A little development economics
Halvor Mehlum, January 29, 2015
1
Background
Many results in economics builds on the premise of diminishing returns in production. In
terms of employment diminishing returns implies that if hours worked is doubled the production is less than doubled. If hours worked is very high at the outset, an additional hour
may even reduce production.
One illustration is given in Figure 3, where L is hours worked and Y is the value of
production. To the left of L∗ Y increases as hours worked increases, but at a declining rate.
To the right of L∗ further increases in L actually lowers production Y . In some contexts
diminishing returns may be an accurate description in other cases not. Diminishing returns,
or diminishing marginal product, may for example be a reasonable assumption on a farm
with a limited stock of capital and land
In the following I will show some important results based on the premise that the marginal
product of labour is indeed decreasing. I will use an example where the return to employment
declines following a linear formula. I could have illustrated all the following qualitative
results using any formula for diminishing return. I choose to use the linear example because
it is simple.
2
Marginal product
The value of the marginal product x captures the contribution to the value of production by
the last of the L hours worked. Total production Y is then the sum of x for the first, second,
third . . . , L’th hour. In Figure 1 the marginal product x is given by a linear relationship.
x is the marginal product, L is total hours worked by all workers, a and b are constants
[Example: a = 10, b = 1]
x=a−b·L
1
(1)
Figure 1: Value of marginal (and average) productivity
value
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∗.
x̄
x
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L
x starts out in a and has a negative slope of b. All employees are assumed equal but the first
hour worked is quite productive while the next is less and so on until L∗ where further employment is unproductive.1 Given the formula above the maximum productive employment
L∗ is found by setting x equal to zero and solving for L∗ as follows
x = 0 = a − b · L∗ ⇐⇒ L∗ = a/b
[Example: L∗ = 10/1 = 10].
Before deriving the value of production Y it is convenient to find the value of the average
product x̄. It tells us what is produced during each hour on average. The average product,
x̄, is the average of each of the marginal products. In the linear case the average product
is the average of the x for the first hour worked and the last hour worked. Think of L as a
large number, say 2000, then the marginal product for the first hour worked is approximately
equal to a while the marginal product for the last hour worked is equal to (a − b · L) . The
average of these is
x̄ =
a + (a − b · L)
b
=a− ·L
2
2
x̄ starts out in the same point as x but is less steep and crosses the horizontal axis at 2 · L∗ .
Now it is easy to find the value of total production. The value of total production Y is
the sum of all the marginal products for all hours worked. Therefore Y is equal to the area
between the x curve and the horizontal axis. An expression for Y is found from the fact
that total production is also equal to the average product times the number of hours worked
b
b
Y = L · x̄ = L · a − L = aL − · L2
(2)
2
2
In Figure 2 these relationships between Y and x and x̄ are illustrated. In the left panel
Y is found as the area under the x curve (a sum of narrow columns one column for each
hour worked.) In the right panel Y is derived as L · x̄. The two diagrams illustrates total
1 For
example as workers are obstructing each other as in a soft ice kiosk with 30 workers.
2
Figure 2: Total production Y as two areas
average product times hours worked L̂
area under the marginal product
value
value
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L
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Figure 3: Total production
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production Y when L = L̂. When L = L∗ , Y reaches it’s maximum. If L is increased further
the x goes negative and there is negative contribution to production. In the right panel, if
L is increased beyond L∗ the rectangle gets wider but less tall. In fact any L to the right
of L∗ gives a smaller rectangle. L = L∗ is the amount of employment that gives the highest
production Y. The value of production Y as a function of employment (see (2)) is given in
Figure 3. Knowing the relationship between the value of production Y, the average product
x̄ and the marginal product x we can investigate a number of important topics.
3
The tragedy of the commons.
Consider a completely isolated village with lots of people/labour L̄ and where the only
production process available satisfies the function given by (1) . For concreteness assume
that the production in question is fisheries in a closed pond with some reproduction. One
fisherman will get a lot of fish, two will get more (but not double), and so on. Maximum catch
is reached with L∗ fishermen. If L exceeds L∗ there will be overfishing and the total catch
will start to decline. Assume first that all decisions are taken by a village council. Their aim
is to get the maximum production in the village and to distribute it evenly between all the
3
villagers. How many workers should be employed in order to get the maximum production?
The answer is simple. The council should employ people until L∗ and share the production
between all the people. (if there are too many workers the solution may be that each worker
works half time).
