Some research has been done on M/M/1 queueing systems with lead-time-and price-dependent demand. However, most of these works assume a constant production cost. It is known that the longer the quoted lead-time the better the firm can manage the production and reduce the cost. The idea of this paper stems from this observation. Indeed, we investigate the lead-time quotation decision in an M/M/1 make-to-order queue while assuming the production cost to be a decreasing function in lead-time. We consider two settings: (1) lead-time is variable but price is fixed, and (2) price and lead-time are both decision variables. For each setting, we provide an approach to find analytically the optimal lead-time and price (if it is a variable). We use the optimal solutions to conduct experiments and derive some insights.