Bogazici University Department of Economics EC 205 Summer 2015 SOLUTIONS TO PROBLEM SET 3 1) When the government imposes a proportional tax on wage income, the consumer’s budget
constraint is now given by:
C = w(1 − t )(h − l ) + π − T ,
where t is the tax rate on wage income. In the figure below, the budget constraint for t = 0, is
FGH. When t > 0, the budget constraint is EGH. The slope of the original budget line is –w,
while the slope of the new budget line is -(1 - t)w. Initially the consumer picks the point A on
the original budget line. After the tax has been imposed, the consumer picks point B. The
substitution effect of the imposition of the tax is to move the consumer from point A to point
D on the original indifference curve. The point D is at the tangent point of indifference curve,
I1, with a line segment that is parallel to EG. The pure substitution effect induces the
consumer to reduce consumption and increase leisure (work less).
The tax also makes the consumer worse off, in that he or she can no longer be on indifference
curve, I1, but must move to the less preferred indifference curve, I2. This pure income effect
moves the consumer to point B, which has less consumption and less leisure than point D,
because both consumption and leisure are normal goods. The net effect of the tax is to reduce
consumption, but the direction of the net effect on leisure is ambiguous. The figure shows the
case in which the substitution effect on leisure dominates the income effect. In this case,
leisure increases and hours worked fall. Although consumption must fall, hours worked may
rise, fall, or remain the same.
2) The firm chooses its labor input, Nd, so as to maximize profits. When there is no tax,
profits for the firm are given by
π = zF ( K , N d ) − wN d .
That is, profits are the difference between revenue and costs. In the top figure below, the
d
revenue function is zF ( K , N ) and the cost function is the straight line, wNd. The firm
1 maximizes profits by choosing the quantity of labor where the slope of the revenue function
equals the slope of the cost function:
MPN = w.
The firm’s demand for labor curve is the marginal product of labor schedule in the bottom
figure on the following page.
With a tax that is proportional to the firm’s output, the firm’s profits are given by:
π = zF ( K , N d ) − wN d − tzF (K , N d )
= (1 − t )zF (K , N d ),
d
where the term (1 − t )zF ( K , N ) is the after-tax revenue function, and as before, wNd is the
cost function. In the top figure below, the tax acts to shift down the revenue function for the
firm and reduces the slope of the revenue function. As before, the firm will maximize profits
by choosing the quantity of labor input where the slope of the revenue function is equal to the
slope of the cost function, but the slope of the revenue function is (1 − t )MPN , so the firm
chooses the quantity of labor where
(1 − t )MPN = w.
In the bottom figure below, the labor demand curve is now (1 − t )MPN , and the labor demand
curve has shifted down. The tax acts to reduce the after-tax marginal product of labor, and the
firm will hire less labor at any given real wage.
2 3) As the firm has to internalize the pollution, it realizes that labor is less effective than it
previously thought. It now needs to hire N(1 + x) workers where N were previously sufficient.
This is qualitatively equivalent to a reduction of z, total factor productivity. The figure below
highlights the resulting outcome: the firm now hires fewer people for a given wage and thus
its labor demand is reduced.
4)
a) Set up the consumer’s problem as usual and plug in the values of exogenous variables:
max 𝑢(𝑐, 𝑙) 𝑠. 𝑡. 𝑐 + 0.75𝑙 = 6.8
!,!
3 For (i), do the problem as usual by setting up the Lagrangian and taking FOCs. Optimal
consumption is 𝑐 ∗ = 4.08 and optimal leisure is 𝑙 ∗ = 3.63.
For (ii), Lagrangian approach will not work since consumption and leisure are perfect
substitutes (indifference curves are straight lines). Since second order conditions are not
satisfied for the perfect substitute case, FOCs of the Lagrangian will not necessarily yield
optimal consumption bundle. In this case, we have a corner solution where consumer either
spends all available time working or does not work at all. For any level of consumption and
leisure, by working an additional unit of time, the consumer loses 𝑎 units of utility and gains
𝑏𝑤 units of utility since one unit of labor is paid 𝑤 units of consumption good and a unit of
consumption good yields 𝑏 units of utility. Thus, consumer spends all his time working if
𝑏𝑤 > 𝑎 and consumer does not work at all if 𝑏𝑤 < 𝑎. Consumer is indifferent between any
consumption-leisure bundle at the budget line if 𝑏𝑤 = 𝑎. In this case, since 𝑏𝑤 = 0.45 >
𝑎 = 0.4, consumer spends all his time working. Thus, optimal leisure is given by 𝑙 ∗ = 0 and
optimal consumption (pinned down by budget constraint) is given by 𝑐 ∗ = 6.8.
For (iii), Lagrangian approach will not work since min function is not differentiable. When
we have the min utility function, optimality requires 𝑐 ∗ = 𝑙 ∗ since if consumption is not equal
to leisure, consumer is wasting the excess consumption or excess leisure. Combining the
optimality condition with the budget constraint, we find optimal consumption and leisure to
be 𝑐 ∗ = 𝑙 ∗ = 3.89.
b) Using the same procedures in part (a) and increasing the wage rate to 1.5,
For (i), 𝑐 ∗ = 11.28 and 𝑙 ∗ = 5.01.
For (ii), 𝑐 ∗ = 18.8 and 𝑙 ∗ = 0.
