* Profits tax to be trimmed for first HK$2 mln
* New subsidised housing scheme to be launched
* Increase in R&D for sputtering tech sector
* Hong Kong needs to be "united, harmonious" society (Adds details, property share price fall and story link)
HONG KONG, Oct 11 (Reuters) - Hong Kong's Chief Executive Carrie Lam warned in her maiden policy speech on Wednesday that the city faced "grave" challenges and must develop a more diversified economy, unveiling a mix of housing and tax relief policies to raise competitiveness.
Hong Kong, one of the world's costliest cities, has battled rising income inequality, the slow implementation of marquee public projects, political tensions with its mainland China and a slide in its competitiveness.
"In the face of competition from other economies as well as the rise of protectionism in recent years, Hong Kong is facing increasingly grave challenges. We must develop a high value-added and diversified economy," Lam said.
Hong Kong has seen its lucrative position as the world's gateway to China eroded as the mainland rapidly builds up its own ports, airlines and financial powerhouses, and opens its markets to foreign investors.
Lam said she would bolster support for small and medium enterprises in the Asian financial hub by cutting company profits tax to 8.25 percent from 16.5 percent for the first HK$2 million ($256,000) of earnings. Earnings after that would be taxed at the current 16.5 percent.
On Hong Kong's soaring housing prices, Lam said that despite a raft of property cooling measures, the government had "no magic wands".
Lam pledged to increase land supply where possible and launch a new subsidised "starter homes" scheme to help families not eligible for cheap-rental public housing. The first phase would provide around 1,000 residential units.
"Even if our housing policy has broad community support, it takes time to find land for increasing the housing supply," conceded Lam, Hong Kong's first female leader.
Hong Kong residents are squeezed into an average living space of just 150 square feet (14 square metres) per person, and apartments are the most expensive in the world, according to a recent UBS report ranking 20 global cities including New York, London, Tokyo and Paris.
Even residents with good jobs and wages have struggled to get on the property ladder.
Chinese President Xi Jinping also voiced concern over the city's property market when he visited on July 1 for the 20th anniversary of the former British colony's handover to China.
Some observers felt Lam's housing initiatives were not bold enough, with local property shares giving up initial gains and closing the day down 1.8 percent.
"Property shares are down because of the lack of mention of farmland conversion to build first homes. But the chief executive cannot be too specific, so the sell-off doesnt reflect the real picture," said Nicole Wong, a property analyst with CLSA.
Lam also said Hong Kong would aim to double expenditure on research and development over the next five years, to 1.5 percent of annual GDP from 0.73 percent, in a bid to bolster its sputtering tech prospects with neighbouring Chinese city Shenzhen having galloped ahead in recent years.
She said the government needed to be "more proactive", and to seize opportunities from China's Belt and Road push to extend trade and transport networks to Europe.
She also said the city needed to better integrate with China's Guangdong province as part of a regional economic development blueprint.
Income inequality is at its highest level in over four decades in the city of 7.3 million people, stoking discontent that has seen large-scale protests in recent years over calls for more affordable housing as well as democracy.
Since taking office on July 1, Lam has sought to heal social divisions amid growing tensions with China, and to forge a softer and more socially engaged leadership style than her predecessor, the staunchly pro-Beijing Leung Chun-ying.
Britain returned Hong Kong to China in 1997 under a "one country, two systems" formula which guarantees wide-ranging autonomy and judicial independence not seen in mainland China.
But tensions have heightened in recent years amid concern over Beijing's interference in Hong Kong, sparking large scale pro-democracy protests and some calls for outright independence from China.
Lam said Hong Kong has the responsibility "to say 'no' to any attempt to threaten our country's (China's) sovereignty, security and development interests," while re-iterating that it was the "constitutional responsibility" of the government to implement new national security laws, known as Article 23.
"We need to have a society that is united, harmonious and caring," she said. ($1 = 7.8045 Hong Kong dollars)
(Additional reporting by Venus Wu, Umesh Desai and Anne Marie Roantree; Editing by Michael Perry and Kim Coghill)