Assume now that the council is closed down. Let each person in the village cater for
themselves and assume that no one owns the pond. It is a common property i.e. commons.
In that case each and every fisherman will, depending on the total number of fishermen,
get the income x̄. The result will be that every of the L̄ persons start as a fisherman. As
L̄ > L∗ , there will be too many fishermen. Note that if the number of people in village
is very high L̄ > 2 · L∗ the outcome will be total extinction of the fish. The number of
fishermen L will grow until 2 · L∗ and at that point there is no more fish.
The tragedy of commons problem is that when production involves a common property
with free access, there will generally be over utilization of the common property. The reason
is that each individual entering gets the average product x̄ which by definition is larger
than the marginal product x. Another way to put it is that each individual do not take
into account that him entering lowers the return to all the others who has already entered.
There is an externality, there are wrong incentives. Note that the tragedy is that there is no
regulation of the utilization of the pond. It is perfectly possible to get optimal harvesting of
a common property if the use is regulated by a village council.
4
Alternative employment
The discussion above was based on the assumption that there was no alternative employment. Assume now that the villagers can travel to the neighboring village and work in a
factory for a fixed wage w. In the case of a council, with the interest of all the villagers
as their priority, the council will decide that a number of workers LA stay in the village
and require that the remaining fraction L̄ − LA of the workers work in the factory for the
wage w. Again, the council wants to maximize total income: the sum of what is earned in
the neighbouring village (w L̄ − LA ) and what is produced Y . As long as w > x income
increases as workers are moved from fishng in the pond to the factory. As long as w < x income increases if workers are moved out of the factory and to fishing in the pond. Maximum
income is reached when
w=x
that is , when the wage in the factory is equal to marginal product in the village. The
optimal allocation of labour is illustrated in Figure 4, where the shaded area captures total
village income. If LA was moved to left or to the right either of the two dark triangles would
be lost and income wold drop, hence LA is the optimum. A village council would realize
4
Figure 4: Optimal allocation of labour
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x
w
LA
L
L̄
L
this and distribute the labour such that LA worked in the village and L̄ − LA worked in the
factory. Eventually all income could be shared equally.
5
Private ownership
The description above could also capture the production in a private firm selling ice cream
in a park. The first worker will have a lot of sales, two will sell more but will not be able
to double the sales and so on - decreasing returns. We can further assume that the owner
of the firm hires labour for a fixed wage w. Now, the owner wants to hire workers in order
to maximize his profits π . Let us call the choice of workers LB . His profits is given by the
value of production Y minus the wages paid w · LB
π = Y − w · LB
As long as w < x profit increases when the owner hires more workers As long as w > x
profit increases if workers are sacked . Maximum profit is reached when
w=x
The optimal hiring of labour is illustrated in Figure 5. In the upper panel the profits π are
drawn. In the lower the profits are equal to the area between the x line and the w-line. The
optimum is LB . If L is moved to left or to the right of LB the net loss would be either of
the two dark triangles and profits would drop, hence LB is the optimum from the owners
point of view.
5
Figure 5: Profit maximizing owner
profits
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This diagram also shows that under these assumptions profits and employment are negatively affected by an increase in the wage. Hence, a wage increase will imply that some
workers are fired. Whether the workers on average are better or worse of is an open question. In classical growth theory (Ricardo, Marx and others) the size of profits determines
investments and hence how the x moves to the right over time. If profits are high x will move
a lot to the right. It follows that low wages in one period generates high profits that in turn
generates high demand for labor (x far to the right) in the next period. Wage moderation
may therefore benefit the workers in the long run as the firms expand. In a globalized world
high profits may attract foreign investments leading to a similar result. However, if in a
globalized world investments go to the country with the best profits opportunities, there
will be a competition for having the highest profits. This may lead to a competition for
having the lowest wages. This mechanism is a variety of social dumping.
6
Sharecropping
Let us now move to agricultural production. Assume that it is not possible for a land owner
to monitor how much the farmers/workers actually work. Without such monitoring the
farmers/workers may go to sleep whenever the land owner looks in another direction. If this
is the case the land owner may let each farmer have a peace of land that the farmer has total
6
Figure 6: Sharecropping
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x
w
γx
LC
LB
L∗
L̄
L
responsibility over. The deal is that the farmer keeps a fraction γ of production and gives
a fraction (1 − γ) of the production to the land owner(for example γ = 50%). Supose that
here also in this case factory employment is an option and consider the case where there is
only one family of farmers so that L̄ is the working hours available from the family members,
LC is the hours worked on the land, while L̄ − LC is the hours worked in the factory. Since
the farmers only keep γ of what they produce the incentive to work on the land is limited.
