The view that Christ's death saves primarily by its example of self-giving love that morally influences sinners to repent and love God in return. No objective transaction occurs on the cross — no satisfaction of justice, no propitiation of wrath, no substitutionary penalty-bearing. The cross is a spectacular demonstration of love intended to melt hard hearts. Associated historically with Peter Abelard (1079-1142), who developed it in part as a reaction against Anselm's Satisfaction Theory, and with 19th-century liberal Protestants (Horace Bushnell, Hastings Rashdall) who found penal substitution morally offensive.
The Moral Influence Theory captures something true and something false. True: the cross does powerfully display God's love (Romans 5:8 — "God shows His love for us in that while we were still sinners, Christ died for us"), and does morally move believers to respond in love (2 Corinthians 5:14 — "the love of Christ controls us"). No Reformed theologian denies this dimension. False: the theory as a complete account of atonement makes nonsense of why the cross was necessary. If God can simply forgive sin on the basis of inspiring sinners to better behavior, why the cross at all? Why not a sermon, a miracle, a book? The brutality of Calvary becomes gratuitous — almost cruel theater — unless something objective was being accomplished. Scripture insists something objective was accomplished: a legal penalty was paid (Isaiah 53:5), wrath was propitiated (1 John 2:2), sin was put away by sacrifice (Hebrews 9:26), the law's curse was exhausted (Galatians 3:13), the enemy was disarmed (Colossians 2:15). The moral-influence dimension only works because the objective accomplishment is real. Remove penal substitution and you still have moving rhetoric — but no actual forgiveness. The Moral Influence Theory is a true melody played on a broken piano.