A public hearing will be held June 24 at City Hall to discuss the proposed $317 million fiscal 2014 budget that relies on a nearly $4 million increase in taxes to avoid a deficit. Members of the grassroots group Brocktonians for Limited Taxation have decried such a hike, saying it can be averted by drawing on some of the approximately $6 million in a stabilization account. Some city officials say the funds may be needed to settle union contracts or for other uses at a time that departmental budgets are already spread too thin. Last week, city councilors voted on budget cuts that included more than $50,000 from the law department, about $100,000 from the treasurer’s office for debt service, and $20,000 from a snow-removal account. New spending included $86,000 for a planning and economic development chief, part of a plan to combine the two departments and draw more business to the city.