Chapter 6.2, Problem 43E

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.

A profit-seeking, independent Taxas nursing home in an urban setting has nurses and aides with an average hourly wage of $20 an hour, a TILE of 8, an occupancy rate of 70%, and $400,000{\text{ft}}^2$ of space.

a. Estimate the nursing home’s annual profit

b. Find the four partial derivatives of P

c. c) Interpret the meaning of the partial derivatives found in part (b).