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Infrequent Stock Transactions. Consider a representative consumer at time t seeking to maximize the sum of discounted lifetime utility from t on, CBS u( CIts ) $=0 subject to the infinite sequence of flow budget constraints Pict + Sial = Sidi-2 + Dial 2 + Y where the notation is as in class: a, is holdings of a real asset (a "stock" ) at the end of period t, S, is its nominal price in t, D, is the nominal dividend that each unit of assets carried into t from period t-2 pays out, Y, is nominal income in t, & is consumption in t, and P is the nominal price of each unit of consumption in t. Note well how the budget constraint is written: it is assets accumulated in period t-2 that pay off in period t - thus, in this model, stocks (for some reason. ..) must be held for two periods, rather than being able to be traded every period. Construct the Lagrangian to compute the stock price S, in period t. Explain intuitively how and why the stock price differs from that in the model studied in class, in which all shares can be traded every period.An increasingly common utility function used in macroeconomic applications is one in which period-t utility depends not only on period-t consumption but also on consumption in periods earlier than period t. This idea is known as "habit persistence," which is meant to indicate that consumers become "habituated" to previous levels of consumption. To simplify things, let's suppose only period-(t-1) consumption enters the period-t utility function. Thus, we can write the instantaneous utility function as u(c, G-1). When a consumer arrives in period t, G_ of course cannot be changed (because it happened in the past). (a) In a model in which stocks (modeled in the way we introduced them in class) can be traded every period, how is the pricing equation for S, (the nominal stock price) altered due to the assumption of habit persistence? Consumption in which periods affects the period-t stock price under habit persistence? To answer this, derive the pricing equation using a Lagrangian and compare its properties to the standard model's pricing equation developed in class. Without habit persistence (i.c., our baseline model in class), consumption in which periods affects the stock price in period t? (b) Based on your solution in part a and the pattern you notice there, if the instanta- neous utility function were u(c, "-1, "-2) (that is, two lags of consumption appear, meaning that period t utility depends on consumption in periods t, t-1, and t-2), con- sumption in which periods would affect the period-t stock price? No need to derive the result very formally here, just draw an analogy with what you found above