I think that Ethiopia's problems go back to the end of the 19th century and were caused by industrialisation and furthered by 'development aid' in the cause of more industrialisation.
How long has Ethiopia been overpopulated for?
In the late 19th century Ethiopia's population was only 4 or 5 million, when Mennilik II began 'modernisation'.
After Mennilik's modernisation programs, which included massive land 'reforms' (enclosures), Ethiopia's population climbed to 10 million. In 1950, as development continued, the population climbed to 18.3 million. In 2010 it is 83 million and climbing rapidly still.
Ethiopia is one of the oldest sites of human civilisation and international trade. It had a variety of populations within, mostly rural, including hill tribes who farmed using ploughs. The various tribal populations mostly kept to their traditional areas and were sustainable within them, although for a long time there have been conflicts between adjacent states and an ancient slave trade along certain routes.
Ethiopia after 'development':
In the late 19th century there were massive land 'reforms':
The land 'reforms' coincided with dispossession and 'integration' of clans and tribes with Mennelik II who was influenced by Russian imperialists. Thrown off their land - particularly those on the plains, the people drifted to the cities, which were growing as agriculture was industrialised and roads and electricity were installed.
This process destroyed local self-government and sustainable local economies where there was probably still men's and women's land, and marriage opportunities would have been kept low through kinship rules that limited approved spouses, , with late menarche the rule, and low fertility through local traditional birth control practices, including contraception, spousal non-cohabitation, abstinence, delay of coitus  and co-operative breeding  (where people live in an extended family around a principal fertile couple, taking the roles of servants, aunts, aunties).
As happened in the Netherlands and in Britain, local populations were disorganised by classes that got control of the food production and government and, lacking the land that had once brought them independence, they were obliged to move to new settlements, with only their labour to sell. This is the kind of system where women and children become slaves to their husbands, fathers, uncles and brothers.
Irrigation was put in place and former grazing ranges were intensively cultivated, which required dispossessing the peoples that had led sustainable lives there for millenia, without overpopulation. Malaria became a huge problem, indicative of land-disturbance and new population movements.
In the 1970s there were more huge land reforms.
In the 1980s there were mass government resettlements.
The 'reform' has not stopped. It has accelerated.
It is not difficult to find many accounts of how foreign aid assisted in all these 'modernisation' projects that led to the dispossession of the peoples of Ethiopia and the surrounding regions and to the current overpopulation problem and enslavement of women, for which more 'development' continues to be recommended, under the malignant pretext that overpopulation was Ethiopia's eternal burden.
What is in it for the countries driving more 'development' in Ethiopia, Somalia, Eritrea, Egypt etc? One of the major reasons for foreign powers to maintain these countries in disorder is so that the foreign powers can continue to dominate the surrounding region with military forces, supposedly to safeguard international investments. The pretence is that these African countries and all those around the Red Sea and to the North of Syria need international aid and management, however it is clear to anyone aware of the petroleum in this region that Ethiopia could be a source of primary wealth, as it was for other reasons in Ancient Egyptian, Greek and Roman times. 
Development, War and Refugees
Australians rightly complain that their country is overpopulated. They also rightly complain that refugee and asylum-seeker numbers are adding to this problem. But they fail to link these problems to the wars the Australian government backs in the regions where the asylum seekers come from. Australia, like the United States in the film, Mother of 7 Billion, adds over-consumption to its overpopulation problem. To maintain our commodity production and our importation of new residents to keep our land-sales up, we rely on huge amounts of petroleum and one of the most important sources of petroleum is the region bordered by the Land of Punt.
If Somalia, Eritrea, Ethiopia and Sudan had maintained their efficient local economies, there would be no excuse for foreign troops and corporations to be in the region, but Somalia, Ethiopia, Eritrea and Sudan would probably all be wealthy through association with the oil producing countries.
Instead the oil producing countries and the regions buffering them are in constant state of flux, overpopulation, poverty, and war and the foreign powers actually rule the region. We hear of Somalian pirates, trying to get control of the oil, but we call our own pirates 'peace keepers'.
 Kevin Myers, "Somalia is not a humanitarian disaster," The Irish Independent, 2008
 Fertility opportunities would have been kept low through kinship rules requiring endogamous marriages that reflected the local environment with up to 8 degrees of incest avoidance and other variations that greatly decreased mating opportunities. See Sheila Newman, Development, Territory, Law: The Rules of Animal and Human Populations, Countershock Press, 2013.
Details of the wide variety of these kinship impediments to marriage and fertility in Ethiopia can be found in Siegbert Uhlig, (Ed.), Encyclopaedia Aethiopica: He-N, Otto Harrassowitz GmbH & Co. KG, Germany, 2007, pp. 55, 405-406, cited by hbdchick at https://hbdchick.wordpress.com/tag/exogamy/. Although the afore-cited Encyclopaedia states that being unmarried is not an option in Ethiopia, this is contradicted by the presence of extended families. "The extended family arrangement is frequently found among Ethiopian families since children, male or female, who have married continue to live with their family. In some cases, unmarried aunts, uncles, cousins and even close family friends continue to dwell with the nuclear family. " http://www.africa-expert.com/ethiopia/family-and-housing/ ftp://ftp.cgiar.org/ifpri/ElizabethB/Ethiopia%20papers/diFalco_Bulte_2013.pdf,
Mhairi A. Gibson and Eshetu Gurmu, "Land inheritance establishes sibling competition for marriage and reproduction in rural Ethiopia"http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3038728/#!po=21.4286: "We use rich anthropological, demographic, and socioeconomic data from five traditional rural villages in Ethiopia (Table S1), where wealth remains positively correlated with reproductive success (characteristic of preindustrial societies)."
