Recent empirical evidence has highlighted how the export patterns of multi-product firms dominate world trade flows, and how these multi-product firms respond to different economic conditions across export markets by varying the number of products they export. In this paper, we further analyze the effects of those export market conditions on the relative export sales of those goods: we refer to this as the firm's product mix choice. We build a theoretical model of multi-product firms that highlights how market size and geography (the market sizes of and bilateral economic distances to trading partners) affects both a firm's exported product range and its exported product mix across market destinations. We show how tougher competition in an export market - associated with a downward shift in the distribution of markups across all products sold in the market - induces a firm to skew its export sales towards its best performing products. We find very strong confirmation of this competitive effect for French exporters across export market destinations. Our theoretical model shows how this effect of export market competition on a firm's product mix then translates into differences in measured firm productivity. Thus, a firm operating a given technology will produce relatively more output per worker when it exports to markets with tougher competition. This productivity gain is further compounded by the effect of competition on the mix of exported products.