The family now wants to maximize total income: the sum of what is earned (w · L̄ − LC )
and their share of what is produced γ · Y . From above you know that the increase in Y by
an increase in L is given by x. It follows that the increase in γY by an increase in L is given
by γ · x. As long as w > γ · x income increases as workers are moved from the land to the
factory. Maximum income for the farmer is reached when
w =γ·x
The equilibrium is illustrated in Figure 6. Since the farmers only keep a fraction of what
they produce they are less willing to work on the land than in the case where they keep
everything. LC is too low. The return to the farmers is the gray area while the return to
the land owner (1 − γ) x is the dark area. the triangle to the right of the dark area is lost.
6.1
Land reform
One way to have the farmer and the landlords interest to go in the same direction would
be to confiscate the landlords land and give it to the farmer. After the reform the landlord
and farmer is now the same person. The land reform has made the farmer into a landlord
7
Figure 7: Labour market with market clearing wage
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that is working on his own land. When this is the case the farmer get everything that is
produced. As a result he will work optimally at a level corresponding to LB .
7
Labour market
Consider a town where there are a number of privately owned firms that each have a technology with diminishing returns. Monitoring of workers is not considered a problem. The
marginal product of all firms taken together is now x. Firms will hire workers until the
marginal product is equal to the wage. Let the number of workers in the town be fixed and
equal to L̄. If there is free competition the competition for workers will ensure that the wage
is equal for all and that supply of labour, L̄ is is equal to demand L. This is illustrated in
Figure 7. In this case the wage level w∗ is the outcome of the interaction between supply
and demand.
Unemployment
If for some reason the wage is set at a level ŵ above the market clearing wage w∗ , there will
be unemployment. The level of ŵ may be the outcome of pressure from the labour unions,
it may be given by law, or it may be the minimum level required for a reasonably healthy
workforce. Unemployment is illustrated in Figure 8. The unemployment U is equal to the
difference between labour supply L̄ and the labour demand (at the fixed wage ŵ).
U = L̄ − L̂
8
Figure 8: Labour market with unemployment
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ŵ
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←
−U −
→
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L̄
L
Note that the expected wage in the town is the probability of getting a job, L̂/L̄, times the
wage.
expected wage=
L̂
· ŵ
L̄
Harris-Todaro(see more below): If the expected wage is high more people may want to move
into town. The result is that L̄ increases and so does U and the expected wage declines.
This is the mechanism that drives many of the results in Harris-Todaros migration model.
Malthus: In a stylized version of Malthus’ view of the world ŵ is determined by reproduction.
If not all workers earn at least ŵ the population will decline. If workers earn more, population
will grow. Hence the labour supply will adjust over time so that supply of labour L̄ always
is equal to the demand L̂ at the wage ŵ.
Investments: As in the analysis of sharecropping, investments in new firms will increase the
marginal productivity of labour, x. When the wage is fixed at ŵ profits will increase. If
there is full employment and the wage is flexible, investments wil raise the wage and profits
may not go up at all. In the first case investments is a good idea. In the latter it may not
be such a great idea for a capitalist even tough the workers gain a lot.
8
Migration
If we consider an entire economy the assumption about employment opportunity in a factory
at a fixed wage becomes unrealistic. If firms are privately owned, the demand for workers
from modern firms should behave according to the profit maximizing principles discussed
above and the wage will decline with increasing employment.
9
Figure 9: Allocation of labour between Agriculture and Manufacturing
value in $
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.A
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x
w
x
w
L
←−−−−−−−−−−−−−−− L̄ −−−−−−−−−−−−−−−→
←−−−−−−− LM −−−−−−−→
Consider an economy where there are two sector and where the demand for labor is
declining as the wage increases. The manufacturing sector (M ) employs LM workers and
has marginal product xM . The agricultural sector (A) employs LA workers and has marginal
product xA . Then if there is full employment in the economy (all L̄ workers have a job) and
if workers move to the sector with the highest wage we have the two conditions
xA = xB = w
and LA + LM = L̄
The equilibrium in this economy is illustrated in Figure 9. The width of the diagram is
determined by L̄, LA is measured from left to right while LM is measured from right to
left. The marginal product in $ in agriculture is drawn from left to right while the marginal
product in $ in manufacturing is drawn from right to left. Note that the wage w is determined
by the interaction between a fixed supply L̄ and demand from two sectors LA and LM . Note
also that this is in effect the Harris-Todaro model of migration in the case where the wage
is completely flexible in both sectors.
