Chapter 6.2, Problem 44E

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

$P\left(w,r,s,t\right)=0.007955w^{-0.638}{r}^{1.038}{s}^{0.873}{t}^{2.468},$

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 43 and 44.

The change in P due to a change in w-when the other variables are held constant is approximately

$\Delta \text{P}\approx \frac{\partial \text{P}}{\partial \text{w}}\Delta \text{w}$.

Use the values of w, r, s, and t in Exercise 43 and assume that the nursing home gives its nurses and aides a small raise so that the average hourly wage is now $20.25 an hour. By approximately how much does profit change?