1. Consider the problem of a value maximizing perfectly competitive firm whose profit function at time t is given by OK? where O denotes the productivity level, K, denotes capital and a is a parameter representing the elasticity of output with respect to capital. The law of motion for capital is standard and is given by Kt+1 = (1 - 8)Kt + It where It is investments in capital at time t and & is the depreciation rate. The price of a unit of capital good is normalized to 1 and investment is subject to smooth convex installation cost given by C ( It, Kt ) = 12, where y > 0 is a constant. Time is discrete and runs to infinity, t = 0, 1... o. Future values are discounted with the factor B. The productivity level for the firm is stochastic and follows an independent and identically distributed (i.i.d) Normal process: at "d N (0, 02 ) , t = 0, 1, 2, ... 0. Productivity 0 is thus assumed to be serially uncorrelated with zero mean (conditional and un- conditional) and constant variance o2. Based on the above information, answer the following questions: 1.2 Write down the Bellman equation. Derive and interpret the optimal investment decision condition. [20%] 1.2 Define marginal Q and provide an economic interpretation. [15%] 1.3 How is average Q related to marginal Q for this firm? [15%]