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4. Consider a Lucas tree economy with additional endowments. We assume that the endowment y can take on two values with equal probability in each period, a high value given by y", and a low value given by y. In addition to the endowment, income has dividends from two types of trees, each of which has a supply of one. The type-h tree produces one dividend when output is high (y = y") and no dividends when output is low. The type- tree produces one dividend when output is low and none when output is high. Therefore, total output in period t inclusive of dividends is yr + 1, where y, = y" with probability 0.5 and y, = y with probability 0.5. Neither the endowment nor the dividend from the tree is storable. (a) Set up the social planning problem where the social planner maximizes the expected present value of utility of consumption subject to c S y, + 1. What is the optimal value for a? Explain. (b) The representative consumer can purchase stock in each type of tree, set , and $41: where the aggregate quantity of each type of stock is unity. One unit of s , (s, , ) will provide one unit of dividend when output is high (low) and no dividends when output is low (high). The representative agent's problem is stated below. max 1 = ( + d; ) s/ + (14 + d) s' + by + yx; lim BE, (d' (a+1) Pitasi+ +1) =0 1=h,l; lim 3" E, (u'(city ) bit+1) = 0; so, to given, where p; is the price of the i'th type of stock, Ry is the gross interest rate, by is a risk-free bond in zero net supply, and , has the interpretation as cash-on-hand. Set up Bellman's functional equation with a multiplier on the cash-on-hand constraint. In the Bellman equation, explicitly take the expectation. (c) Write the first order conditions for an interior solution. (d) Use the envelope theorem to substitute out the marginal value of r next period. Also, substitute equilibrium values for consumption and dividends, based on the realization of the endowment state. (e) Solve for the gross interest rate when current output is high. Compare its value with its value when the current output is low. Explain intuitively. (f) Define it = pr (y' ) 1 () + 1) + p" (2") 12 (2" + 1) B = P (1) 1 (8 + 1) + p(2) 1(2"+1). where pr (y' ) is the price of the type-h stock when output is low, and other prices are analagous. The prices with tildes have the interpretation of marginal utility-weighted prices for the type-h and type- stock. Use your first order equations to construct a difference equation for the marginal utility-weighted price of the type-h stock and solve it. You can infer the solution for p, without actually solving. Provide an intuitive interpretation of the solutions. (g) Now use first order conditions and the solutions for the prices with tildes to solve for of (y" ) and pr (u') .Endowments: Endowments are contingent on which of the 3 possible employment states the worker is in. When unemployed, the worker receives income b per period. Also with probability a he gets an offer of employment at a wage w ~ F(.) on [0, w] where w > b. When employed, the worker faces two possibilities. With probability & the job is destroyed and he moves to unemployment. Or, with probability ), he is moved to "temporary lay-off" (in that case he is not occupied in the job but the job still exists). When on temporary lay-off, the worker receives b each period and with probability a draws a wage offer from F(.). The difference from unemployment is that he can also get recalled back to his old job at his old wage. This happens with probability p. It is also possible, with probability 6, that while he is on temporary lay-off his former job gets destroyed in which case he looses any connection to the firm and becomes unemployed. If a temporary lay-off worker gets hired by someone other than his former employer the connection with the former employer is lost. Note: in each state the events that can happen to him are assumed to be mutually exclusive. This requires that a + 6 + At p 0 units of the non-storable consumption good. Old people receive de where o > 0. (a) Write down the problem of a Social Planner who weights each generation equally and solve for the implied efficient stationary allocation. b) Write down and solve the problem faced by a generation t household given a market for inside money. (c) Define a competitive equilibrium. Write down the market clearing conditions and solve for the endogenous variables (including any prices) associated with the stationary allo- cation. (d) In terms of the model parameters, obtain the lowest value of o (call it of) for which the laissez-faire competitive equilibrium is Pareto efficient. Explain your answer. (e) Now suppose o 0 (U) where Cf represents consumption, Gr is government spending, and 1- & is leisure, with unity as the time endowment and &, as labor supply. Output is produced only with labor according to: (Y) The government's budget constraint is given by (G) and the agent's budget constraint by Bit1 = (1 + r) Be + M (1 - 71) -C, (FBC) where By are bonds, 7, is the output tax rate. (a) Derive the first order condition on By+1 and solve for the Euler equation between current and future consumption. Derive the first order condition on labor and use it to solve for the equilibrium relationship between labor and consumption. (b) Aggregate equation (FBC) across agents imposing the equlibrium values for B, and Bi+1. (Note that there are no government bonds since the government's budget constraint is always balanced). Use the resulting equation, together with equation (Y), to solve for f, as a function of parameters and possibly 74. Then substitute into equation (Y) to solve for output as a function of parameters and possibly 7. Explain how an increase in the tax rate affects labor supply and output.1. In this question, we will use the framework of the real business cycle model to study con- sumption in the face of anticipated productivity shocks. Welfare of the representative agent is given by FEo [InGi + , In(1-4)1. 0 b. Whenever employed, the wage (in the same job) is subject to change with probability y. New wages are also drawn from F(.). Note: (i) the worker does not get a choice to stay on the old wage but can quit to unemployment, (ii) this is not a one-time wage adjustment - as long as the worker remains employed at the firm he is subject to further wage adjustments. Also when employed with probability & the job is destroyed and the worker moves to un- employment. Assume that the events of getting a wage adjustment and job destruction are mutually exclusive (i.e. not independent) and that ) + 9