have a profit of $1352 while in the Stackelberg model we will obtain income of $1521. The difference between the two scenarios is $169$ ($1,521-1,352$), which is less than the $200 investment that makes our product on the market earlier than the rival firm.

Therefore, as this investment is greater than the benefits that we could obtain from being in a leading position in the market, it is not convenient for us to invest the $200 and therefore we must stay in the Cournot model.