Download Intermediate Macroeconomics, Sciences Po, 2014 Zsófia Bárány

Intermediate Macroeconomics, Sciences Po, 2014
Zsófia Bárány
Answer Key to Problem Set 6
1. Static Macro Model: Consider the simple static macroeconomic general
equilibrium model introduced in the lecture, in which a representative
consumer decides on consumption and leisure, and thus labor supply,
a representative firm decides on labor demand and production, and the
government has to respect a budget constraint.
(a) Using a diagram, determine the effects of an increase in total factor productivity on output, consumption, employment, and the real
wage.
Answer:
Figure 1: Effect of higher total factor productivity
Improvements in total factor productivity affect the economy by
giving a access to a bigger set of consumption-leisure bundles. For
a given K and N , output Y is higher. The production possibility
frontier Y = zF (K, h − l), which is implied by the time constraint
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l + N = h and the production function Y = zF (K, N ), and the consumption possibility frontier (C = Y −T = zF (K, h−l)−G) both expand. Moreover, the marginal product of labor M PN = zFN (K, N )
increases at a given K and N . Therefore the consumption possibility frontier shifts out and becomes steeper (its slope is the negative
of the M PN ), i.e. it pivots around the point (l = h, C = −G).
Since there are no market failures, the competitive equilibrium is
Pareto efficient (first welfare theorem), and the allocation can be
found by maximizing the welfare of the representative consumer
subject to the consumption possibility frontier (as a social planner
would do). The optimal choice of leisure and consumption is such
that the consumer’s indifference curve is tangent to the consumption possibility frontier.
The outward shift of the consumption possibility frontier has two
effects:
i. substitution effect: as the M PN , which in equilibrium is equal
to the real wage (from firm’s optimality condition), increases,
the cost of leisure increases (or the return to work increases).
Therefore the households substitutes away from leisure to consumption.
ii. income effect: as z increases, income (Y ) increases1 Since both
leisure and consumption are normal goods, the household wants
to increase both.
Overall, C, w, Y increase, but the change in N = h − l is ambiguous.
Figure 1 shows a case in which the income and substitution effects
on leisure exactly cancel out, and the economy moves from point F
to point H.
(b) Now modify the model: Suppose that total factor productivity, z, affects the productivity of government production just as it affects private production. That is, suppose that the government collects taxes
which are then turned into government-produced goods according
to G = z · T , so that z units of government goods are produced for
each unit of taxes collected. With the government setting G, an increase in z implies that a smaller quantity of taxes are required to
finance the given quantity of government purchases G. Under these
circumstances, using a diagram determine the effects of an increase
in z on the equilibrium, treating G as given.
Answer:
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The decomposition of labor and capital income is irrelevant since the household owns the
firm.
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Figure 2: Effect of higher total factor productivity when T = G/z
When T = G/z, the consumption possibilities frontier is given by
C = Y − T = zF (K, h − l) − G/z. Note: for z = 1 the frontier
is the same as in the previous part of the problem set. When z increases, the private sector is more productive (as in part (a)), but
now there is also an effect on the government: the government is
able to produce the same about of goods G with lower taxes, which
generates an additional increase in income for consumers, and the
vertical intercept of the consumption possibilities frontier increases.
This implies an additional income effect, which tends to raise consumption and leisure demand. Therefore, relative to part (a), there
is a larger increase in consumption and an increase in leisure.
Assuming initially z = 1 and thus T = G, figure 1b shows the original frontier (labelled P P F1 , with equilibrium at point A), reproduces the result of part A (labelled P P F2 , with equilibrium at point
B), and the new outcome with frontier P P F3 and the competitive
equilibrium at point C. At point C the slope of P P F has increased,
which stimulates a further increase in the real wage.
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(c) Consider again the standard model (with T = G). Suppose there is
a natural disaster that destroys part of the economy’s capital stock.
Determine the effects on output, consumption, employment, and
the real wage, with reference to income and substitution effects.
Answer:
Figure 3: Effect of a natural disaster
The only impact effect of this disturbance is to lower the capital
stock. Therefore, the consumption possibility frontier shifts down
and the marginal product of labor falls (PPF is flatter). This is shown
in figure 3. The economy starts at point A on P P F1 . The reduction in
the capital stock shifts the production possibilities frontier to P P F2
and reduces its slope. Flatter P P F implies lower real wage at a
given level of employment (the marginal product of labor is lower)
which makes consumers substitute towards leisure. If we wanted
to isolate this effect, we would move to a hypothetical point B on
the figure below, where agents consume less of the consumption
good and enjoy more leisure. To see this, recall that in the equilibrium the slope (in absolute value) of an indifference curve (M RS)
must be equal to the wage rate. At point A, the slope of the indifference curve would now be higher than the equilibrium wage rate
(which is equal to M PN and the absolute value of the slope of the
dotted P P F ), which means that the representative consumer would
be willing to ‘pay’ more for an additional increase in leisure than
they have to ‘pay’ due to forgone income (M RS > w). This makes
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agents move to point B on the same indifference curve. Because the
production possibilities frontier shifts down, there is also an income
effect. The income effect implies less of both goods: less consumption and less leisure (more work). On net, consumption must fall,
but leisure could decrease, remain the same, or increase, depending
on the relative strengths of the income and substitution effect. The
situation depicted on the figure below represents the case when the
income effect dominates and the household attains less leisure in
the new equilibrium point C.
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