For (iii), 𝑐 ∗ = 𝑙 ∗ = 7.52.
After a change in wage rate, substitution effect causes consumption to rise and leisure to fall.
Income effect causes both consumption and leisure to rise since consumer is richer. For (i),
income effect dominates substitution effect so that optimal leisure rises in response to wage
increase. For (ii), substitution effect is very strong since goods are perfect substitutes.
However, we see no change in leisure since leisure was already zero even before the wage
increase. For (iii), there is no substitution effect since consumption and leisure are perfect
complements. Thus, consumption and leisure both rise as a result of the income effect.
c) Using the same procedures in part (a) and increasing taxes to 8,
For (i), 𝑐 ∗ = 2.88 and 𝑙 ∗ = 2.56.
For (ii), 𝑐 ∗ = 4.8 and 𝑙 ∗ = 0.
For (iii), 𝑐 ∗ = 𝑙 ∗ = 2.74.
An increase in taxes create a pure income effect, which makes the consumer poorer and
causes optimal consumption and optimal leisure to fall in all cases, since consumption and
leisure are normal goods.
4 5)
i) The firm maximizes profits choosing labor:
max 𝐾 ! 𝑁 !!! − 𝑤𝑁
!
First order condition yields 1 − 𝑎 𝐾 ! 𝑁 !! = 𝑤. Rearranging this equation gives the optimal
!
!!! !
.
!
∗
labor demand 𝑁 = 𝐾
ii) Profit in this case is given by 𝐾 + 𝑁 − 𝑤𝑁. This is the case where capital and labor are
perfect substitutes in production. Since profit function is linear in 𝑁, first order condition does
not necessarily give the optimal labor demand in this case. Realize that for any level of labor,
employing one more unit of labor gives the firm 1 − 𝑤 units of profits. Thus, optimal labor
demand is infinity if 1 − 𝑤 > 0 (or 𝑤 < 1), is zero if 1 − 𝑤 < 0 (or 𝑤 > 1) and can be any
nonnegative number if 1 − 𝑤 = 0 (or 𝑤 = 1).
iii) This is the case where capital and labor are perfect complements so that optimality
requires 𝑁 ∗ = 𝐾 if labor demand is positive. However, firm will choose to employ no labor if
𝑤 > 1 by the same logic in part (ii). Optimal labor demand is 𝑁 ∗ = 𝐾 if 𝑤 < 1 and optimal
labor demand can be any number between 0 and 𝐾 if 𝑤 = 1.
6) a) As in class, use the three-step analysis to find the effects of a decline in 𝐾 on endogenous
variables:
Step 1: Direct Effect (what happens if nobody changes behavior)
𝐾! < 𝐾!
𝐾! < 𝐾!
𝑦! < 𝑦! 𝜋! < 𝜋! Step 2: Indirect Effect (who is affected and changes behavior) Firms: 𝐾! < 𝐾!
𝐹! 𝑧, 𝐾! , 𝑁 < 𝐹! 𝑧, 𝐾! , 𝑁
Consumers: 𝜋! < 𝜋!
𝑁!! < 𝑁!! Negative income effect: Consumption falls, leisure falls. Step 3: Equilibrium (what happens to equilibrium prices and quantities) Labor market equilibrium: Since labor demand curve shifts inwards and labor supply curve shifts outwards, real wage falls and the effect on employment is ambiguous. Goods market equilibrium: Since output falls (due to a fall in K, assuming N unchanged) and government spending remains unchanged, consumption falls since 𝑌 = 𝐶 + 𝐺 by income-­‐expenditure identity. Summarizing: After a decline in capital stock: Output falls, consumption falls, real wage falls, effect on employment is ambiguous. 5 b) Changes in the capital stock are not likely candidates for the source of the typical business cycle. While it is easy to construct examples of precipitous declines in capital, it is more difficult to imagine sudden increases in the capital stock. The capital stock usually trends upward, and this upward trend is important for economic growth. However, the amount of new capital generated by a higher level of investment over the course of a few quarters, of a few years, is very small in comparison to the existing stock of capital. On the other hand, a natural disaster that decreases the stock of capital implies lower output and consumption, and also implies lower real wages, which are all features of the typical business cycle contraction. 7) a) In the context of our model, representative consumer’s welfare is measured by utility, which is increasing in consumption and leisure. We covered the effects of an increase in government spending and an increase in TFP (which yields qualitatively the same effects with an increase in capital stock -­‐ this also answers the small homework question on the effects of an increase in capital stock -­‐ since capital and TFP enters into the model in the same way through the production function, qualitative effects of an increase in TFP and an increase in capital stock is the same) in class. From that analysis, we know that consumption and leisure falls after an increase in government spending. We also know that consumption rises and the effect on leisure may rise or fall due to an increase in capital stock. In this question, an increase in government spending is also associated with an increase in capital stock. Thus, combining the effects from our analysis in class, consumption may rise or fall and leisure also may rise or fall after an increase in government spending in the context of this question. Thus, it very well could be the case that consumption and leisure could increase after an increase in government spending, thus it could be the case that an increase in government spending increases consumer welfare. b) We already discussed in part (a) that the effect on consumption and hours worked are ambiguous. From our analysis in class, we know that both an increase in government spending and an increase in capital stock cause output to increase. Combining the effects, we thus have that output increases unambiguously after an increase in government spending when government spending has positive productivity effects. 6