Calvin had a way of cutting to the heart of things when he wanted to. In chapter 6 of Book 3 of the Institutes he discusses the Christian life, the object of God’s regenerating (life-giving) work in our hearts by the Spirit, a life lived in obedient harmony with God’s righteousness. He points out that, over the years, various moral philosophers have given capable enough accounts of what we ought to do and why we ought to do it. (3.6.1) Now, they’re good as far as they go, but, of course, scripture gives far better reasons, rooting our motive for righteousness more securely, among other reasons, in God’s own holiness, our desire to be in communion him, and a desire to be numbered among those inhabitants of the holy city. (3.6.2) But Calvin goes further and says that, as great as these are, scripture gives us a deeper reason still:
And to wake us more effectively, Scripture shows that God the Father, as he has reconciled us to himself in his Christ [cf. 2 Corinthians 5:18], has in him stamped for us the likeness [Hebrews 1:3] to which he would have us conform. Now, let these persons who think that moral philosophy is duly and systematically set forth solely among philosophers find me among the philosophers a more excellent dispensation. They, while they wish particularly to exhort us to virtue, announce merely that we should live in accordance with nature. But Scripture draws its exhortation from the true fountain. It not only enjoins us to refer our life to God, its author, to whom it is bound; but after it has taught that we have degenerated from the true origin and condition of our creation, it also adds that Christ, through whom we return into favor with God, has been set before us as an example, whose pattern we ought to express in our life. What more effective thing can you require than this one thing? Nay, what can you require beyond this one thing? For we have been adopted as sons by the Lord with this one condition: that our life express Christ, the bond of our adoption. Accordingly, unless we give and devote ourselves to righteousness, we not only revolt from our Creator with wicked perfidy but we also abjure our Savior himself.
Then the Scripture finds occasion for exhortation in all the benefits of God that it lists for us, and in the individual parts of our salvation. Ever since God revealed himself Father to us, we must prove our ungratefulness to him if we did not in turn show ourselves his sons [Malachi 1:6; Ephesians 5:1; 1 John 3:1]. Ever since Christ cleansed us with the washing of his blood, and imparted this cleansing through baptism, it would be unfitting to befoul ourselves with new pollutions [Ephesians 5:26; Hebrews 10:10; 1 Corinthians 6:11; 1 Peter 1:15,19]. Ever since he engrafted us into his body, we must take especial care not to disfigure ourselves, who are his members, with any spot or blemish [Ephesians 5:23-33; 1 Corinthians 6:15; John 15:3-6]. Ever since Christ himself, who is our Head, ascended into heaven, it behooves us, having laid aside love of earthly things, wholeheartedly to aspire heavenward [Colossians 3:1 ff.]. Ever since the Holy Spirit dedicated us as temples to God, we must take care that God’s glory shine through us, and must not commit anything to defile ourselves with the filthiness of sin [1 Corinthians 3:16; 6:19; 2 Corinthians 6:16]. Ever since both our souls and bodies were destined for heavenly incorruption and an unfading crown [1 Peter 5:4], we ought to strive manfully to keep them pure and uncorrupted until the Day of the Lord [1 Thessalonians 5:23; cf. Philippians 1:10]. These, I say, are the most auspicious foundations upon which to establish one’s life. One would look in vain for the like of these among the philosophers, who, in their commendation of virtue, never rise above the natural dignity of man.
-John Calvin, Institutes 3.6.3
To sum up: Why does Calvin say we obey? Because God has saved us in Christ.