In this question we will solve a simplified version of the Grossman model.

Suppose utility is U(Ht,Zt) = Zt Ht. The production function for health is Ht = Mt + Ht1 and production for the home good is Zt = Jt (Hint: Notice these are not func- tions of TH or TZ). The price of Mt is 2$ per unit and the price of Jt is 1$ per unit. The individual works a fixed amount of time, T W = 5 at wage w = 2$ per unit of time (Hint: we are assuming TW is fixed and not a choice). Previous periods health, Ht1 = 1. There is no time constraint in this example.

For the remaining, we focus on a single period and assume the future does not impact decisions today.

What is the budget constraint in this period?

Now we will solve for the optimal allocation of health and the home good. Steps;

(a) Calculate the first order conditions of the utility function and set them equal to each other.

(b) Combine this with the budget constraint and solve for the optimal level of H and Z.