final10 - Final Exam EC720.01 Math for Economists Boston...

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Unformatted text preview: Final Exam EC720.01 - Math for Economists Boston College, Department of Economics Peter Ireland Fall 2010 Due Tuesday, December 14, 2010 at 11:00am This exam has three questions on five pages; before you begin, please check to make sure your copy has all three questions and all five pages. Each of the three questions is worth 20 points; as indicated at the beginning of the semester, this 60-point final will account for 60 percent of your course grade and the problem sets will account for the remaining 40 percent of your course grade. This is an open-book exam, meaning that it is fine for you to consult your notes, textbooks, and other references when working on your answers to the questions. I expect you to work independently on the exam, however, so that the answers you submit are yours and yours alone. 1. Growth and Pollution, Part I This question asks you to use the Kuhn-Tucker theorem to solve a static constrained optimization problem that can be used to think about patterns of pollution and environmental quality seen when looking across a sample of countries at different stages of economic development. Suppose, in particular, that a country has k > 0 units of physical capital that can be used to produce output of a single consumption good using any one from a continuum of technologies indexed by z ∈ [0, 1]. Lower values of z represent “cleaner” technologies that generate less pollution and higher values of z represent “dirtier” technologies that generate more pollution. People dislike pollution, but they also like consumption; and because dirtier technologies can be used to produce output at a lower cost, a trade-off arises between economic growth and environmental quality. To make these ideas more specific, suppose that the country’s k units of capital, when used with technology z , produce c = Akz units of output, where A > 0 is a parameter governing the overall level of productivity across all available technologies. Hence, according to this specification, more capital used with dirtier technologies yields more output. Suppose also, however, that the same k units of capital, when used with the same technology z , generate p = Akz β units of pollution, where β > 1 is a parameter governing how the use of dirtier technologies causes environmental quality to deteriorate. Suppose, finally, that a representative consumer in this country gets utility u( c) = c 1− σ − 1 1−σ 1 from consuming c units of output but suffers disutility ￿￿ B v ( p) = pγ γ when the level of pollution is given by p, where the preference parameters satisfy σ > 0, B > 0, and γ > 1. A social planner for this economy takes the stock of physical capital k as given, and chooses the levels of consumption c and pollution p to maximize the representative consumer’s total utility subject to the constraints imposed by the description of the technologies from above. Although there are a number of ways to formalize the social planner’s problem, perhaps the easiest is to start by observing that because preferences over consumption are such that limc→0 u￿ (c) = ∞ and because the cleanest technology z = 0 yields no output, the constraints c ≥ 0 and z ≥ 0 will never bind at the optimum and can therefore be dropped from further consideration. On the other hand, for certain parameter configurations at least, it may be optimal to chose the dirtiest technology z = 1; hence the constraint 1 ≥ z must still be accounted for. Substituting the technological relations c = Akz and p = Akz β into the utility and disutility functions u(c) and v (p) then allows the social planner’s problem to be written as one with a single choice variable and a single constraint: ￿￿ (Akz )1−σ − 1 B max − (Akz β )γ subject to 1 ≥ z. z 1−σ γ a. Define (write down) the Langragian for this problem. b. Next, write down the full set of first-order conditions, constraints, nonnegativity conditions, and complementary slackness conditions that, according to the Kuhn-Tucker theorem, characterize the value z ∗ that solves the social planner’s problem together with the associated value of the Lagrange multiplier. c. Now use your results from part (b), above, to solve for the optimal choice of z ∗ as a function of the model’s parameters: k , A, β , σ , B , and γ . d. Next, use your result from part (c), above, to obtain an expression that shows how total pollution depends on the model’s parameters k , A, β , σ , B , and γ when z ∗ is chosen optimally. e. Finally, in order to obtain a sharper economic characterization of your result from part (d), above, suppose that the preference parameter σ is larger than one: σ > 1. Under this extra assumption, what does your result imply for the relationship between the economy’s level of economic development, as measured by its stock of physical capital k , and its optimal choice of pollution? Looking at a cross-section of countries, under what circumstances will more developed countries have better environmental quality? Under what circumstances will more developed countries have worse environmental quality? 2 2. Growth and Pollution, Part II This question asks you to use the maximum principle to characterize the solution to a dynamic version of the problem that you just considered in a static context. The dynamics are added to the model by allowing the social planner to increase the capital stock k over time by allocating output to investment instead of consumption. Suppose that time is continuous and the horizon is infinite, so that periods can be indexed by t ∈ [0, ∞). Let k (t) denote the economy’s stock of physical capital at time t and, extending the specification from above, suppose that by using a technology indexed by z (t) during time t, the economy produces Ak (t)z (t) units of output but also generates p(t) = Ak (t)z (t)β units of pollution where, as before, A > 0 and β > 1 are parameters. Now, however, suppose that in addition to choosing an optimal technology z (t) at each date t ∈ [0, ∞), the social