Potatoes Paradox - Sherwin Rosen, 1999, University of Chicago
Potatoes Paradox - Sherwin Rosen, 1999, University of Chicago
Potatoes Paradox - Sherwin Rosen, 1999, University of Chicago
Sherwin Rosen
University of Chicago and the Hoover Institution
Price and quantity data prove that Irish potatoes in the 1840s were
not Giffen goods. Intertemporal trade-offs required by the fact that
a sizable fraction of the potato crop is needed for seed crops can
produce unusual market dynamics. The Irish experience is well
described by a normal demand model in which a permanent de-
cline in the productivity of seed potatoes was at first mistaken as
a transitory crop failure. These mistakes provoked ‘‘oversaving’’ of
seed crop in a population in dire circumstances. With the benefit
of hindsight, consumption of seed crop capital was warranted. Er-
roneous expectations of potato productivity by growers delayed
necessary agricultural adjustments and contributed to the catastro-
phe later on.
I. Introduction
Potatoes are very interesting goods. They have been an important
food staple in the world since the eighteenth century, serving as the
main (nonprotein) source of food energy for many people at various
times and places (Salaman 1949; Heiser 1990), yet they also have
been implicated in one of the worst famines in recorded history.
And because of their importance in the diets of many poor people,
potatoes are a leading practical example of an inferior good, if not
S294
potato paradoxes S295
a dreaded Giffen good. But what makes them especially interesting
for economics is the little-known fact that potatoes are capital goods as
well as consumption goods. Intertemporal elements of choice affect
supply decisions. Some economic consequences of this fact are pur-
sued in what follows.
Potatoes are not produced commercially from seed. They are re-
produced vegetatively. A significant portion of the crop is withheld
from final consumption, and potato buds (or ‘‘eyes’’) are replanted
for next year’s crop. Seed crop capital is an important component
of total production in modern conditions, depending on climate,
soil quality, and agricultural technology. In parts of Russia today,
seed potatoes may account for as much as 25 percent of the crop.
In the United States it is 7–9 percent. It probably was on the order
of 15 percent in nineteenth-century Ireland during the famine era
(Bourke 1993).
Adjustments in seed stock inventory affect supply decisions in un-
usual ways that can confuse demand and supply responses in many
situations. Dynamical responses to shocks can lead to bizarre results.
For example, when demand increases, short-term consumption can
fall (Rosen 1987). When the quantity supplied decreases, consump-
tion can increase even if demand is downward-sloping, an effect that
might be wrongly confused with the Giffen paradox.
Near the 150th anniversary of the Great Irish Famine of 1845–47,
it is appropriate to draw out these implications and test them on
available data. Another reason for economists to study that unfortu-
nate episode is that the appearance of the fungus p. infestans on the
Irish potato crop of 1845 virtually duplicates the conceptual experi-
ment needed to analyze dynamic market responses to an unusual
kind of production shock. The pest was unknown prior to that time,
and its arrival proved permanent. Pathologies are interesting to
study when the main causal elements dominate everything else. The
blight was an entirely exogenous shock that overwhelmed all other
factors affecting demand and supply of the major food crop in Ire-
land. Section II sketches an abstract economic model of the potato
market. Later sections interpret some of the Irish data in its light.
The major outlines of the disaster are well described by a model
in which a permanent adverse decline in potato productivity was at
first mistaken for a transitory one. The blight was completely out of
the realm of European agricultural experience in 1845. In the first
year of the famine, anticipation that productivity would return to
normal induced growers to carry over excess seed inventory for
planting the next year. It is clear in hindsight that in the first year
seed stocks should have been eaten instead. Since potatoes were so
important in the diets of rural dwellers and smallholders, gross
S296 journal of political economy
oversaving by people in such dire circumstances had catastrophic
consequences later on. And it delayed substitution to other crops
that could have eased, though hardly eliminated, the suffering.
