This paper explores how farm productivity affects poverty, and how various factor market constraints affect farm productivity. The empirical analysis draws on representative surveys of farm households in Kilimanjaro and Ruvuma, two cash crop growing regions in Tanzania. We find that poorer households do not only possess fewer assets, but are also much less productive. We find that agricultural productivity directly affects household consumption and hence overall poverty and welfare. Stochastic production frontier analysis indicates that many farmers are farming well below best practice in the region. Analysis of allocative efficiency suggests that family labour is substantially over utilized, a sign of considerable excess labour supply. Use of intermediate inputs on the other hand is well below what is commensurate with the estimated value of their marginal productivities. An important reason for low input use is lack of credit to purchase inputs, but difficult access to the inputs themselves, being connected to the economy, and food security and self insurance considerations are also important impediments. Easy access to credit is positively associated with being a member of a savings association or being in a contractual arrangement with a cooperative or firm. The findings support a continuing emphasis on increasing agricultural productivity in designing poverty reduction policies. Better agronomic practices and increased input use will be crucial in this strategy. Financial constraints might be relieved through fostering institutional arrangements facilitating contract enforcement and institutions that facilitate saving by the households themselves. They may also be relieved by the provision of more adequate consumption safety nets.