# W5: Agency and Fixed Prices

Assume \(B(x)=4x^{1/2}\), \(NB^{0}=0\), anc \(c=1\). Further assume that prices are fixed administratively at, \(\bar{p}=2\). Note that, in this case, we work only off of the patient’s net benefit constraint.

## Question 1:

**What is the physician’s and patient’s optimal amount of care provided?**

The physician maximizes profit, \(\pi = px - cx\), subject to the constraint, \(4x^{1/2} = px\). Since \(p\) is fixed, the physician simply sets \(x\) as high as possible, which means they set \(x\) to the point where the constraint binds. With \(p=2\), this means \(4x^{1/2} = 2x\), and solving for \(x\) means \(16x = 4x^{2} \longrightarrow 16=4x \longrightarrow x=4\).

The patient cares about maximizing net benefit, \(4x^{1/2} - 2x\). This is maximized at \(x\) such that \(2x^{-1/2}-2=0 \longrightarrow x=1\).

## Question 2:

**The government is considering increasing the price to \(\bar{p}=3\). What are the new optimal levels of care for physicians and patients at this new price?**

At this new price, the physician would choose \(x\) such that \(4x^{1/2} = 3x \longrightarrow 16x = 9x^{2} \longrightarrow 16=9x \longrightarrow x=\frac{16}{9}\). The patient would choose \(x\) such that \(2x^{-1/2}-3=0 \longrightarrow x=4/9\).

## Question 3:

**How would the price change affect the difference between the patient and physician’s optimal amounts?**

First, note that the amount of care decreases as the administrative price increases (opposite of the standard supply curve). Similarly, the deviation between the patient and physician optimal amounts decreases.