I'm struggling with this formula.

kt+1 CtBasics on dynamic versus static models. In most ntroductory courses, you are focusing on static models. The equilibrium of supply and demand, the intersection of the indifference curve and the budget constraint. the price quantity decision of a monopolist all are static decisions. A dynamic model is needed when choices made today influence the choice set available in the tuture. State variables: evolve over time in response to decisions following some kind of transition process (Markov for example). Decision / choice / control variables: decisions made in a time period. In continuous time we use optimal control models (Hamiltonians) and in discrete time we use dynamic programming models (Bellman equations). We solve dynamic models for a steady state where there 1s no mnternal foree in the model that waill lead to further change 1n the state variables. We then look at the path to the steady state. Starting from an initial state. do the forces internal to the model force us towards or away from this steady state? Can have multiple steady states. Can have convergent paths and divergent paths. Sedentarisation zone 1680808Extensive pastoralism zone Recentery zone Costly accumulation zone 5 4 i = 10years In (Herd size, (+1) (solid) i = 1 year (dashed) 0 0 2 3 4 5 6 7 In (Herd size, I) Nadaraya-Watson estimates using Epanechnikov kernel with bandwidth (h = 1.5) Fig. 4. Nonparametric Estimates of Expected Herd Size Transition Functions Lybbert et al. (2004)Define ct as consumption, U(ct ) as the utility of consumption, kt as the capital stock, and g(kt ) as the growth of capital in period t produced as a function of capital stock in period t (births in a herd, interest on an account, production from a machine). In my work. I work a lot with livestock, so say k is the size of the herd, c is how many you eat, and g(k) records how many are born. Beta is the discount factor, like (1/1+r). V[K+] = max U(C+) + B * V[kt+1] ct K+ +1 = k+ + g(k+) - Ct These you can solve for a steady state consumption level and herd size (where the marginal utility of consumption equals the discounted shadow value of capital stock; Take V[k,] = max U(c) + B * V[k+ + g(k;) - ct] Ct au av And the first order condition for solving this is thus: act dkt+1 Marginal utility of consumption equals the discounted shadow value of a marginal unit of capital in the future. Assume concavity of the value function and utility function; low k means high shadow value, implies low c. From a high capital stock, reverse the logic, low marginal utility of consumption (hence high consumption) low shadow value of capital (high herd size). K and c will increase together according to this simple model