Would innovation and productivity decrease or increase under democratic communism relative to capitalism?
On the one hand, the existence of employment of last resort (EoLR), where a person could make a living wage working 60 percent of a standard week (24 hours per week) in a relative easy job, erases the trade-off that capitalism enforces on workers, i.e. work as hard as possible as long as possible or starve. I think that the frenetic pace of those employed under capitalism would diminish considerably. We could expect a decrease in labor intensity and labor hours per person under democratic communism. Furthermore, it is impossible to get white-boy wealthy under democratic communism. A strongly progressive income tax and the abolition of private ownership of the means of production puts an upper bound on individual wealth. (The people democratically decide where that upper bound is. It's possible the people could decide to make some people as wealthy as Bill Gates, but unlikely, and what the people giveth, the people can taketh away.) Thus, democratic communism cuts the bottom and the top off of capitalist incentives: no one starves, and no one can become obscenely wealthy. To the extent that these incentives work, by design, democratic communism loses them, and what they incentivize.
On the other hand, EoLR means that no one is involuntarily unemployed. Total employment should increase. Note that because the government directly manages natural monopolies, many people working under EoLR will be doing directly productive things. And, of course, the provision of public goods, which are difficult to make immediately profitable under capitalism, should increase.
In addition, a lot of production will still be essentially private (although worker-owned and -operated). Businesses still have the producer-surplus incentive: a more efficient firm incurs less than the overall socially necessary cost, and thus returns more money to its workers. Inefficient firms will not have enough money to pay its workers competitive wages (or the EoLR wage), and will shed workers or go bankrupt. Moreover, both the quantity and cost of investment is decided by the people; the people can, if they choose, accept a low or even negative rate of return to pursue social goals of making production more efficient (as well as making life "better"), and EoLR (and other social welfare measures that a democracy can implement) will ensure that falling real labor costs will not collapse aggregate demand. Democratic communism replaces the marginal profitability of capital with marginal social benefit. Social benefit is, of course, more difficult to measure, but so what? We may already be at the point (secular stagnation) where the marginal profit of capital is approaching zero; if we're not already there, Marx argues that we will be there sooner or later.
It's also the case that the capitalist incentives for innovation don't directly incentivize the people who actually innovate. Work or starve kills creativity, and engineers and scientists, even creative advertising and marketing workers, do not receive great wealth; the wealth goes to their bosses, who let a little trickle down to the actual innovators. If innovation is good, or to the extent that particular innovations are good, then individuals in a democracy should want to innovate. The only need to privatize wealth would be to innovate contrary to what the majority (or a sizable minority) actually wants.