5. In a certain kingdom, the workforce consists only of the clan of dwarves and the clan of elves. Historically, in this kingdom, dwarves and elves have always worked separately, and no enterprise has ever hired both at the same time. The aggregate supply of labor resources of the dwarves is represented by the function $w_{\text {dwarves }}^{S}=1+L / 3$, and the aggregate supply of labor resources of the elves is $w_{\text {elves }}^{S}=3+L$. The inverse function of the aggregate demand for dwarf labor is $w_{\text {dwarves }}^{D}=10-2 L / 3$, and the inverse function of the aggregate demand for elf labor is $w_{\text {elves }}^{D}=18-2 L$. The king, who recently ascended to the throne, is very concerned that the wage rates of his subjects are different, so he issued a law stipulating that the wages of elves and dwarves must be the same and that workers in the kingdom should not be discriminated against based on clan affiliation. At the same time, the king believes that regulatory wage rate control will have a negative impact on the economy of his kingdom and requires all his subjects to behave in a perfectly competitive manner. By what factor will the wage of the group of workers whose wage was lower before the king's intervention increase, if firms in the kingdom are indifferent to hiring elves or dwarves?
# Solution:

Before the law was introduced, the wages of elves and gnomes were determined by the condition of equality of supply and demand in each market:

$w_{\text {gnome }}^{S}=w_{\text {gnome }}^{D}$, from which $1+\frac{L}{3}=10-2 L / 3$ and $L_{\text {gnome }}=9$ and $w_{\text {gnome }}=4$

$w_{\text {elf }}^{S}=w_{\text {elf }}^{D}$, from which $3+L=18-2 L$ and $L_{\text {elf }}=5 w_{\text {elf }}=8$

After the law was introduced, all market participants must behave in a perfectly competitive manner, and the total demand for labor resources must equal the total supply of labor resources. Therefore,

$L_{\text {gnome }}^{S}+L_{\text {elf }}^{S}=L_{\text {elf }}^{D}+L_{\text {gnome }}^{D}$, where

$L_{\text {gnome }}^{S}+L_{\text {elf }}^{S}=\left\{\begin{array}{l}3 w-3, \text { if } w \leq 3 \\ 4 w-6, \text { if } w>3\end{array}\right.$

$L_{\text {elf }}^{D}+L_{\text {gnome }}^{D}=\left\{\begin{array}{l}9-w / 2, \text { if } w \geq 10 \\ 24-2 w, \text { if } w solutions. All transformations and calculations are correct. The correct answer is obtained. | $\mathbf{1 0}$ |
| The correct sequence of all steps in the solution is provided. There are gaps in the justification of the choice of number combinations (for example, one extra solution (0+4) is not considered, pairs of digits are not divided by 2, etc.) or a computational error or typo that does not affect the further course of the solution. As a result of this error or typo, an incorrect answer may be obtained. |  |
| The problem is not solved, but its solution is significantly advanced, i.e.:  - a significant part of the solution is performed correctly, possibly inaccurately  (for example, part of the combinations is considered and calculated correctly); |  |
| - the other part is either not performed or performed incorrectly, possibly |  |
| even with logical errors (for example, most combinations are not considered |  |
| or extra combinations are considered, combinations are calculated incorrectly, cases of erasing the same digit (2+2, |  |
| 5+5, 8+8) are not considered, etc.); |  |
| In this case, the solution may be incomplete. |  |

All cases of the solution that do not meet the above criteria for scoring 3 and 6 points.