# Task 1. Maximum 20 points

Let's say in country A there are only two firms engaged in research and development in a certain field. Currently, each of them independently decides whether to participate in the development of a new technology. It is known that if a firm develops a new technology, it will bring it $V$ monetary units of income in the future, provided the second firm is either unsuccessful or does not participate in the development. If both firms develop the new technology, each will receive $0.5 \mathrm{~V}$ monetary units. The investment costs for research and development for each firm will be IC monetary units regardless of the success of the development. Let the probability of success for each firm be $\alpha, 0<\alpha<1$. Then we can write the expected income of each firm as $\alpha(1-\alpha) V+0.5 \alpha^{2} V$, if both firms participate in the development, or $\alpha V$, if only the given firm participates in the development.

(a) Under what conditions will both firms participate in the development?

(b) How many firms will participate in the development if $V=16, \alpha=0.5$, and $I C=5$?

(c) Let's assume that social welfare can be assessed as the total profit of the firms. Will the number of firms found in part (b) be socially optimal? Comment on the result obtained.
# Solution and Grading Scheme:

(a) For each firm to decide to participate in the development, it is necessary that the expected profit from participation for each firm is greater than 0 (we use the formula for expected income in the case where both firms are involved in development):

$\alpha(1-\alpha) V+0.5 \alpha^{2} V-I C \geq 0$ (4 points)

(b) Substitute the known variables into the inequality from part (a) and check its validity:

$$
\begin{gathered}
0.5^{2} \cdot 16+0.5 \cdot 0.5^{2} \cdot 16-5 \geq 0 \\
4+2-5>0 \text { (4 points) }
\end{gathered}
$$

It makes no sense to check the equilibrium where only one firm is involved in development, as two inequalities must be satisfied simultaneously:

$$
\alpha V-\text { IC } \geq 0 \text { and } \alpha(1-\alpha) V+0.5 \alpha^{2} V-\text { IC }<0
$$

We know that the second one is not satisfied. (2 points)
(c) Calculate in which case the total profit will be greater:

1) both firms participate in development:

$\pi_{1}+\pi_{2}=2\left((1-\alpha) V+0.5 \alpha^{2} V-I C\right)=2(6-5)=2$ (3 points)

2) one firm participates in development.

$\pi_{\text {one }}=\alpha V-\mathrm{IC}=0.5 \cdot 16-5=8-5=3$. (3 points)

We see that it is more efficient for one firm to be involved in development. This is due to the fact that in the case of two firms participating, the duplication of research costs leads to a greater increase in costs than the increase in potential revenue for the firms due to the increased probability of discovery. (4 points)