Tax evasion and R&D subsidy in a mixed market

In this paper I consider a mixed duopoly where a private firm competes with a public one. The private firm maximizes its profit and the public firm maximizes social welfare. R and D investments of firms are subsidized by the government, profits of both firms are taxed and the private firm might evade taxes. It is shown that tax evasion (decision not to declare part of profit) directly affects a rate of the optimal R and D subsidy and makes it dependent on a profit tax rate. While making a choice on the rate of the R and D subsidy the government takes into account both positive and negative effects of the subsidy on social welfare. On the one hand the R and D subsidy results in a decrease in total production costs because an allocation of total output between two firms is improved; on the other hand it results in an increase in social costs of tax evasion. The optimal R and D subsidy decreases with an increase in a profit tax rate and increases with an increase in tax evasion costs. An equilibrium allocation is not efficient, but an increase in tax evasion costs leads to the reduction of difference between efficient and equilibrium welfare levels.