Satisfaction of divine justice
Several inadequate atonement theories have been propounded throughout church history. They are classified as inadequate mainly because they focus on just one point (which may be true) to the detriment and neglect of the others.
Having said that, it is true to say that the atonement of Calvary was made God-ward primarily and man-ward secondarily. It is certainly vicarious and representative, in that Christ died in the stead of sinners (1 Peter 2:24; 3:18); more than that it is a satisfaction.
Paul pinpoints the central cause of the cross and says that it was to vindicate the righteousness of God, to show that in forgiving and justifying sinners, God is still just. He is able to do it and He is right in doing it. "But now the righteousness of God without the law is manifested, being witnessed by the law and the prophets, even the righteousness of God which is by faith of Jesus Christ unto all and upon all them that believe...whom God hath set forth to be a propitiation through faith in his blood, to declare his righteousness..." (Romans 3:21-31).
Holiness (and hence absolute justice and the upholding and honouring of His law) is God's fundamental attribute. Therefore it is not only to be expected but it is everywhere emphasized that Christ's death supplies the necessary satisfaction for sin so that it may be cancelled and forgiven, and sinners received into God's presence.
The Law sent Jesus to the cross. The law and the cross are intimately related. The (broken) law spells the need of atonement, and in the atonement the law is implied.
Man, having dishonoured God's law, has incurred His displeasure and condemnation. God rightly exacts the penalty of a broken law. He cannot act capriciously, freeing sinners from guilt and condemnation as long as the law is not honoured. If He were to do so, He would be denying Himself, that is, denying His intrinsic and inherent holiness. God must punish sin, much more so than the state is bound to punish criminals.
God's justice is that moral excellence which demands the righteous distribution of rewards and punishments which renders it certain, under the government of God, that obedience will be rewarded and sin punished.
Christ's death was a penal satisfaction. Sinners could not make amends for their own sins; therefore God, impelled by His love, provided Christ the Substitute to become personally responsible for sins He did not commit, that is, the sins of others. In this way God's justice is honoured, seeing that Christ voluntarily bore the sins of many (Isaiah 53).
Christ's death is also a satisfaction to the Law of God. The law demands far more than the punishment of sin, and therefore satisfaction to the law includes more than the satisfaction of vindicatory justice. Under the gospel believers are delivered from the law (Romans 8:1ff.). Not because the law is abrogated, but because Christ obeyed the law on their behalf and took its punishment upon Himself. By His obedience and sufferings, by His whole righteousness, active and passive, He did and endured all that the Law demands. This He did as our representative and substitute.