class: center, middle, inverse, title-slide # Module 2: Physician Agency and Treatment Decisions ## Part 1: Physician Agency ### Ian McCarthy | Emory University ### Econ 372 --- <!-- Adjust some CSS code for font size and maintain R code font size --> <style type="text/css"> .remark-slide-content { font-size: 30px; padding: 1em 2em 1em 2em; } .remark-code, .remark-inline-code { font-size: 20px; } </style> <!-- Set R options for how code chunks are displayed and load packages --> # Definition Physicians are better informed about treatment decisions than patients, and so there exists some <b>agency</b> relationship between the two. For many conditions, patients can't treat themselves even if they wanted to. --- # Setup - Denote quantity of physician services by `\(x\)` - Denote benefit of services to patient by `\(B(x)\)` - Patients pay (and physicians receive) a price of `\(p\)` for each unit of service `\(x\)` - Physicians incur cost `\(c\)` for each unit of care - Net benefit to patients is `\(NB(x)=B(x)-px\)` - Physicians must choose quantity of care at least better than the patient's outside option, `\(NB(x)= B(x)-px \geq NB^{0}\)`. --- # Solving the model With this framework, how much care will be provided? (i.e., what is the optimal value of `\(x\)`) <br> -- Solve the model in two steps:<br> 1. Physician will provide minimum surplus to keep the patient, `\(NB(x) = B(x)-px = NB^{0}\)` 2. Substitute into physician profit function, `\(\pi=(p-c)x = B(x)-NB^{0} - cx\)`, and solve for `\(x\)`<br> --- # Solving the model This two step approach applies when prices and quantity of care are variable. If the physician cannot set price, then we just work off of the constraint, `\(B(x)-\bar{p}x=NB^{0}\)`. -- Why? This is a corner solution...can't just take a derivative. --- # In-class Problem: Physician agency Denote the quantity of care consumed by `\(x\)`, and denote by `\(B(x)\)` the function that determines the benefit of care to the patient. Assume that the patient must pay the full price of care, `\(px\)`, so that their net benefit is `\(NB=B(x)−px\)`. Further assume that the physician can choose both `\(x\)` and `\(p\)`. 1. Find the patient’s optimal `\(x\)`. 2. Draw the marginal benefit function on a graph and note the price and patient's optimal quantity. Assume that `\(B'(x)>0\)` and `\(B''(x)<0\)` (i.e., the marginal benefit function is positive and downward sloping). 3. Find the physician's optimal `\(x\)` assuming `\(NB^{0}=0\)`. 4. Add the physician's optimal `\(x\)` to your graph and interpret the difference. --- # Physician agency in a graph <img src="02-agency1_files/figure-html/tikz-agency-1.png" style="display: block; margin: auto;" /> --- # Example Assume `\(B(x)=8x^{1/2}\)`, `\(NB^{0}=2\)`, and `\(c=1\)`:<br> 1. Find the physician's optimal level of `\(x\)` and `\(p\)`. 2. Find the patient's optimal level of `\(x\)`. 3. Draw this graphically. --- # Answer First let's re-write the constraint such that `\(px = 8x^{1/2}\)` and `\(\pi = 8x^{1/2} - 2 - x\)`. The first order condition for `\(x\)` is then `\(4x^{-1/2} -1 =0\)`, which is satisfied at `\(x^{*}=16\)`. Substituting this into the constraint, `\(8x^{1/2} - px=2\)` yields `\(p=\frac{15}{8}\)`. But if they could choose on their own, the patient would prefer to maximize their net benefit. This would occur at `\(4x^{-1/2}=p\)`, which yields `\(x=(32/15)^{2} \approx 4.5\)` at `\(p=15/8\)`.