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Reading assignment: Sections 10-12, 28. 1. Consider an agent who values consumption in period O and 1 according to the following utility function: of is a discount factor (a $ 1) which indicates that the agnet prefers to consume today more than be can tomorrow. Suppose that the agent is given a total wealth today of w and that he may save any portion of this money in order to consume tomorrow. If he saves money he is paid interest r. This the agents budget constraint is: Itr - =0 (a) By renaming variables, notice that this problem is identical to the regular cobb douglas problemn. (b) Solve for the agents marshallian demand functions. (c) At what relation of & and r will the agent consume the same amount in each period? (d) Suppose that instead of being given a fixed endowment, the agent instead has access to a production technology that can make two products s, if that can be sold at prices p., p- The technology is limited by inputs so that at most: Pty=2 Solve for the optimal amount of r and y produced that maximize the agents profits. (e) Suppose that r is the amount of production that can be made in period 0 and y is the amount of production made in period 1. Suppose that pr =1, p, = 1,r= 0 Why can we seperate the production decision from the consumption decision in this problem? Give a value for s such that consumption is held constant in each period and the agent exactly consumes his production. 2. The legrangian multipliers in economies are often called "shadow costs". In this problem we will try to see where this term comes from and build intuition on how to use them to answer economic relevant questions. (a) Lets start with a very simple problems. W = max 2r- 12 I ST : ICKConsider Ramsey's model with human capital based on Lucas' (1988) paper, with the following modified human capital formation technology: h = shy h, " (1 - 4,) he is the average human capital in the economy in period t and y E (0,1). How does the value of y affect growth rate in the economy and its possible divergence from the welfare maximizing growth rate?Consider a variation to the baseline Solow growth model without population or technological progress. The per-capita production function is given by yt = f(kt) = kta, where 0