(Requires Some Calculus)In the Following, Assume That X_t Is Strictly

question 28

Essay

(Requires some calculus)In the following, assume that X_t is strictly exogenous and that economic theory suggests that, in equilibrium, the following relationship holds between Y^* and X_t, where the "*" indicates equilibrium.
Y^* = kX_t
An error term could be added here by assuming that even in equilibrium, random variations from strict proportionality might occur. Next let there be adjustment costs when changing Y, e.g. costs associated with changes in employment for firms. As a result, an entity might be faced with two types of costs: being out of equilibrium and the adjustment cost. Assume that these costs can be modeled by the following quadratic loss function:
L = λ₁(Y_t - Y^*)² + λ₁(Y_t - Y_t-1)²
a. Minimize the loss function w.r.t. the only variable that is under the entity's control, Y_t and solve for Y_t.
b. Note that the two weights on Y^* and Y_t-1 add up to one. To simplify notation, let the first weight be θ and the second weight (1-θ). Substitute the original expression for Y^* into this equation. In terms of the ADL(p,q)terminology, what are the values for p and q in this model?

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