It is convenient to derive the PPF equation for all \alpha at once. First, equate aggregate demand and aggregate supply:

10 - 2d - P = 2 + P - d - \alpha v,

so equilibrium price level is P = \frac{8 - d + \alpha v}{2} = 4 - \frac{d}{2} + \frac{\alpha v}{2}, and thus, equilibrium GDP is Y = 2 + P - d - \alpha v = 6 - \frac{3d}{2} - \alpha v/2. Plugging v = 2(1 - d) in the last equation, we get

Y = 6 - \frac{3d}{2} - \alpha (1 - d) = 6 - \alpha - \left( \frac{3}{2} - \alpha \right) d.

Health, in its turn, depends on the level lockdown as follows:

H = 3 - \sqrt{v}/2 = 3 - (1 - d) = 2 + d.

Note that since d \in [0,1] H ranges from 2 to 3.

To derive the PPF, we eliminate d from the last two equations by expressing it, say, in terms of H and plugging the result into the equation for Y. Namely, d = H − 2, so finally

Y = 6 - \alpha - \left( \frac{3}{2} - \alpha \right)d = 6 - \alpha - \left( \frac{3}{2} - \alpha \right)(H - 2) = 9 - 3\alpha - \left( \frac{3}{2} - \alpha \right)H.

Now it is easy to answer parts (b) and (c).

For \alpha=1 the PPF equation is Y = 6 − H /2. It is a downward-sloping line with extreme points (2, 5) and (3, 4.5) (recall that H takes values from 2 to 3). The first point corresponds to d=0 and the second to d=1.

For \alpha=2 the PPF equation is Y = 3 +H /2. It is a upward-sloping line with extreme points (2, 4) and (3, 4.5). The first point corresponds to d=0 and the second to d=1.