If, due to investments, the demand for labor in manufacturing went up, the xM line
would shift up and in the new equilibrium workers would have moved from the rural villages
and to the town where manufacturing takes place (LM up and LA down).
Unemployment (Harris-Todaro model) If for some reason the wage in town is set at a
level ŵ above the market clearing wage w, there will be unemployment. The level of ŵ may
10
Figure 10: Harris-Todaro: allocation of labour between Agriculture , Unemployment, and
Manufacturing
value in $
value in $
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ŵ
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wA
LA
LM
←−− U −−→
be the outcome of pressure from the labour unions. It may be given by law. Or it may be
the minimum level required for a reasonably healthy, efficient workforce. Unemployment is
illustrated in Figure 10. The unemployment U in town is equal to the difference between
labour supply L̄ − LA and the labour demand LM (at the fixed wage ŵ).
U = L̄ − LA − LM
The size of U is determined by how many workers that find it worthwhile to hang around
in the town even without a job. The reason for hanging around is that there is a chance of
getting a job in the modern sector at the wage ŵ as long as you stay in town. One simple
formulation assumes that workers will move to town until the average wage in town (for
the LM employed and the U unemployed) is equal to secure wage in agriculture wA = xA .
Assuming for simplicity that the unemployed earns nothing, the average wage in town is
wAV =
0 · U + ŵ · LM
U + LM
For a fixed ŵ and LM , wAV will be lower the higher is U. When U is zero then wAV = ŵ.
Note that
(U + LM ) · wAV = LM · ŵ
hence, for fixed ŵ and LM , the wAV curve will be the collection of all combinations of
(U + LM ) and w that multiplied together yields the fixed number LM ∗ ŵ. In other words
11
it is a hyperbola. The equilibrium is found where
wAV = wA
as illustrated in Figure 10.
9
Comparative advantage and international trade
An influential theory for international trade goes back to Ricardo and Smith. It encompasses
the concept of comparative advantage. Comparative advantage can be illustrated in the
following model. Consider an economy with two sectors: The manufacturing sector (M )
employs LM workers and has marginal product xM and price pM . The agricultural sector
(A) employs LA workers and has marginal product xA and price pA . Assume that the
marginal product in each sector is decreasing in the employment (diminishing returns) and
assume that each sector has optimal level of employment given the wage w. This implies
that the value of the marginal product in $ (e.g. xA · pA ) is equal to the wage w. For both
sectors this implies that
xA · pA = w
and xM · pM = w
Assume also that there is full employment so that
LA + LM = L̄
where L̄ is the total labour supply. The equilibrium in this economy is illustrated in Figure
11. The width of the diagram is determined by L̄, LA is measured from left to right while
LM is measured from right to left. The marginal product in $ in agriculture is drawn from
left to right while the marginal product in $ in manufacturing is drawn from right to left.
Note that the wage w is determined by the interaction between a fixed supply L̄ and demand
from two sectors LA and LM . Note also that this is in effect the Harris-Todaro model of
migration in the case where the wage is completely flexible in both sectors.
Now, knowing the value in the local currency ($) does not tell us that much. We get
more information if we measure everything in terms of units of goods. With two goods we
could use either good as the good that we use when measuring. Let us use the agricultural
good (bags of corn) when measuring the value of production and the value of the wage. By
dividing on both sides of the equilibrium conditions by the price of food pA the equilibrium
conditions becomes equal to
xA = w/pA
and xM · pM /pA = w/pA
The value of the marginal product measured in bags of corn should in each sector be equal
to the value of the wage measured in bags of corn. Then Figure 9 will change to Figure 12.
12
Figure 11: Allocation of labour between Agriculture and Manufacturing
value in $
value in $
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x ·p
w
x
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w
L
←−−−−−−−−−−−−−−− L̄ −−−−−−−−−−−−−−−→
←−−−−−−− LM −−−−−−−→
Figure 12: The wage income meassured in bags of corn
bags of corn
bags of corn
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x
x
w/pA
LA
13
· p /p
w/pA
Figure 13: Trade liberalization and structural shifts
bags of corn
bags of corn
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w/pA
· (p /p )
w/pA
LA
The figure is almost the same but now the interpretation is different. Note for example that
area below the w/pA line is total wage income in terms of bags of corn. Hence, it captures
how much the workers (and their families) can eat and is thus a measure of standard of
living.