A. Technological Structure
Let S t be the size of the seed crop, g the net reproduction rate, Q t
total output, and C t consumption of the good. The production func-
tion is
Q t⫹1 ⫽ (1 ⫹ g)S t , (1)
with g ⬎ 0. Potatoes can be stored for no more that 10 months, so
seed crops withheld from final consumption are all that is available
for producing output in following years. In nineteenth-century Ire-
land, the 15 percent seed/output ratio meant that one seed potato
produced about 1/.15 ⫽ 7.5 potatoes next season, implying g ⫽ 6.5.
In the United States today it is 12–15 to 1, so g is in the 11–14 range.
Total output is used for seed crop and consumption:
Q t ⫽ S t ⫹ C t. (2)
B. Behavioral Relationships
To simplify, assume no intertemporal elements in consumption de-
mand from year to year (but see below on seasonal variations). Then
market demand is C t ⫽ C(p t), where p is the price of the good. Sup-
ply is determined by farmers’ decisions either to sell their crops for
immediate consumption or to hold some of them over the season
for seed, replanting, and selling or planting crop next year. When
labor inputs are ignored,1 the return from selling is the current price
1
Labor inputs are important for potatoes. Consider eq. (1) as a reduced form in
which other inputs have been optimized out and their factor prices impounded in
g. In reality, productivity g varies with soil fertility on each farm. At the market level,
limited amounts of higher-quality land make g a decreasing function of aggregate
output, so (1) must be interpreted as a local approximation. Because it is so simple,
I choose to model rising supply price as arising from increasing marginal holding
costs rather than from diminishing returns directly, but they are (approximately)
equivalent for local dynamics (see below).
potato paradoxes S297
p t . Withholding a unit of stock and replanting it produces 1 ⫹ g
units next season. They can be sold for expected price p t⫹1 . Let k(S )
be the direct marginal cost of planting and storage, with k ′(S ) ⱖ 0,
and let r be the rate of interest. With G ⫽ (1 ⫹ g)/(1 ⫹ r ), the
marginal return from replanting is Gp t ⫹1 ⫺ k(S ). If the crop is simul-
taneously consumed and replanted for future production, profit
must be equal at both margins, or
p t ⫽ Gp t ⫹1 ⫺ k(S t ). (3)
The crop is renewed if G ⬎ 1. Otherwise, renewal is not economic
and cultivation ceases.
The intertemporal arbitrage condition in (3) is one restriction on
the observed paths of stocks and supply prices. When the demand
curve, production function, and storage-consumption-production
identity are combined, stock-flow dynamics imply another restric-
tion:
St⫹1 ⫺ S t ⫽ gS t ⫺ C(p t⫹1). (4)
C. Market Equilibrium
If demand and supply are unchanging, long-run equilibrium values
of p, C, Q, and S are defined by (2)–(4), with p t ⫽ p t⫺1 ⫽ ⋅ ⋅ ⋅ ⫽ p
and S t ⫽ S t⫺1 ⫽ ⋅ ⋅ ⋅ ⫽ S. Equation (3) becomes
k(S)
p⫽ . (5)
G⫺1
The long-run supply price of the good is the marginal planting and
carrying cost of inventory grossed up by the inverse of the net inter-
nal rate of return from growing the crop. Renewal implies that this
rate of return, G ⫺ 1 ⫽ (g ⫺ r )/(1 ⫹ r ), is positive. When marginal
cost is increasing in S, equation (5) defines a rising curve in the (p,
S ) plane.
At these steady-state values, equation (4) describes the seed inven-
tory required to sustain long-run consumption:
C(p)
S⫽ . (6)
g
Equation (6) defines a decreasing curve in the (p, S ) plane if the
good is normal and an increasing or backward-bending curve if it
is Giffen in part of its range (see figs. 1 and 2).
