3. A commodity $q$ is traded at price $p$ in a competitive market with price-taking consumers and
firms.
There are $N$ identical consumers each with income $Y = 50$. Each consumer has a utility
function over numeraire consumption $c$ and commodity $q$ given by:
$u(c, q) = c + 12q - \frac{1}{2}q^2$
There are $M$ identical firms each with cost function given by:
$c(q) = 2 + 8q + \frac{1}{2}q^2$
The number of firms is fixed in the short run, but in the long run firms can freely enter or
exit the market. Thus, the number of firms is flexible in the long run.
a. Prove that the long run equilibrium price is equal to:
$p = \delta + 2$
b. Prove that in the long run equilibrium, the total output of the commodity is
$Q = N(10 - \delta)$
and that the output per consumer is:
$q_d = 10 - \delta$
c. Prove that in the long run equilibrium, the utility of each consumer is:
$u = 50 + \frac{1}{2}(10 - \delta)^2$
d. Real GDP is a measure of real output per person. Using this model as an example,
explain how:
i. Population growth (an increase in $N$)
ii. Productivity growth (a decrease in $\delta$)
affect real GDP, real GDP per capita, and consumer well-being.
