AP/ECON2400A (Fall 2012) W.Ho
Example 5: A competitive equilibrium with distortionary taxes
Consider a one-period, closed economy consisting of a large number of identical con-
sumers, a large number of identical ▯rms, and a government. Each economic agent acts as a
price taker in perfectly competitive markets. The economic behaviors of the consumers and
▯rms are as described in Example 3 with the following modi▯cations.
▯ The consumers faces a labor income tax rate, t, instead of the lump-sum tax, T.
▯ The production function is linear in labor and does not need capital input, Y = zN.
Questions:
1. Illustrate how the ▯rm determines the optimal labor demand and output supply deci-
sions. What is the condition that has to be satis▯ed?
2. Write down the budget constraint of the consumer.
3. Illustrate how the consumer determines the optimal consumption and leisure decisions.
What is the condition that has to be satis▯ed?
4. What are the four conditions that a competitive equilibrium must satisfy for this
economy?
5. Draw a diagram with the PPF to illustrate the Pareto optimal allocation.
6. Illustrate why the competitive equilibrium of the economy is not Pareto optimal. AP/ECON2400A (Fall 2012) W.Ho { Example 5 2
Answers
1. The ▯rm’s real pro▯t is ▯ = Y ▯ wN = zN ▯ wN. The ▯rm’s optimal labor demand
will be perfectly elastic at w = MN = z, and its optimal pro▯t is zero, ▯ = 0.
2. Given the labor tax rate t, the consumer’s budget constraint is C = (1▯t)w(h▯l)+▯.
From question 1, we have ▯ = 0, and the consumer’s budget constraint becomes
C = (1 ▯ t)w(h ▯ l):
3. The objective of the consumer is to maximize U subject to C = (1 ▯ t)w(h ▯ l): The
consumer tries to achieve the highest feasible level of utility. Given that the indi▯erence
curves are downward sloping and convex, the optimal consumption bundle (point H)
is given by the point where the budget constraint is tangent to an indi▯erence curve.
H
At this point, we have the condition, MRSl;c= w(1 ▯ t): (See Figure 1)
Figure 1: Optimal consumption-leisure choice
C
6
budget line
w(1 ▯ t)h rslope=▯w(1 ▯ t)
@
@
@
@
@
C▯ @r
@
@ indi▯erence curve
@ slope=▯MRS l;c
@
@ r - leisure, l
0 l▯ h
4. The four conditions that a competitive equilibrium must satisfy are
(a) The representative consumer chooses C and N to maximize utility, U, taking

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