This paper investigates the topping-up scheme in health insurance when both public and private firms use linear contracts. First, the case with identical consumers is analyzed. The optimal public coverage is derived both when the firms play simultaneously and when they play sequentially. In the former case consumers are over-insured, whereas, in the latter case, the second-best allocation is obtained. Then, consumers' heterogeneity is introduced: consumers differ in their wage rate and labour supply is endogenous. It is assumed that public coverage is uniform and health expenditures are financed by linear taxation. Results show that, in the sequential game, the optimal public coverage is negative and consumers are under-insured.