We use linked employer-employee microdata for New Zealand to examine the relationship between firm-level productivity, wages and workforce composition. Jointly estimating production functions and firm- level wage bill equations, we compare migrant workers with NZ-born workers, through the lens of a derived "productivity-wage gap" that captures the difference in relative contribution to output and the wage bill. Whether we look at all industries using a common production function, or separately estimate results for the five largest sectors, we find that skilled and long-term migrants make contributions to output that exceed moderately-skilled NZ-born workers, with that higher contribution likely being due to a mix of skill differences and/or effort which is largely reflected in higher wages. Conversely, migrants that are not on skilled visas are associated with lower output and lower wages than moderately-skilled NZ-born, also consistent with a skills/effort narrative. The share of employment for long-term migrants has grown over time (from 2005 to 2019) and we show that their relative contribution to output appears to be increasing over the same period. Finally, we present tentative evidence that high-skilled NZ-born workers make a stronger contribution to output when they work in firms with higher migrant shares, which is suggestive of complementarities between the two groups or, at least, positive mutual sorting of these groups into higher productivity firms.