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ECON 2400
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Wai Ming Ho
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AP/ECON2400A (Fall 2012) W.Ho
Example 3: The One-Period, Closed-Economy Macroeconomic Model
1. Consider a one-period-lived consumer having preferences given by the utility function,
U(C; l ), where C is the quantity of consumption, and l is the quantity of leisure. As-
sume that the consumer’s preferences can be characterized by a map of indi▯erence curves
which are downward sloping and convex. The consumer is endowed with h units of time,
works at the market real wage, w, receives real dividend from ▯rms, ▯, and pays a lump-sum
tax, T, to the government. Assume that ▯ > T.
(a) Write down the budget constraint of the consumer.
(b) Illustrate how the consumer determines the optimal consumption and leisure decisions.
What is the condition that has to be satis▯ed?
(c) Holding ▯ and T constant, illustrate how a decrease in the lump-sum tax, T, a▯ects
the optimal consumption and leisure decisions of the consumer. Explain your answer.
(d) Suppose now that the consumption good and leisure are perfect substitutes in the
consumer’s preferences, with U(C; l ) = al+bC, where a and b are positive constants.
Illustrate how the consumer determines the optimal consumption and leisure decisions.
2. Consider a ▯rm which is endowed with K units of capital, and faces the production function
Y = zF(K;N). Y is the output of consumption goods, K is the quantity of capital input, N
is the quantity of labor input, and z is a parameter measuring the total factor productivity.
The marginal product of each input is always positive but diminishing as the input increases,
holding other things constant. The market real wage is denoted by w. The objective of the
▯rm is to maximize its real pro▯t, taking as given the values of w, K, and z.
(a) Illustrate how the ▯rm determines the optimal labor demand and output supply deci-
sions. What is the condition that has to be satis▯ed?
(b) Suppose now that there is a decrease in the capital input, K. Holding w and z constant,
illustrate how the decrease in K a▯ects the ▯rm’s labor demand, output supply, and
real pro▯t. Explain your answer.
3. Consider a one-period, closed economy consisting of a large number of identical consumers,
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 Questions 1 and 2. (The consumers’ indi▯erence curves are downward
sloping and convex.) The government’s purchase of the consumption goods is ▯nanced by
lump-sum taxes imposed on the consumers, G = T.
(a) Write down the equation for the production possibilities frontier (PPF) of the economy.
(b) What are the four conditions that a competitive equilibrium must satisfy for this model?
(c) Draw a diagram with the PPF to illustrate the competitive equilibrium of the economy.
Label and describe your diagram. AP/ECON2400A (Fall 2012) W.Ho { Example 3 2
Answers
1. (a) The consumer’s budget constraint: C = w(h ▯ l) + (▯ ▯ T).
(b) The objective of the consumer is to maximize U subject to C = w (h ▯ l) + ▯ ▯ T:
That is, 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 E) is given by the point where the budget constraint is tangent to an
indi▯erence curve. At this point, we have the conditi= w: (See Figure 1a)
l;c
(c) The original optimal consumption bundle is point E. Now, a decrease in the lump-sum
tax, T, raises the non-labor income of the consumer, ▯ ▯ T. Given that there is no
change in w, a decrease in T shifts the budget constraint upward parallelly, and it
has a positive pure income e▯ect on the consumer. As both consumption good and
leisure are normal goods, the optimal demands for C and l increase. The new optimal
consumption bundle is given by point H where the new budget line is tangent to an
indi▯erence curve, Ml;c= w: The consumer is now at a higher utility level. (See
Figure 1b)
Figure 1a: Optimal consumption-leisure choiceFigure 1b: A decrease in T (winewT > T
C
C 6
6
budget line wh + ▯ ▯new slope=▯w
wh + ▯ ▯ T wh + ▯ ▯ [email protected]=
@ slope=▯w @==
@ @ @
@ @ @
@ @ @
C▯ @
▯ @ new @ @rH
C @ rE C▯ @r
@ [email protected] @
@ indi▯erence curve @@
@ ▯ ▯ new @ @q
@q slope=▯MRSl;c
▯ ▯ T ▯ ▯ T @q
- leisure, l -
0 l▯ h 0 l new h leisure, l
(d) Given that U(C;l) = al+bC; C and l are perfect substitutes in consumer’s preferences.
The marginal utility of each good is constlnt, MU = a cnd MU = b, al;cMRS
a=b. The indi▯erence curves are linear with slope equal to ▯a=b. The consumer will
choose the consumption bundle that yields the highest attainable utility level. If a=b >
w, then MUl> w ▯ MU c and the optimal consumption point will be point A with
C = ▯ ▯ T and l = h. If a=b < w, thel MU < w c MU , and the optimal consumption
point will be point B with C = wh + ▯ ▯ T and l = 0. (See Figures 2a and 2b.) AP/ECON2400A (Fall 2012) W.Ho { Example 3 3
.
Figure 2a: The cab> with Figure 2b: The case< wth
b
C C
6 r
wHh + ▯ ▯ @
@ ee @ indi▯e

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