Suppose there is a perfectly competitive market where

4. Suppose there is a perfectly competitive market, where consumers maximize utility v ( q, α )+ m
subject to budget constraints w = ( p + t ) q + m , where t is a tax on consumption of q , and firms have
costs C ( q ) = cq . Assume that v
αq
( q, α ) ≥ 0. (i) Characterize a perfectly competitive equilibrium in
the market, and show how tax revenue — tq
∗
( t, α ) — varies with t and α . (ii) Suppose α changes
and the government adjusts t to hold tax revenue tq
∗
( t, α ) constant. Use the implicit function
theorem to show how t ( α ) varies with α . (It might help to think about the elasticity of q with
respect to t , ε =
t
q ∗
∂q
∗
∂t
)
We did this in class.
5. When an objective function f ( x, y ) depends on two controls, x and y , and is subject to
a linear constraint c = ax + by , the problem can be simplified to a one-dimensional problem by
2

solving the constraint in terms of one of the controls,
y =
c − ax
b
and substituting it into the objective to get
max
x
f
parenleftbigg
x,
c − ax
b
parenrightbigg
Derive FONCs and SOSCs for this problem. What assumptions guarantee that the SOSC’s hold?