Chapter 6.2, Problem 46E

Nursing facilities. A study of Texas nursing homes found that the annual profit $P$ (in dollars) of profit-seeking, independent nursing homes in urban locations is modeled by

$P\left(w,r,s,t\right)=0.007955w^{-0.638}{r}^{\text{1}.038}{s}^{0.873}{t}^{2.468}$

where $w$ is the average hourly wage of nurses and aides (in dollars), $r$ is the occupancy rate (as a percentage), $s$ is the total square footage of the facility, and $t$ is the Texas Index of Level of Effort (TILE), a number between 1 and 11 that measures state Medicaid reimbursement. [Source: K. J. Knox, E. C. Blankmeyer, and J. $R$. Stutzman, “Relative Economic Efficiency in Texas Nursing Facilities,” Journal of Economics and Finance, Vol. 23, 199–213 (1999).] Use this information for Exercises 45 and 46.

46. The change in $P$ due to a change in $w$ when the other variables are held constant is approximately $\Delta P\approx \frac{\partial P}{\partial w}\Delta w$

Use the values of w, $r$, $s$, and $t$ in Exercise 45 and assume that the nursing home gives its nurses and aides a small raise so that the average hourly wage is now $40.25 an hour. By approximately how much does profit change?