With equation (5), equation (3) can be arranged to read as
p t⫹1 ⫺ p t ⫽ (1 ⫺ G )(p t⫹1 ⫺ p) ⫺ [k(S t) ⫺ k(S)]. (7)
Fig. 1.—Normal transient dynamics
2
Linearizing k(S ) and C(p) in the neighborhood of the steady state yields a pair
of linear difference equations. In a stable system the associated characteristic equa-
tion has two real roots, whose product is 1 ⫹ r. In the general solution the unstable
root larger than 1 ⫹ r is eliminated by the boundary condition. Its effect is seen
only in the forward-looking part of the particular solution. Notice that in this system,
it makes no difference whether the p manifold is increasing because of rising inven-
tory costs or directly diminishing returns, as asserted above.
S300 journal of political economy
this case. One is stable and the other is unstable. If the Giffen effect
is so strong that the p curve is flatter than the S curve in figure 2,
the phase arrows near that intersection reflect the fact that the asso-
ciated linear system has explosive cycles (complex eigenvalues with
modulus greater than 1 ⫹ r ). The market is Hicksian unstable (and
the implied cycles are inefficient). This node is labeled A in figure
2. The equilibrium is stable only when the rising S curve cuts the p
curve from below (point B in the figure). Here the stable arrows
in the neighborhood of equilibrium slant in the opposite direction
compared to figure 1. If initial inventories are excessive, price must
be driven up to encourage the extra consumption needed to work
them down to steady-state levels. If initial stocks are too small, price
must be driven down to discourage consumption and encourage ac-
cumulation. From such stuff are paradoxes made.
namics caused by decisions about seed inventory do not necessarily imply Giffen
goods.
S302 journal of political economy
elasticity of demand and ⑀ as the price elasticity of long-run supply.
Comparative statics on (5) and (6) yield
∂ log p [1 ⫺ (r/g)] ⫹ ⑀
⫽⫺ ⬍0 (9)
∂g ⑀(g ⫺ r )[1 ⫺ (η/⑀)]
and
∂ log S η ⫹ [1 ⫺ (r/g)]
⫽⫺ . (10)
∂g (g ⫺ r )[1 ⫺ (η/⑀)]
As shown above, market stability requires 1 ⫺ (η/⑀) ⬎ 0.
Equation (9) shows that long-run price always decreases in g. The
sign of (10) is ambiguous. The derivative (∂ log S )/∂g is positive or
negative according to whether the elasticity of demand exceeds or
falls short of 1 ⫺ (r/g). If demand is sufficiently elastic, S falls when
g falls, and steady-state consumption decreases so much that steady-
state inventories decrease in spite of the fact that more seed crop is
required to produce any level of output. When demand is suffi-
ciently inelastic, steady-state seed inventories increase when g falls.
Since consumption does not fall so much, the smaller productivity
of seed requires a larger seed stock to sustain it. In this case, inter-
temporal stock adjustments exacerbate price and consumption
movements. Notice that if demand is sufficiently elastic, the adverse
shock conceivably could cause price to fall and consumption to in-
crease initially. In that case, dumping inventory on the consumption
market smooths the necessary price and consumption adjustments
to the new steady state rather than worsening them.
4
To put the disaster in proper perspective, the famine killed at least 12 percent
of the population over a three-year period. Another 6–8 percent migrated to other
countries. In terms of the percentage of population affected, the 1845–48 famine
is one of the largest ever recorded. Other famines have killed more people in total
because the affected populations were larger, not the percentage of exposure. For
instance, the 30 million or more people who perished in the Chinese famine of
1958–62 were 5 percent or 6 percent of the population. In comparison to other
disasters in living memory, World War II casualty rates (military and civilian) in all
countries except Yugoslavia were much smaller than 12 percent (Wright 1965).
5
Pre- and postfamine gross yields per acre in Ireland do not differ much because
acreage on less productive land fell drastically in the latter period. Since aggregate
yields contain these changing composition effects, inferences about productivity
changes directly attributable to the blight are more secure from relative price data
than from aggregate quantity data.
6
I am indebted to Peter Solar for allowing me to reproduce his data. The figures
shown pertain to the Waterford market. Solar’s prices from the Belfast market and
Ó Gráda’s (1993) time series from the Dublin market are similar to what is shown
in fig. 4a.
