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The Black Death affected Europe and China alike.
In Europe, the Black Death resulted in increased social mobility for the peasants and, as a result, labor-saving innovations were introduced. All these laid the foundation for the European Renaissance later (see the above Wiki page).
Why didn't the Black Death have the same social and economic effects in China?
Firstly, there's no reason to think it didn't. In fact, it's fairly routine for afflicted regions to receive government subsidies in Imperial China. As early as AD 218, the Book of the Later Han records that Cao Cao issued an edict ordering welfare measures for survivors of a plague in the preceding winter. The next year, the Record of the Three Kingdoms mentions Cao's contemporary and rival Sun Quan abolished all land rents in the province of Jin, also in response to an epidemic.
Closer to the Black Death period, in 1408, the Ming court ordered rice stipends for survivors in Fujian, and exempted stricken families from their corvée obligations. So in a very literal sense, surviving peasants often directly benefited in the aftermaths of plagues in China as well.
Secondly, labour scarcity is an inherently temporary effect unless population ceases growing altogether. As the Wikipedia article you cited says,
For English peasants, the 15th century was a golden age of prosperity and new opportunities . . . A century later, as population growth resumed, the peasants again faced deprivation and famine.
Moreover, population recovery can be greatly hastened by immigration. This did not really happen in Medieval Europe for a variety of reasons including political fragmentation and feudal interests. In China, however, a tradition of unified central authority allows labour redistribution to happen efficiently and over great distances. For example, in 1410, the county of Shaowu in Fujian reported that 12,000 families had been killed off by the plagues of the preceding years. Whereas English peasants would've moved into vacated houses or be paid to till abandoned farmlands, the Chinese imperial government simply responded by settling convicts on the vacant farms.
Thirdly, for whatever reason, the plague in China was not, for lack of a better word, as concentrated as the Black Death was in Europe. The reason for the socioeconomic upheaval in Europe is because there was so many deaths happening in so short a time when the plague first struck in 1348. In England, for instance,
That demographic crisis was the death of upwards of a third of the population in the first outbreak of the Black Death in 1348–50. Beneficed clergy seem to have lost 40 percent of their numbers; tenants in chief, 27 percent . . . The 1348-50 epidemic seems to have been quite as severe as has been thought: it quite plausibly killed a third to a half of the population.
Palmer, Robert C. English Law in the Age of the Black Death, 1348-1381: A Transformation of Governance and Law. Univ. of North Carolina Press, 2000.
As many as 170 epidemics might have hit Ming China, but they were all localised to usually no more than one or two provinces at a time, and the death toll was spread out over many years. Imperial population records reveal no serious gaps of the sort experienced by Europe as a result of these plagues. In other words, an outbreak may create a local shortage of labour, but nationally there always remained a huge surplus of labour relative to the available land.
Sources:
All these laid foundation for the European Renaissance
It's important to remember that there have been plenty of deadly plagues in Europe but only one Renaissance. The Plagues of Justinian or Antonine both caused a level of depopulation similar to the late medieval Black Death. So while the latter may have been a contributory factor to Renaissance, it is not a direct or causal relationship.
To a certain extent, the plague brought about the downfall of the Yuan Dynasty[1] and allowed the founding of the Ming Empire.
You might want to read a bit about the Ming Dynasty. It was one of the most innovative empires in worldwide history. However, instead of innovating science and technology, they innovated governmental administration via meritocratic exams, and arts and literature.
Information about Ming Innovations in statecraft and painting and other stuff was transmitted to Europe and the rest of the world starting with Matteo Ricci and the Jesuits towards the end of the Ming. It can be said - I need a solid source for this, will update soon - that centralized European states were loosely based on the Ming administrative methods, as reported by Ricci.
[1] Wikipedia says that
outbreak of plague in the Chinese province of Hubei in 1334 claimed up to eighty percent of the population. China had several epidemics and famines from 1200 to the 1350s and its population decreased from an estimated 125 million to 65 million in the late 14th century.
The Yuan dynasty went from ~1270-~1370, so this huge population loss is during this same time. All these deaths are not entirely attributable to the plague, hence the disclaimer phrase "to a certain extent". Note that its a 50% population loss, and a population loss of this extent leads to huge social instability.
