LONDON, June 5 (Reuters) - Oil prices rose by more than $1 a barrel on Monday after top crude exporter Saudi Arabia pledged to cut production by a further 1 million barrels per day (bpd) from July to counter macroeconomic headwinds that have depressed markets. The voluntary cut is on top of a broader deal by the Organization of the Petroleum Exporting Countries (OPEC) and allies including Russia to limit supply into 2024 as the OPEC+ producer group seeks to boost flagging oil prices. "The immediate market impact of this Saudi cut is likely lower, as drawing inventories takes time, and the market likely already put some meaningful probability on a cut today," the bank's analysts added