When aging seniors begin to need basic assistance with daily living, they can essentially choose from one of the following to get the long-term care they need:
(1) receiving assistance at home from a loved one;
(2) hiring a caregiver from a homecare agency; or
(3) moving into an assisted living facility. While approximately 70% of seniors over 75 years of age obtain help from a loved one in the US, home care agencies (HC) and assisted living facilities (ALF) are growing, lucrative industries. ALFs and HC services provide quality senior care and assistance for those in the aging elderly US population who have the ability to lessen the burden on their children by paying for expert long-term care services with their home equity, pensions, retirement savings, and/or government funding.
ALFs naturally compete with HC agencies for seniors and it is typically the adult daughter who decides if her aging parent will either move into an ALF or hire an in-home caregiver. Presumably, an adult daughter will choose the option that cultivates the most health and happiness to her aging parent at the lowest cost (especially in the current economic climate), and the goal of an ALF is to maximize revenue while keeping occupancy rates high by not losing seniors to HC companies. However, ALFs (known in signaling games as the sender (Source 1), since they send a price signal to the adult daughter) can vary greatly in quality (i.e. 'good' or 'bad') and HC quality is more stable (See Note). Ideally, operators of 'good' ALFs would signal their high quality to adult daughters with high prices, but because HC is a valuable alternative and there are 'bad' ALF that could raise their prices to falsely signal quality, the 'good' ALF operator has to carefully set its rates. This uncertain price-quality signaling between high revenue for the ALF and optimal benefit to the senior resident can be analyzed using game theory, particularly an extensive form signaling model, to help owners and operators of ALFs answer the question:
How should I price my assisted living facility to profit and show high quality, while still attracting residents?
As with any theoretical model, many assumptions about the 'game' must be made in order to solve. First, we will presume that there is uncertainty for the adult daughter regarding the quality of ALFs and a facility can be either 'good' (G) (higher benefit), with a probability of (p), or 'bad' (B) (lower benefit), with probability (1-p). Moreover, the benefit a senior receives from an in-home caregiver is more constant and simply provides a benefit of (HC). Second, ALFs can charge a high monthly rate (H) or a low monthly rate (L) and HC companies charge a constant amount, (K). The following values will be used as a numerical example to represent the costs and benefits of various senior living choices:
(G) = 6 (arbitrary benefit value of a 'good' ALF)
(B)= 3 (half of the benefit value of a 'good' ALF because of lower quality care levels)
(HC)=6.5 (highest benefit value option, assuming seniors would rather stay at home at receive care)
(H)= goal to solve for (H); typical high priced ALFs in the US charge $3,000-$6,000/month)
(L)= 1.5 (represents the cost of typical low priced ALFs charging $1,500/ month)
(K)= 5.5 (represents typical monthly caregiver costs charged by HC companies of $5,500/month)
(p)= 0.5 (assume that 1/2 of ALF are 'good' regarding the health and happiness provided to the senior)
Consequently, a set of parameters and a graphical representation of this game can be created from these assumptions. Therefore, the adult daughter who wishes to maximize her aging parents utility (benefit minus cost) would order her preferences for care options (highest to lowest utility) as follows:
(G - L) > (G - H) > (B - L) > (HC - K) > (B - H)
Thus, the adult daughter would first like a 'good' ALF at a low cost, second, a 'good' ALF at a high cost, third, a 'bad' ALF at a low cost, fourth a HC agency at the typical cost (K), and lastly, a 'bad' facility at a high cost.
There exists a specific price at which the adult daughter may choose a high priced ALF in hopes that it is 'good' (G - H), in spite of the risk that the ALF is 'bad' (1 - p) and gets (B - H). Subsequently, the total utility is: (p)(G - H) + (1 - p)(B - H). The adult daughter will then only choose a high priced ALF if the utility is greater than the utility from a HC agency, illustrated by the equation:
(p)(G - H) + (1 - p)(B - H) > (HC - K)
For a numerical application, assume the values from above to solve for (H) (See Figure 2 for detailed calculations) and the resulting highest price an adult daughter is willing to pay for the potential benefit of a 'good' ALF at high price, with the risk of that the ALF could be 'bad' factored in, calculates to H
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