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9. One-shot DMP model (inspired by Albrecht, Navaro and Vroman (2010]) Time: One period Demography: A unit mass continuum of workers and a larger mass of firms who can create individual vacancies. Workers are ex ante heterogenous, they are indexed by y ~ F on [0, 1] (which will be their individual productivity level). F(.) is a continuous distribution with density f(.). Preferences: Both firms and workers are risk neutral. Productive technology: An employed worker of type y produces y units of the consumption good. If a worker does not find employment they will receive b > 0 units of the consumption good. Creating a vacancy costs a units of the consumption good. Matching technology: All workers begin the period unemployed. The probability that a worker meets a firm is m(v) where v is the measure of vacancies created. The function m(.) is twice differentiable, strictly increasing and strictly concave. We have m(0) = 0, m'(0) = 1 and lim,-.. m(v) = 1. Let ,(v) = m'(v)u/m(v), the elasticity of m(v). Then, assume further that q(v) 0. Households receive labor income and profits from firms, and pay labor income taxes to the government. Households can store their assets Ky, and earn zero returns on the stored assets. As usual, assume that assets held at the beginning of period # + 1, Ki+1, are chosen in period . Note that capital is used only as a storage device, and not as a factor of production. Households face the usual initial, non-negativity and No-Ponzi-Game conditions. The government collects taxes from labor income and maintains a balanced budget rule: (BB) Taxes are driven by government spending. which follows an AR(1) process around the log of its steady state value: 3 = In (G/G..) =09-It, Ogdcl, (TS) where { } is an exogenous i.i.d. process, and G.. > 0 is steady state government spending. Individual consumers and producers are sufficiently small to take r, as well as G, as given. (a) Define a competitive equilibrium (b) Write down the firm's problem and find the first order conditions that maximize profits (c) Write down the household problem in a recursive form (Bellman's equation and constraints), and find the first order conditions that maximize household utility. (d) Let lower-case letters with carats " " denote deviations of logged variables around their steady state values, and letters with subscript,, denote steady state values. Log linearize the wage equation (the FOC of the firm), Euler equation, and the equations that characterizes the labor-leisure trade-off. Note that to analyze the next question, you had better to substitute 7 and W, as functions of L, and G- (e) Suppose the economy is hit by a temporal expansionary fiscal policy shock (e, > UnderSection 1. (Suggested Time: 45 Minutes) For 3 of the following 6 statements, state whether the statement is true, false, or uncertain, and give a complete and convincing explanation of your answer. Note: Such explanations typically appeal to specific macroeconomic models. 1. As long as we have existence, the number of equilibria does not matter when it comes to comparative statics 2. Without immigration certain jobs in developed economies would not get done. 3. Raising tariffs on goods and services imported from the rest of the world will increase the average real wage of US workers. 4. The repeal of the Affordable Care Act will raise aggregate output in the long-term. 5. Output and inflation are positively correlated. 6. In dynamic general equilibrium macroeconomic models, the equilibrium gross interest rate (one plus the interest rate) tends to equal the inverse of the discount factor.7. Consider an economy composed of heterogeneous agents who can live for a maximum of two periods. Let s denote the survival rate from the first to the second period. Agents derive utility from consumption c according to utility function In(c) if alive, and derive zero utility if deceased. Agents discount utility of the second period with a discount factor of B. In the first period, an agent receives an endowment w from a distribution with a probability density function f (w), and chooses consumption cy, savings k, and the amount of annuity a. The gross return on savings is (1 + r). Each unit of annuities is sold at a uniform unit price of q, and gives one unit of goods in the second period if the agent survives to the second period, and zero units of goods otherwise. In the second period, living agents choose how much to consume, which is denoted by c2. Formally, the life-cycle problem for an agent who receives wage w is written as follows: max In(c ) + AsIn(c2) C,ca.ka subject to ci + k + qu = w = (1+r)k+ a Annuity contracts are offered by an insurance firm which earns zero profit: (1 +r) / a(w)af(w)d(w) = (1+o) / sa(w)f(w)d(w) where a(w) denotes the choice of annuity for agents of wage w, and o is a loading factor, measuring the operation cost of the insurance firm. Note that the return of premium revenue is the same as that of private savings. For simplicity, we further assume 3(1 + r) = 1. (a) Write down the Bellman's equation for the first period. You need to be clear about the state and choice variables. (b) Solve the agents' problem and derive the first order conditions. Note that you should have inequalities for these first order conditions, since the optimal choice is not an internal solution. Intuitively explain the meaning of the left hand side and right hand side of each condition. (c) Suppose there is no operation cost of the insurance firm and the loading factor is zero 0 =0, show that agents will have zero savings. (d) Under what conditions about o, would agents choose to have zero annuities? (e) If o= 0, how would the elimination of the annuity market affect welfare (ex-ante utility of agents)? (f) If agents optimally choose to have zero annuities, would the elimination of the annuity market affect welfare (ex-ante utility of agents)? 