A tax-payer makes a gross income of $40, which he must decide to declare or hide. An inspector must decide to audit the tax-payer or pass. If the tax-payer is not audited, he pays to the inspector an income tax equal to 25% of $40 if he declared his income, and $0 if he hid it.

If the tax-payer is audited, he pays to the inspector an income tax of 25% of his true gross income plus a penalty equal to 10% of his true gross income. The inspector incurs a cost of $5 in case she decides to audit the tax-payer.

(a) Write down the strategic game between the tax-payer and the inspector that is associated with this situation. Does this game have a Nash equilibrium in pure strategies? Explain.

(b) Compute the mixed-strategy Nash equilibrium of this game.