Solving consumption models with multiplicative habits

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Precautionary Saving and Precautionary Wealth (2006, background): “For example, consider an agent with additive habit formation (as distinct from multiplicative habits, cf. Carroll (2000)), for whom reduced consumption not only increases assets but reduces the size of the consumption habit, and so unambiguously leads to more willingness to bear risks.”
Estimated General Equilibrium Models for the Evaluation of Monetary Policy in the Us and Europe (2005, background): “+ 1 1 + υβ εt − ζt) and b st = b wt + b Nt − b yt which reflects labour’s share of income and is equivalent to the log-linearised marginal cost of production given our Cobb-Douglas production function, (8).”
Growth, habit formation, and catching-up with the Joneses (2005, methodology): “Finally, by taking the derivative of (6) with respect to the Lagrange multiplier λt, the solution to the optimization problem involves also to satisfy the resource constraint (4) , and λt · [f(kt)− ct − (1 + n) kt+1 + (1− δ) kt] = 0, (15)”