Dynamic tax evasion with audits based on visible consumption

We solve the problem of a representative agent who maximises the expected present utility of his intertemporal consumption under the assumption that an optimal fraction of his wealth is hidden to the tax authorities (we show conditions under which evasion is expedient). Evasion affects the capital dynamics in two ways: the growth rate of capital increases because some taxes are not paid, but when caught evading the consumer has to pay a fine (proportional to evasion). Consumption can be allocated between ordinary goods and so-called visible goods. The latter are used by the Government for targeting the audit, since they are considered like an indicator of consumer's income. In fact, the probability of being caught is a function of the distance between the actual and the presumed consumption in visible goods. We find a closed form solution to the dynamic optimization problem and show how fiscal and audit parameters affect the optimal evasion and the optimal allocation between the two types of consumptions.