This is a great question. The answer will entirely depend on the context. Let me start with an answer of "yes". If top quality is not important for a product then why not make things easier, so that productivity can be increased by sacrificing quality. For instance, products that do not need to last or products that are only used temporary do not need the highest craftmanship. In these cases, quality can be sacrificed. Now let me answer "no." For some products the sacrifice of quality may actually hurt productivity. If a product is expected to last and there are shortcuts in quality, then this can cause great problems down the line, which can decrease productivity. For example, what if there is a recall on a car, because it was not done well at first? Productivity will be hampered!
Productivity is defined as quantity produced per unit of input. The input used for calculating productivity could be any factor of production such as material, equipment, manpower or some combination of different types of factors of production. But most often productivity refer so productivity of manpower.
Whether productivity can be improved by sacrificing will depend on two factors. One, to what extent production can be increased by sacrificing quality. Also, to the extent reduction in quality will lead to acceptance of the output produced.
In some cases, minor changes concessions in quality requirement can result in substantial increase in rate of production. In other cases this difference may not be much. For example, Take the case of making packets containing 100 grams of groundnuts. If the the groundnuts have to be weighed very accurately, the process could be slow. This process can be speeded up considerably by designing a container of suitable size which will accommodate on average 100 grams of ground nuts. Using the system the rate of packing can be improved considerably. However this process will involve higher variation in the weight of groundnuts in each packet.
Coming to the second factor, the acceptability of the reduced quality level, when increase in production by relaxing quality is more than offset by reduction in value of out put due to poor quality, production can be increased by sacrificing quality. Otherwise productivity may actually decrease. For example, in case of the packets of groundnuts a variation of, say, 2% in weight of individual packets is not going to reduce the value of packets. It will be still possible to sell the packets at the designated price of 100 grams packets. But 2% variation in dimension of precision product would mean production of too many components that will be unusable and therefore will be rejected. IN a case like this perhaps trying to increase productivity be relaxing quality will not be a very good idea.