7. Suppose you live in a small town and are very familiar with all its residents. Since the townspeople, quite rightly, consider you to be economically savvy, some of them have come up with the idea of you opening a recruitment agency where you alone would represent all employable residents in the labor market. For this, the townspeople would pay you $1 \%$ of their labor income, and there would be no other way for them to find employment. You have calculated that your costs for placing the townspeople in jobs would be $T C(L)=\frac{1}{3200} L^{2}$, where $L$ is the number of labor hours the townspeople will work at the enterprises. (The townspeople trust you so much that they are willing to work for any wage if you represent their interests to the employer.) The town has only one key enterprise that produces a finished product using the technology $q=16 \sqrt{L}$ (workers from other towns do not come to the enterprise) and sells it on a perfectly competitive market at a price of 1.

After some thought, you realized that there are two possible negotiation schemes with the enterprise:

1) You name the wage rate $w$, and the enterprise decides how much labor (in hours) it will use at this wage rate.
2) The enterprise names the wage rate $w$, and you decide how much employment (in hours) you will offer the townspeople.

Whichever scheme is implemented, all employable residents of the town work the same number of hours.

(a) Which scheme would you personally prefer, and how much would your recruitment agency earn in this case?

(b) Is it really beneficial for the townspeople to have you represent their interests? Justify your answer.
# Solution:

(a) It is more advantageous for you to choose the scheme where you have a leadership position: you make the decision first, anticipating how the company will then act. Let's show this.

1) Let's predict how the company will respond after you name the wage rate, and take this into account when we choose the wage rate ourselves.

The company maximizes its profit:

$16 \sqrt{L} - wL \rightarrow \max _{L}$

You can take the derivative or use a substitution to bring the function to the form of a downward-opening parabola and then find its vertex: $L = \frac{64}{w^2}$

You maximize your profit:

$0.01wL - \frac{1}{3200}L^2 \rightarrow \max _{w}$ given that $L = \frac{64}{w^2}$, i.e., substituting the amount of labor, we get your problem.

The optimal $w = 2$, then $L = 16$ and the agency's profit is 0.24.

2) In this case, the company tries to anticipate my move:

$0.01wL - \frac{1}{3200}L^2 \rightarrow \max _{L}$ or $wL - \frac{1}{32}L^2 \rightarrow \max _{L}$.

We get $L = 16w$.

The company maximizes its profit, taking into account how much labor I will offer them:

$16 \sqrt{L} - wL \rightarrow \max _{w}$ given $L = 16w$, i.e., $16 \cdot 4 \sqrt{w} - 16w^2 \rightarrow \max _{w}$ or $4 \sqrt{w} - w^2 \rightarrow \max _{w}$.

Taking the derivative, we get $w = 1$.

Then $L = 16$ and the agency's profit is $0.01wL - \frac{1}{3200}L^2 = 0.08$.

We indeed see that being the leader is more advantageous.
(b) If the residents were hired directly, the company would simply be a monopsonist in the labor market and would maximize its profit. In the case where you act as an intermediary, the residents would gain more market power - they would be hired on more favorable terms overall. However, the agency charges a fee for this, and if the residents formed a union with lower costs (including lower transaction costs), it would be more advantageous for them.

## Grading Criteria:

(a) Intermediate and penalty points:

- 5 points if only one scheme is correctly calculated and the profit is found.
- 10 points if both options are correctly calculated
- If the maximum is not justified by any method - minus 1 point
- 3 points if only the profit maximization problem of one agent is solved with a fixed variable (price) of the second agent

(b) Intermediate points:

- 1-5 points if there are arguments about the disadvantage/advantage of monopsony/union + justification (depending on the completeness of the justification).

15 points - for a complete, correct, and justified solution to the problem.