Another likely indication of earlier prevalence of local birth control methods would be family-size trends in older women who may have had more links to traditional villages. The results varied according to tribal origins as well as religion. A survey in 1986 showed that childlessness varied from 20% to 3% in the earliest cohorts (women aged 59 or older) compared to 12% to 2% in the most recent cohorts (women aged 30-39). Although much of this was attributed to successful treatment of sexually transmitted diseases the article concludes that there is more to it than that.- See more at: http://www.popline.org/node/372469#sthash.OatvN3Yl.dpuf
 John Knowles (Lead author), Report, "History of Birth Control Methods," Katharine Dexter McCormick Library and the Education Division of Planned Parenthood Federation of America, 2012, http://www.plannedparenthood.org/files/2613/9611/6275/History_of_BC_Methods.pdf
 Cooperative breeding is similar to the extended families described (which existed in most pre-WW2 societies) in note  Traditional Ethiopian societies were comparable to traditional European ones in many ways, so I don't mean for the following references to imply that they were all hunter-gatherers, However, for some modern evolutionary research of the phenomenon in terms of inclusive benefits etc in human societies, see, for instance, Smaldino, Paul E., Newson, Lesley, Schank, Jeffrey C., Richerson, Peter J., "Simulating the Evolution of the Human Family: Cooperative Breeding Increases in Harsh Environments," http://www.plosone.org/article/info%3Adoi%2F10.1371%2Fjournal.pone.0080753 and Hill, Kim, and Hurtado, A. Magdalena, "Cooperative breeding in South American hunter–gatherers, " School of Human Evolution and Social Change, Arizona State University, PO Box 872402, Tempe, AZ 85287-2402, USA, http://rspb.royalsocietypublishing.org/content/early/2009/08/18/rspb.2009.1061.full.pdf+html
[5a] Sheila Newman, Demography, Territory, Law 2: Land-tenure and the Origins of Capitalism in Britain, Countershock Press, 2014.
 David N. Weil, "The Impact of Malaria on African Development over the Longue Durée,"
Brown University and NBERhttp://www.indiana.edu/~econdept/workshops/Fall_2010_Papers/Malaria_and_Early_African_Development.pdf: "The current geographic distribution of malaria impact may not accurately represent the historical burden of the disease. Packard (2007) gives numerous examples of how human activity has changed the intensity of malaria. Clearing forests and introducing irrigation often produces the marshy environment needed for breeding of Anopheles mosquitoes. When large tracts of land are consolidated and controlled by a small number of landowners, people living on these lands are often deprived of the tools for properly tending to it and for avoiding malaria infection. This problem is exacerbated when seasonal labor is used, as migrants may come from non-malarious zones and lack immunity, then return to these zones carrying malaria with them. As for Africa, he notes that “while climate and the presence of highly efficient vectors contributed to the persistence of malaria, conditions of production played an equal if not greater role." Webb (2008) similarly describes how the introduction of agriculture led to malaria endemicity in lowlands throughout the Eurasian landmass. Population density is also important to the nature of malaria burden. Packard argues that in the African contexts, low population densities associated with mobile hunter-gatherer populations would have been unable to support the P. Falciparum infection because of the speed with which the disease exits or kills the human host. The emergence of Anopheles gambiae is also believed to closely associated with the development of agriculture. For all these reasons, the current distribution of malaria in Africa today is potentially a poor proxy for the historical distribution." See also malaria references in Sheila Newman, Demography, Territory, Law 2: Land Tenure and the Origins of Capitalism in Britain, Countershock Press, 2014.
 "Ethiopia, 30 years after the famine," by David Smith (23 October 2014). "Three decades after images that shocked the world, country has become darling of the global development community – and the scourge of the human rights lobby." http://www.theguardian.com/world/2014/oct/22/-sp-ethiopia-30-years-famine-human-rights
 "This is pure common-sense commercial logic,” ... “There are numerous discoveries along the East African Rift in other nations and also it ends basically in Yemen, which is a known oil-bearing territory." "http://www.bloomberg.com/news/2014-07-21/gazprombank-s-gpb-global-signs-oil-exploration-deal-in-ethiopia.html; Ogaden Basin in Ethiopia: http://en.wikipedia.org/wiki/Ogaden_Basin; "Africa's next Frontier - Ethiopia?" http://oilprice.com/Energy/Crude-Oil/Africas-Next-Oil-Frontier-Ethiopia.html]
 And some of us can see that the same thing is happening here, with 'development' as has happened to Ethiopia.
 I spoke at a screening of Mother of 7 billion in 2013 and was uncomfortable at some of the assumptions made in the film. Most difficult for me was the assumption that Ethiopians had always had huge families. There was seemingly no inkling that Ethiopia had ever been anything except poor and overpopulated - of the sudden leap from 4 million to 83 million. When one considers Ethiopia's remarkably long and impressive history, this is quite an oversight. It was because of this film that I started writing an article on Ethiopia's long and different past, but it has taken me until today to get this short version up.