planner must also choose how to optimally divide up output Ak (t)z (t) into an amount c(t) to be consumed and an amount Ak (t)z (t) − c(t) to be invested. If, in addition, the parameter δ , with 0 < δ < 1, measures the depreciation rate of physical capital, then the stock of capital evolves according to ˙ Ak (t)z (t) − δ k (t) − c(t) ≥ k (t) for all t ∈ [0, ∞). With the additional dynamics introduced through physical capital accumulation complicating the problem, assume now that the constraints 1 > z (t) and z (t) > 0 can be ignored, either because there are ever “dirtier” technologies than the dirtiest z = 1 considered before or because the configuration of parameters insures that these constraints will not bind at the optimum. Then, with the specification for the utility over consumption u[c(t)] = c ( t ) 1− σ − 1 1−σ with σ > 0 and the disutility from pollution ￿￿ B v [p(t)] = p( t ) γ γ with B > 0 and γ > 1 the same as before, the social planner’s problem can be stated as follows: given the initial capital stock k (0), choose continuously differentiable functions c(t), z (t), and k (t) for all t ∈ [0, ∞) to maximize ￿ ￿￿ ￿ ￿∞ c ( t ) 1− σ − 1 B − ρt βγ e − [Ak (t)z (t) ] dt 1−σ γ 0 subject to ˙ Ak (t)z (t) − δ k (t) − c(t) ≥ k (t) for all t ∈ [0, ∞), where ρ > 0 denotes the discount rate. 3 a. Write down the Hamiltonian for the social planner’s problem. b. Next, write down the first-order conditions and the pair of differential equations that, according to the maximum principle, characterize the solution to the social planner’s problem. c. The conditions you derived for part (b), above, should form a system of equations involving the not only optimal choices of c(t), z (t), and k (t) but also the associated values for a new variable, corresponding to the Lagrange multiplier on the capital accumulation constraint that you would have introduced into the problem if you had decide to solve it using the Kuhn-Tucker theorem instead. Use one of your optimality conditions to eliminate this new variable from the system, so as to obtain a smaller set of three equations involving only the model’s parameters and the three variables c(t), z (t), and k (t) that enter into the original statement of the planner’s problem. d. The smaller system you derived for part (c), above, implies that starting from any initial value of k (0), there are unique values of c(0) and z (0) that place the economy on a saddle path that converges to a steady state, in which c(t) = c∗ , z (t) = z ∗ , and k (t) = k ∗ for all t. Use your answers from part (c), above, to derive a system of three equations that can be used to solve for the steady-state values c∗ , z ∗ , and k ∗ in terms of the model’s parameters: A β , δ , σ , B , and γ . Note: you don’t have to actually solve for the steady-state values; just write down the three-equation system that would, in principle, allow you to find these values. 3. Stochastic Growth This problem asks you to use the guess-and-verify method to characterize the solution to a version of the stochastic growth model. Suppose that time is discrete and the horizon is infinite, so that periods can be indexed by t = 0, 1, 2, .... Let output produced during each period t = 0, 1, 2, ... be given by yt = At kt , where kt is the beginning-of-period capital stock and At is a random shock to productivity. By consuming ct units of output at each date t = 0, 1, 2, ..., the representative consumer gets expected utility ∞ ￿ E0 β t ln(ct ), t=0 where the discount factor satisfies 0 < β < 1. Letting δ , with 0 < δ < 1, denote the depreciation rate for physical capital, the evolution of the capital stock is described by At kt + (1 − δ )kt − ct ≥ kt+1 , a constraint that must hold for all t = 0, 1, 2, ... and all possible realizations of the productivity shock At at each date t = 0, 1, 2, .... 4 Assume, finally, that the productivity shock At gets realized at the very beginning of period t, before ct and kt+1 are chosen, and that the total return on capital, accounting for depreciation, K Rt = At + 1 − δ follows the first-order autoregressive process K K ln(Rt+1 ) = (1 − ρ) ln(R) + ρ ln(Rt ) + εt+1 , where the parameter ρ, satisfying 0 < ρ < 1, measures the degree of serial correlation in the K productivity shock, the parameter R, satisfying R > 0, determines the average value of Rt , and where εt+1 is a serially uncorrelated shock, satisfying Et εt+1 = 0 for all t = 0, 1, 2, ..., K that gets realized at the beginning of period t + 1. Hence, when ct and kt+1 are chosen, Rt K is known but Rt+1 is still viewed by the representative consumer as random. The representative consumer’s problem can now be stated as: given the initial capital stock k0 , choose contingency plans for ct , t = 0, 1, 2, ..., and kt , t = 1, 2, 3, ..., to maximize E0 ∞ ￿ β t ln(ct ), t=0 K subject to the constraints (which are more conveniently rewritten in terms of Rt ) K Rt kt − ct ≥ kt+1 K for all t = 0, 1, 2, ... and for all possible realizations of Rt at each date t = 0, 1, 2, .... a. Write down the Bellman equation for the representative consumer’s problem. b. Next, guess that the value function takes the time-invariant form K K v (kt , Rt ) = E + F ln(kt ) + G ln(Rt ) where E , F , and G are unknown constants to be determined, and, using this guess, write down the first-order and envelope conditions that help characterize the consumer’s optimal decisions. c. Use your results from part (b), above, to solve for the constant F in terms of the model’s parameters β , R, and ρ. d. Use your results from parts (b) and (c) to derive a pair of equations that can be used to construct the optimal choices for ct , t = 0, 1, 2, ..., and kt , t = 1, 2, 3, ..., for a given K value of the initial capital stock k0 , a given set of realizations for Rt at each date t = 0, 1, 2, ..., and for given values of the model’s parameters β , R, and ρ. 5 ...
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This note was uploaded on 02/29/2012 for the course ECON 720 taught by Professor Ireland during the Fall '09 term at BC.

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