S304 journal of political economy
Fig. 4.—a, Irish potato prices per hundredweight in the Waterford market.
b, Prices of potatoes at Waterford relative to wheat and oats.
7
Earlier, Dougan (1982) analyzed the unusual dynamic implications of Giffen
goods using the static Hicksian stability framework. This analysis is a natural exten-
sion of his.
potato paradoxes S307
been greater in Ireland than elsewhere,8 imports and intertemporal
rearrangements of world consumption across countries could have
made up the Irish shortfall. The blight was not nearly so extensive
in other European countries (though their exposure was much
smaller as well), so the enormous price increase observed in Ireland
implies substantial limitations on the international potato trade.
Since potatoes consist mostly of water, transportation costs per unit
of nutrient value are large compared to more commonly traded
foods such as grain. The high land transport costs of the mid-nine-
teenth century made potato trade relatively unimportant, but not
zero. Irish potatoes had been exported, and even in 1845 some were
sent to the low countries reporting blight. Nonetheless, imports of
potatoes for Irish consumption were trivial.
Trade in seed crop is more interesting economically. Potatoes are
so productive as capital that seed crop requires a much smaller vol-
ume of trade and involves lower costs per unit of value compared
to potatoes for consumption. In fact high-yielding new varieties of-
ten were exported. The Irish potato was known throughout the
world and remains so today. But the key issue is imports. My analysis
plays off the need to abstain from current consumption and retain
seed to produce next year’s crop. Full consumption of seed crops
in any period drives production to immediate extinction in a closed
system. Significant importation of seed crop breaks the sharp con-
straint that links current consumption of seed inventory to smaller
future production. Small amounts of seed potatoes were imported
prior to the blight by some of the larger growers seeking higher-
yielding varieties with better taste. Yet during the famine years, there
is no evidence of significant imports of seed potatoes. The Irish po-
tato economy was essentially closed.
The standard economic arguments for restricted trade and lim-
ited intertemporal substitution revolve around capital market imper-
fections or, in more fashionable terminology, ‘‘liquidity con-
straints.’’ The people who suffered most from the famine were rural
dwellers: smallholders, tenants, and laborers operating largely in a
cashless, subsistence economy. Self-production of food on small
plots of land accounted for a very high proportion of their incomes.
Most potatoes were produced and consumed by families on tiny
plots of land. Prior to the famine, such production sustained a vigor-
8
The blight was endemic in South America, where the potato originated, but
Europe was blight-free because p. infestans could not survive the long sailing voyage.
The shorter steamship voyage changed all that (McNeill 1976). It is ironic that much
later the blight destroyed the South American crop, and blight-resistant varieties
imported from Europe are grown there today. Nothing is left of the native South
American varieties (Heiser 1990), though strains of the blight are still active.
S308 journal of political economy
ous, rapidly growing population.9 But that system had tragic conse-
quences.
Population pressure in rural Ireland caused land prices to rise
earlier in the nineteenth century. And declining food prices after
1825 reduced farm wages and made it desirable for rural folk to
economize further on food. The land economies afforded by the
immense yield per acre of potato cultivation help but do not fully
account for the increasing concentration and specialization of pota-
toes in rural diets immediately prior to the famine. The low income
levels of these people, their self-sufficiency and lack of transferable
wealth, and the huge losses they suffered from declining potato pro-
ductivity all worked to shut down the market for imported seed crop.
And their prior self-sufficiency in food and lack of financial wealth
made it impossible to import much of anything else.
10
It often has been noted that pigs were part of the rural family, as it were, an
observation that still applies in rural areas of many poor countries today. If in the
United States pigs are ‘‘corn with legs,’’ in Ireland of that time pigs were potatoes
with legs. Longfield (1834, pp. 249–56) presents a lucid early analysis.