Until now we have not discussed how the prices are determined. Assume for simplicity
that they are determined by domestic demand in a situation with no trade. Now, what happens if the country opens up for trade? Assume that the trade implies that both agricultural
and manufacturing goods can be bought and sold at the world market at fixed prices. Let
T
the relative prices at the world market be (pM /pA ) and let the relative prices without trade
I
T
I
be (pM /pA ) ,where T indicates trade and I indicates isolation. If (pM /pA ) < (pM /pA ) ,
manufacturing goods are relatively cheaper at the world market and it makes sense to start
exporting agricultural products and importing manufacturing products. In this case the
country has comparative advantage in the production in agricultural goods. For a poor
country, this may be a realistic response to trade liberalization. Anyhow, this is the view of
the IMF:
I would like to see the markets of the industrialized countries opened. So that
these countries, with people who are poor and can produce agriculture, are able
to export. That’s what would get them out of poverty. Stanley Fischer (2001)
”The new rulers of the world” Carlton Television Ltd.
T
I
Lets assume that it is indeed the case that (pM /pA ) < (pM /pA ) , then in Figure 12
the xM · pM /pA curve would shift downwards as illustrated in Figure 13 where the dotted
I
curve is the one that existed in the case without trade, ( xM · (pM /pA ) .) Three effects
are immediate. Demand from labour in manufacturing decreases and labour is shifted to
14
agriculture (i.e. urban to rural migration). As a result the wage (in terms of bags of corn)
drops. Profits in agriculture increase while profits in manufacturing decrease. Another way
to put it is that owners of land gain while owners of manufacturing capital lose.
The wage in terms of corn drops. Does that mean that the workers lose? Not necessarily.
If workers also consume manufacturing goods they will benefit from the lower price and that
will compensate somewhat the welfare loss and possibly give a gain.
For the country as a whole there is a gain. The gain is captured by the shaded triangle. The triangle captures bags of corn in excess if the country after trade reform and
reallocation of labour wants to consume exactly what they did before the trade reform. The
argument is as follows. After trade reform, if the country do not reallocate labour, they can
consume the same amount of manufactures and agriculture as before - that’s trivial. But
without reallocation of workers the marginal productivity will be higher in agriculture than
in manufacturing. The difference is the height of triangle at the left hand side. As workers
are moved from manufacturing to agriculture this gap is reduced until the new equilibrium.
The total gain, measured in bags of corn, from realocating workers is captured by the shade
triangle. To sum up: Given the assumptions of this model (not all are spelled out) A trade
reform that leads to increased agricultural production has the following implications
1. Owners of capital in manufacturing lose (a lot)
2. Owners of land gain (a lot)
3. The labour may win or lose. If they primarily consume agricultural goods they lose.
4. The country as a whole gains. This gain is small compared to the different groups
gains and losses. it is a triangle versus rectangles.
If trade reform instead implied increased production of manufactures and less production
of agriculture, all the gains and losses would change sign, so that owners of capital win,
owners of land lose, while labour probably win if they primarily consume food. Also in this
case the country as a whole would gain.
Why is this interesting? First of all, for better or worse, it is an influential framework for
thinking about trade. The arguments about comparative advantage pops up everywhere.
Second, the model illustrates starkly the conflict between distribution and over-all gains.
The reshuffled income between groups are larger in size than the over-all gain. This result
is often forgotten. Third, it can be a good starting point for discussing trade reform under
other assumptions. For example: What if the workers own land? What if capital can be
moved from manufacturing to agriculture? What if there is unemployment? Is it possible to
compensate people? Is it possible to form alliances in favor of/ against trade liberalization?
15
Should we trust governments of poor countries when they claim that their country wants
trade reform? Should we trust Norwegian farmers when they claim that they are against
import of food because of a concern for the poor countries? What if there is increasing
returns in manufacturing (learning by doing)?
10
Sustainable debt
There are many countries in the world that have debt levels that are far above what they
possibly can handle. For these countries some kind of debt reduction is unavoidable. Given
that some countries should get debt relief it is essential to decide what debt level a country
can possibly handle. One important principle in debt relief is that it should be once and
for all so that the country do not end up in similar problems after a year or ten. The relief
should bring the debt down to a level that is sustainable.
