In all questions of this nature, we warn the querier that we cannot give a conclusive answer after hearing only one side, as even two honest people can have different viewpoints of the same events. While we often refuse to answer at all and say that the two sides must either be heard in beit din or employ some other permitted dispute resolution framework, we cannot ignore your question of whether you should make an issue of the matter at all. Therefore, we will briefly discuss general sources and factors.
The client has the advantage in a disagreement between a client and a worker over the amount that was set for payment, due to the rule that one who wants to extract payment requires proof.[1] The client would be obligated, however, to take a serious oath that he does not owe more than he admits. Thus, if the client claims that he is uncertain how much he owes, he should have to pay, because he is unable to take that oath.[2] Nonetheless, when he is incapable of knowing how much he has to pay, the rule that one who cannot take the oath must pay does not apply.[3]
In a situation such as yours, in which it is clear that the side that is claiming payment will know how much time he put in and the one is being asked to pay will not know, the rules are somewhat different. According to a mishna that discusses a particular agent who claims he incurred expenditures on behalf of someone,[4] as well as another mishna that discusses the case of a husband who made improvements in his wife’s field before divorce,[5] the plaintiff swears about how much he spent and is then reimbursed. The Mordechai,[6] Maharik,[7] and Rama[8] understand this as a broad rule regarding plaintiffs who know about their expenditures and defendants who do not. The plaintiff is believed and receives payment upon taking an oath supporting the veracity of his claim.
The above appears contradicted by the halacha that one who seeks reimbursement for expenditures that resulted from unreasonable steps taken by the other litigant must actually prove how much he spent; merely taking an oath is not sufficient to justify his claim.[9] The S’ma[10] distinguishes between cases in which the plaintiff worked for the benefit of the other side, in which case he takes an oath and is reimbursed, and cases in which he acted against the other side’s will, in which case he must prove his expenditures. The Shach[11] distinguishes between cases in which the defendant requested of the plaintiff to make the outlays and cases in which the plaintiff acted on his own accord. Part of the logic is that when Reuven asks Shimon to do something that deserves reimbursement without demanding proof from the outset, he is in effect granting trust in the veracity of Shimon’s charge.
The obligation to pay wages is equivalent to that of paying expenses. In your case, the S’ma and Shach would agree that you should believe the technician, who worked on your behalf and based on your request. Certain cases could arguably be exceptions. One is when you have strong grounds to believe the technician is lying.[12] Another is if the technician should have informed you when he figured out the extent of the cost, enabling you to decide whether it was worthwhile to have the computer fixed.[13]
It is generally best to research a professional’s reliability before you hire him, and if you have heard favorable reports, to trust him. While it is your prerogative to not use this technician in the future, refusing to pay in full is a drastic step. Some situations may lend themselves to expressing (in a menschlich way) your displeasure and suggesting that your willingness to use him again depends on a reduction in price. There are so many unclear factors that it is difficult to give firm advice as to what to do, and without hearing the other side it is certainly wrong to even attempt to tell you who is right.