There include several different forms of capital punishment....

why are we the only first world country that still has capital punishment.

In the United States only 38 states have capital punishment statutes.

Many countries that use capital punishment have now adopted lethal injection, because it's thought to be less cruel for the offender and less brutalising for the executioner.

The laws in the United States have change drastically in regards to capital punishment.

Capital punishment is wrong for many reasons....

mained relatively infrequent until the early 1990s; the frequency then escalated rapidly to a peak of 40 in 2000. Thereafter, there has been drop-off to about 20-25 per year. plots the homicide rate in Texas (as well as California and New York) over the same period. The pattern for all three states closely resembles the U.S. national trend. From 1974 through the early 1990s the Texas homicide rate rose then fell and then rose again before falling steadily from 1991 to the early 2000s, when it leveled off. For the period from 1976 to 1991, there is no apparent relationship between the homicide rate and the frequency of execution. However, the steady decline in the homicide rate since 1991 does correspond with the dramatic increase in executions that occurred in the early 1990s. Thus, if the early 1990s is assumed to be the demarcation of Texas shifting to a dramatically higher use of capital punishment, the data are consistent with that shift having a deterrent effect.

Muhlhausen are two articles that support capital punishment as a deterrent of crime.

Such data do not exist. This gap is potentially a serious one for studying deterrence. If the severity of noncapital sanctions for murder is correlated with the legal status or the frequency of use of capital punishment, failure to account for the severity of noncapital sanctions may result in serious bias in estimates of deterrent effect. If, for example, capital punishment jurisdictions tended also to impose more severe imprisonment sanctions than noncapital jurisdictions, a reduced level of homicide in such jurisdictions may be attributable to these other features of their sanction regime and not to the death penalty. Or, if capital punishment jurisdictions are otherwise

The American culture’s morals have accepted the death penalty as a reasonable punishment.

Capital punishment has been a way of punishing people for many years.

I’m very sorry, but I think the 2,000 year tradition is something of a myth. Before Pope Innocent I’s permission in 405 for public officials to use torture and capital punishment there was nothing handed down in prior tradition (as Innocent I himself states). Some of the patristic sources cited by Feser and Bessette do not really show support for capital punishment (not even in principle). The teaching against euthanasia is definitive, and has been confirmed as such. Church tradition in support of capital punishment was never definitive. This is why the last three popes have all called for the end of the practice.

Capital punishment also has great flaws.

The claim that the traditional Catholic teaching regarding the death penalty may be superceded is in part a function of the assessment of what constitutes ordinary magisterium. But on this point, it is very difficult to find an account of the death penalty as not being ordinary magisterium that would not largely evacuate the ordinary magisterium of its content. The unanimous teaching of the Fathers—despite their Christian aversion to judgments of blood—that the penalty may be just, itself ought be sufficient (given the teaching of Trent) to indicate that this is not possible. The Church has never accepted that it is a properly papal power to abrogate ordinary magisterium, any more than it is a marital power to contracept or abort. On a grave moral matter, for two millennia, the Church has not only taught that the penalty may be just in some cases, it has also–in the Papal states–under direct papal authority applied the penalty, and approved many catechisms articulating this teaching, a teaching also articulated in various papal statements. Moreover, the middle term of the claim of development–human dignity–is the very reason that Scripture and the tradition take as justifying the penalty. I.e., 1)the unjust violation of the dignity of the person in murder merits the gravest penalty, and where consistent with the medicinal requirements of the common good such violation may justify the application of the penalty as expressing the moral judgment that some crimes clearly merit death; 2)the human person being created in the image of God, is such that just application of this penalty does not of necessity prevent the person from achieving his final end, if suffered with contrition and in expiation of sin; 3)it is the dignity of the human person that calls for punishment for crime as such–we do not tend to have trials assessing the knowledge and culpability of beasts when they kill human beings. Genesis 9:6 does not say that because man is created in the image of God the death penalty is not permitted, but indicates that it is the very dignity of the rational creature that founds the rationale for the death penalty when that dignity is criminally violated. There is a sense in which the death penalty is “per se contrary to the Gospel”–but not in the sense of injustice or absolute contrariness to human dignity. That sense is that both in the beginning, and in the beatific vision, death has no place. The eschatological witness of the Church in teaching that death has been swallowed up in victory is essential. But simply speaking, given the reality of the Fall and of horrifically evil action, the Church has taught for two millennia that killing—in the death penalty, in war, and in defense—may be just. It simply is not clear how a Roman Catholic can be obligated on these matters to become a Waldensian or Mennonite. On a lower but still important level of discourse, such a judgment would also seem to deny philosophically certain judgments regarding the transcendence of the common good, and its superiority over private good in any genus. When Christ was told by the “good thief”—whose crime may very well have been something like looting and murdering—that while Christ did not merit the death penalty, he himself did, Christ did not instruct him to the contrary. Rather, it was He who moved him to that contrition and desire for eternal life so as to open for him the portal to Heaven.

The recidivism rate for capital punishment is zero.

It says..that we don't value the victim's life enough to punish the killer fully."Lord Justice Denning, Master of the Rolls of the Court of Appeals in England said to the Royal Commission on Capital Punishment in 1950: