In Acts 16:31 the word "believe" is in the aorist tense (undefined action). In Acts 16:34 the word "believe" is in the perfect tense. What conclusions can be drawn exegetically from the change in verb tenses? In light of 16:34, would it be accurate to conclude that the "believe" of 16:31 was a command to continually believe?
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We should back up first to v. 30 and revisit the question the jailer is asking Paul and Silas:
It is to this question Paul and Silas are replying in v. 31.
The answer is πίστευσον, which is a 2nd person singular aorist active imperative. The imperative implies a command or request. They are given an object in which to believe ("the Lord Jesus") as well as an outcome or goal of belief ("and you will be saved", σωθήσῃ is 2nd person singular future passive indicative):
As you pointed out, the aorist says the least possible about the verb 'to believe.' All we can say from this passage is that in order for the jailer (and his household) to be saved, he needed to believe in the Lord Jesus.
Continuing on in this context to v. 34, the jailer is rejoicing in his belief:
πεπιστευκὼς is a nominative masculine singular perfect active participle, so the second translation is technically more literal. However, the first translation captures more of the nuance of the perfect participle in this context. It should be noted that the perfect participle is generally antecedent with reference to the main verb. In other words, the emphasis here is not on the fact that he had come to believe (along with his household), but rather that he rejoiced (the belief was the circumstance for rejoicing).
In light of all this, you really can't argue from these passages that there is a command to 'continually believe.' This is a narrative where Paul and Silas simply told the jailer what he must do in order to be saved: believe upon the Lord Jesus. After he and his household hear the word of the Lord2 and are baptized, the jailer rejoices because he has come to believe.
It seems that you may be confusing the meaning of the perfect (as do many). I would highly encourage you to read a modern intermediate grammar such as that by Daniel Wallace or David Black. Here is a quote from Wallace's Greek Grammar Beyond the Basics (pp. 573-574) that you may find helpful in understanding the perfect tense:
1 It should be noted that most translators translate πανοικεὶ at the end of this clause in order to imply that the entire household both rejoiced and had come to believe (as evidenced by their baptism in v. 33). I chose to translate it literally in both instances since moving it to the end of the clause is an interpretive decision that is not under direct consideration in this question.
2 Or "the word of God," there is a manuscript discrepancy here.
Paul and Silas and the Philippian jailer were probably on different wavelengths, as it were, when in response to the jailer's question, "Sirs, what must I do to be saved?" they responded by saying "Believe in the Lord Jesus, and you will be saved, you and your household."
The jailer was likely thinking of his immediate circumstances and the dilemma he faced should the prisoners entrusted to him escape. He was so paralyzed with fear (16:29) that he contemplated suicide (v.27), but Paul wisely assured him his prisoners would not be escaping. Still fearful that the news of the event would find its way to his superiors, perhaps through his employees who had been ordered to bring light into the jail cell (v.29), the jailer's primary concern at that point was for his own life.
Just as the guards at Jesus' tomb would have been punished for allowing Jesus to "escape" from the tomb had the chief priests and elders not bribed the soldiers, given them a "cover" story, and assured them they would also bribe Pilate if necessary, so also would the Philippian jailer have been punished. If, perhaps, he had been "on probation" for a previous mistake or indiscretion, his punishment would have been death. We simply do not know.
In conclusion, the jailer's--and his family's--belief in the Lord Jesus did not occur immediately after the earthquake but only after he had brought Paul and Silas into his home (which was likely attached to the prison) and heard them speak the word of the Lord. As evidence of his conversion to Christ, the jailer demonstrated the "fruit of repentance" by washing the missionaries' wounds and by feeding them.
Interestingly, Paul and Silas did not tell the jailer to join a church and only after having taken some pre-baptismal classes be baptized! No, they baptized him and his family right away. The "having believed" in verse 34, then, indicates the transformation that had occurred in the jailer's (and his family's) life after hearing the word of God and believing it. The "undefined action" you allude to vis a vis the first "belief" became the "concrete action" of the second "having believed" and repented, complete with a kind of "restitution," an indication of genuine conversion.