S310 journal of political economy
that the Irish stock was 55 percent smaller in 1847 than at the agricul-
tural census of 1841. Though data for the years in between are miss-
ing, it seems probable that these adjustments tempered some of the
human distress caused by crop failure. Spotty export records indi-
cate that pig and cattle shipments were unusually high in 1847 com-
pared to 1835, the closest earlier year for which records are available.
Notice that the principle of comparative advantage applies to the
rationality of exporting animals for cash rather than eating them:
the proceeds could be used to purchase cheaper substitutes, such
as Indian corn, which was imported into Ireland during the famine,
but hardly in sufficient quantity to avoid starvation.11 It is probable
that large numbers of pigs also were directly consumed. Of course
these buffers and substitution possibilities are greater for transitory
shocks than for permanent ones. They would have been more im-
portant in 1845–46 than later, but the data are not available to tell.
Schultz (1938) estimated the elasticity of demand of potatoes for
human consumption as ⫺.65 in the United States during 1875–95.
Potato budget shares for the average 1840 Irish consumer would
have been much larger than for the average 1875 American con-
sumer, tending to raise the elasticity. Fewer available food substitutes
tend to lower it. No direct estimates of the demand elasticity of pota-
toes for animal feed are available from any source. The stock of pigs
in Ireland probably fell by more than 50 percent during the period
when the price of potatoes doubled. Use of substitute feeds would
have reduced demand for potato feed by more than 50 percent, so
a lower bound for the arc elasticity of feed demand is in the ⫺1/3 to
⫺1/2 range. Since alternative feeds were in such limited supply, it is
implausible that feed demand was much more elastic than this.
VII. Conclusion
As horrific as it was, the main production aspects of the Irish famine
can be analyzed in fairly conventional economic terms. Adapting a
familiar dynamic model does a tolerable job of explaining the main
time-series data on the consumption and production of potatoes.
This application is possible because potato technology is so simple
and because the event that precipitated it is so transparent in hind-
sight. Most empirical dynamic economic models require greater
elaboration and detailed technical assumptions about unobserved
exogenous disturbances. Issues of that kind do not arise here be-
cause the timing of the blight’s arrival is precise and its conse-
quences eclipse everything else.
The most drastic consequences of the blight were caused by tragic
miscalculation that a permanent shock was transitory, as previously
emphasized by Mokyr (1983) and Ó Gráda (1993). This delayed re-
organizations of agricultural production to more productive crops.
These faulty perceptions perhaps are understandable, if not ‘‘ratio-
nal,’’ given the information that was available to people in those
days. Potatoes had supported a large and growing population for
many years with relatively little risk. Decisive scientific identification
of blight as the true cause of the productivity decline came only years
after its first occurrence. The huge effects of agricultural science on
agricultural productivity in general occurred almost a century later.
The new point added by this analysis is that the mistake provoked
oversaving by a population in dire circumstances, certainly in the
first year and possibly in later ones as well. The famine may have
been prolonged by a perceived need to conserve seed stocks even
after its permanent nature was recognized. However, the high dis-
count rates in the affected population, made even higher by sickness
and starvation, suggest a limited role there. Instead, potato cultiva-
tion was scaled way back.
The adverse consequences of the potato blight were heightened
by the extraordinary importance of potatoes in the self-production
and consumption of food among large rural Irish families. It caused
the equivalent of a sudden 50 percent or more increase in the price
of food for at least one-third of the population. That translates to
a massive 25–30 percent unexpected decline in their standard of
living. An event of that scale would cause misery even in a rich popu-
lation.
potato paradoxes S313
Since the Giffen paradox is not useful for understanding the Irish
experience, is it asking too much for future writers of elementary
texts to find another example? Fictions have no place in the teaching
of economics. If any paradox remains for understanding the eco-
nomics of the famine, it is how human fertility decisions, population
pressures, and the social and economic system of Ireland at that time
encouraged such large families, such specialized personal food pro-
duction, and such shockingly undiversified diets. Perhaps this is a
case in which economic analysis is limited by not being able to ac-
count for tastes.
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