An exhaust gas purifier equipped with a catalyst converter including an early-activation catalyst converter and a pre-catalyst converter for purifying harmful exhaust gas discharged form an engine evaluates a characteristic of the exhaust gas purification rate of the catalyst converter from the cooled condition to the warmed condition of a catalyst to diagnose a deterioration of the early-activation function of the catalytic converter. The purifier compares a rising part of the evaluated characteristic of the exhaust gas purification rate with a rising part of a reference characteristic and judges that the early-activation catalyst converter is deteriorated if a difference between these rising parts is beyond a predetermined value, and compares a steady state part of the evaluated characteristic of the exhaust gas purification rate with a steady state part of the reference characteristic and judges that the pre-catalyst converter is deteriorated if a difference between these steady state parts is beyond a predetermined value.