* Steel and iron ore open with modest gains
* Overall steel demand remains tepid
* Mills buy more higher-grade iron ore, lifting prices
SHANGHAI, Jan 2 (Reuters) - Chinese steel futures ticked higher on the first trading day of the new year, but concerns remain about tepid consumption in the world's top consumer.
Steel demand across the country is seasonally waning as construction activities slow due to cold weather.
"The sharp fall in physical steel prices in the last week of 2017 has encouraged some limited restocking by traders, which stabilized the physical market, but the overall demand remains weak and commercial inventories keep rising," said a trader in Shanghai.
Stockpiles of construction product rebar held by Chinese traders in big cities rose for a second week to 2.973 million tonnes, up 1.5 percent from a week ago, according to data tracked by SteelHome website. SH-TOT-RBARINV
The most active rebar on the Shanghai Futures Exchange SRBcv1 edged up 0.8 percent to 3,824 yuan a tonne by 0235 GMT, after declining 2.9 percent last week.
Steel mills in 28 major producing cities are cutting production until mid-March as part of the government's efforts to combat smog, leading them to turn to higher-grade iron ore to reduce pollution while still maximizing output.
Iron ore on the Dalian Commodity Exchange DCIOcv1 rose 1.7 percent at 539 yuan.
Coke DCJcv1 inched up 0.4 percent to 1,995.5 yuan a tonne, while coking coal DJMcv1 slipped 0.2 percent to 1,312 yuan.
Iron ore for delivery to China's Qingdao port .IO62-CNO=MB settled almost unchanged at $72.61 a tonne on Dec. 29 from the previous day, according to Metal Bulletin.