A little development economics
Halvor Mehlum
1
Background
Many results in economics builds on the premise of diminishing returns in production. In terms of employment
diminishing returns impliesthat 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. I will return with an example of that
below. 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 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)
Figure 1: Value of marginal (and average) productivity
value
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x̄
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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
L
L̂
L∗
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
b
a + (a − b · L)
=a− ·L
x̄ =
2
2
x̄ starts out in the same point as x but is less steep and crosses the horizontal axis at 2 · L∗ .
The value of total production Y is the sum of all the marginal product 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 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.
1 For
example as workers are obstructing each other as in a soft ice kiosk with 30 workers.
2
Figure 3: Total production
Y
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3
L∗
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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 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
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 is 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 (w L̄ − LA )
and what is produced Y . As long as w > x income increases as workers are moved to the factory. As long as
w < x income increases if workers are moved out of the factory. Maximum income is reached when
w=x
3
Figure 4: Optimal allocation of labour
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A
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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 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 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. The owner of the firm hires labour for a fixed wage w. The owner wants to hire LB workers in order
to maximize his profits π . 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.
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
4
Figure 5: Profit maximizing owner
profits
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Figure 6: Sharecropping
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x
w
γx
LC
LB
L∗
L̄
L
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
Assume now 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
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%). 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 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 get 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 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 (or hire workers) optimally. At LB
6
Figure 7: Labour market with market clearing wage
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7
L
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
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 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 all workers
do no earn 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.
7
Figure 8: Labour market with unemployment
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8
L̄
L
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.
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 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.
If, due to investments, the demand for labor in manufacturing went up, the xM line would shift to the
left 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 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
8
Figure 9: Allocation of labour between Agriculture and Manufacturing
value in $
value in $
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←−−−−−−−−−−−−−−− L̄ −−−−−−−−−−−−−−−→
←−−−−−−− LM −−−−−−−→
Figure 10: Harris-Todaro: allocation of labour between Agriculture , Unemployment, and Manufacturing
value in $
value in $
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wA
LA
LM
←−− U −−→
9
ŵ
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 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 11 will change to Figure 12. The figure is almost the same
10
Figure 11: Allocation of labour between Agriculture and Manufacturing
value in $
value in $
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←−−−−−−−−−−−−−−− L̄ −−−−−−−−−−−−−−−→
←−−−−−−− LM −−−−−−−→
Figure 12: The wage income meassured in bags of corn
bags of corn
bags of corn
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LA
11
· p /p
w/pA
Figure 13: Trade liberalization and structural shifts
bags of corn
bags of corn
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LA
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
T
and sold at the world market at fixed prices. Let the relative prices at the world market be (pM /pA )
I
and let the relative prices without trade be (pM /pA ) ,where T indicates trade and I indicates isolation. If
T
I
(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 curve is the one that existed in the
I
case without trade, ( xM · (pM /pA ) .) Three effects are immediate. Demand from labour in manufacturing
decreases and labour is shifted to agriculture (i.e. urban 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 possible give a gain.
For the country as a whole there is a gain. The gain is captured by the shaded rectangle. The rectangle
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
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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.
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? 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
13
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 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.
14