Consider the market for stress tests as a diagnostic service for detecting heart problems. There are I consumers in this market. Let:

x_i = denote the consumption of stress tests for consumer i.

m_i = denote consumption of other goods for consumer I (the numeraire).

Each of the I consumers is endowed with 4 units of the numeraire good that they can choose to consume or invest in the production of stress tests. The value of the numeraire good is $1 (p_m = 1). The consumers have no endowments of stress tests. The price of a stress test is denoted p.

There is 1 producer of stress tests. The technology to transform the numeraire input into the leather wallet output is:

y = , where z is the quantity of the numeraire good used as an input to production and y is the output supplied of stress tests, (z,y) ³ 0.

The utility function of consumer I for all I = 1,…,I is:

U_i(x_i,m_i) = m_i + ln x_i

Each consumer receives 1/I of the profit of the single firm.

(a) Solve for the competitive equilibrium allocation. (p*, x*, y*).

(b) Give 3 reasons why this model might not represent the market for stress tests well. You can cite Arrow.

(c) How might government intervention bring us to a preferred allocation that represents a competitive equilibrium as well?

(d) How might you improve on this model to find a competitive allocation that makes more sense?