8. Seigniorage in Sidrauski model with log-linear preferences Time: Discrete, infinite horizon Demography: A continuum, mass normalized to 1, of (representative) infinite lived con- sumer/worker households. There is a large number, mass N, of firms owned jointly and equally by the households. Preferences: The instantaneous household utility function over, consumption, cy, and real money balances, my, is yIn(c.) + (1-y) In(m;) where y E (0, 1) and In(.) indicates the natural logarithm function. The discount factor is B E (0, 1). Technology: Aggregate output, Y, = F(Kt, It ) where K, is the aggregate capital stock and Ly = 1 is the aggregate labor supply. The function F(.,.) is twice differentiable, strictly increasing in both arguments, concave and exhibits constant returns to scale. It will be helpful to use f (k) as the output per worker where k, is the capital stock per worker. Capital depreciates by a factor o in use each period. Endowments: Each household has one unit of labor and an initial endowment of capital ko. Each also has an initial nominal money holding Ho. Information: Complete, perfect foresight. Institutions: Competitive markets in each period for capital, the consumption good, labor and money. There is a government that has to meet the exogenous sequence of per capita expenditures, g. To do so, it issues new money and buys output from the market. Its policy instrument is the nominal money growth rate, o, such that #, = (1 + @) H-1. There are no cash transfers to households. The government injects money entirely by purchasing goods in the goods market for newly generated cash. (a) Using Mi+1 as period f nominal money demand, A, as the period t price of the consump- tion good, r, as the rental rate on capital and w, as the wage paid per effective unit of labor, write down and solve the household's problem.7. Diamond "style" OG model with concerned parenting Time: discrete, infinite horizon, t = 1, 2, ... Demography: A mass N, = N for all t of newborns enter in period f (i.e. no population growth). Each person is born into a household or dynasty. Everyone lives for 2 periods except for the first generation of old people who live for one. Preferences: for the generations born in and after period 1, U(at, (2#1, $1+1) = m(C) + Buz(Cat+1) + Bu. (8#+1) where car is consumption in period f and stage i of life and s,+1 is the amount of old age consumption good given up in the process of providing human capital h for the young person in their household. For the initial old generation (c2, ) = us(Q,) tu,(s;). The utility functions, u.(.) / = 1, 2, s are all twice differentiable and concave. The function u,(.) represents the extent to which a parent cares about the quality of their child's future skill set. Productive technology: The production function available to firms is F( H, N) where H is the human capital stock and / is the number of workers employed. F(.,.) is a constant-returns-to-scale standard neoclassical production function which satisfies the Inada conditions. (You may find it convenient to use the implied per young person production function, f(h) where h is the human capital stock per worker.) Foregone consumption (or saving) by the old is converted one-to-one into the human capital of their child. Children cannot create human capital on their own account. So, the only way they can accumulate human capital is if their parents spend the required resources on their behalf. Endowments: Everyone has one unit of labor services when young. (The old survive by renting their human capital.) The initial old share an endowment, H, of capital so they have hi units each. Institutions: There are competitive markets every period for labor and human capital. (You can think of a single representative firm which takes wages and interest rates as given.) (a) Write down and solve the problem faced by the individuals born in period t. (b) Write down and solve the representative firm's problem in period t. (c) Write down the market clearing condition for human capital and define a com- petitive equilibrium. (d) Solve for an equation that characterizes the dynamics of the human capital stock. Now suppose 1 (c) = uz(c) = In(c), u.(s) = Bs where B is a constant and f (h) = And where A is a constant. (e) Obtain any steady state value(s) of human capital per worker, h', and characterize the dynamical properties. 15 (f) Draw the phase diagram (g) Now, without doing the work, state how the model analysis would differ if each dynasty started out with a different initial human capital endowment say distribe uted uniformly on [0, (a.A) ]? & Consider the following variant of the Lucas tree model. The preferences of the repre- sentative consumer are E. ( 3'0, I(ci)), 0