Sustainable debt can be defined in a number of ways. One definition is that it is a level
of debt such that the debt does not grow faster than GNI. Hence it should be possible to
hold the DEBT/GNI ratio constant over time. That implies that if the growth rate of GNI
is 3% a year the growth rate of DEBT should be no more than 3% a year.
The growth of the debt is determined by many factors. Two important factors that I
will focus on is the 1) export surplus and 2) interest on the existing debt (r∗DEBT where
r is the rate of interest). Abstracting from other factors it follows that
Growth in DEBT=r ∗ DEBT − export surplus
Assume that the expected growth rate of GNI is g. From the definition above, the DEBT
is sustainable if the growth in DEBT is equal to g∗DEBT. Assume that the export surplus
realistically can be a fraction a times the GNI. Hence, a requirement for sustainable debt
(DEBTSUST ) is that
g ∗ DEBTSUST = r ∗ DEBTSUST − a ∗ GNI ⇐⇒
DEBTSUST
a
=S =
GNI
r−g
Sufficient debt relief then implies bringing the debt level down to the level DEBTSUST , or
to put it differently bringing the debt to GNI ratio down to S. What determines S? Assume
that a = 0.01 (that the export surplus is expected to be positive and equal to 1% of GNI)
and assume that r = 0.07 (7%) then
S=
0.01
0.07 − g
If the growth rate is zero (g = 0) then S = 14%, if g = 2% then S = 20%, if g = 5% then
S = 50%, if g = 6% then S = 100%. (If the growth rate is large, g > 7%, then a can be
16
negative. That implies that if the growth rate of GNI is higher than the interest rate it
is possible to have a small export deficit at the same time as the debt is sustainable. The
important condition is that debt does not run faster than the GNI).
Anyway, the main message is that the answer to the question ”What level of debt is
sustainable?” depends critically on the assumptions regarding r, a and g. If one for example
is too optimistic with regards to the growth rate g and expect g = 6%, a debt ratio of 100%
may appear sustainable. Debt relief based on that premise will prove not to be sufficient
if g turn out to be 2% instead (in which case only a debt ratio of 20% is sustainable). In
the ongoing debate between IMF Word Bank and the heavily indebted countries about debt
relief the assessment of what the sustainable debt level is for each country is one of the
main points. Note that the countries that want maximum debt relief is best served by being
pessimistic with regards to growth and export surplus potential.
Problems
1. Consider the case where the marginal product curve is
x1 = 25 − 5 · L1
(P1)
a) Draw the curve
b) Derive the marginal product when L1 = 4
c) Shade the total product when the number of workers is L1 = 4 and assess the value.
d) Derive the expressions for the average product and the total product for all levels of L1 .
Use the result to calculate the total product when L1 = 4. Compare with c)
e) Derive the value of L1 for which x1 is zero.
f) Assume that the total available labour is L̄ = 10. Under central command what is
optimal value of L1 ?
g) Assume that there is alternative employment with fixed marginal product equal to 5.
Under central command what is optimal value of L1 (and L̄ − L1 ?)
h) Under f) and g) what is the resulting L1 if P1 is a commons resource . Illustrate the
efficiency loss.
i) Under f) and g) what is the outcome if P1 is a factory owned by a profit maximizing
owner setting L1 and paying as little as possible. Illustrate the income distribution between
factory owner and workers.
2. Consider the case where total product is independent of employment L1 .
17
Y = 10
(P2)
a) Draw the total product curve
b) Derive the expressions for the average product for all L1 .
c) Assume that there is alternative employment with fixed marginal product equal to 5.
Under central command what is optimal value of L1 (and L̄ − L1 )? What will be the
resulting L1 when P2 is a commons resource
3. Consider the case where there are two alternative sectors with profit maximizing owners
in both sectors and marginal product as follows
x1 = 10 − 2L1
(P3a)
x2 = 5 − L2
(P3b)
a) Assume that total labour supply is L̄ = 9. Derive the equilibrium allocation of labour.
b) Assume that sector 1 is turned into a commons resource.
Derive the equilibrium
allocation of workers.
c) illustrate with shading how going from a) to b) affected efficiency and income distribution.
4. Consider Figure 12.7 in Ray’s book. Draw a marginal production curve and two wagecurves (opportunity cost) that illustrates the exact same message regarding L∗ and L∗∗ .
18