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1. In an in finite horizon capital/consumption model, if kt and ct are the capital stock and consumption at time t, we have f(kt) = ct+kt+1 for t ≥ 0 where f is a given production function, and the total utility to be maximized is
where U is a given period utility function and β ? (0; 1) is a discount factor. Rephrase this as a standard (in finite horizon) control problem and write its Bellman equation.
2. Consider the discrete time control problem:
subject to x0 = x; xt+1 = g(t; xt; ut) for t = 0; : : : ; T - 1 (here f; g are C1, xt; ut ? R, x ? R given). Rewrite this as a Lagrangian optimization problem with 2T +2 variables (x0; : : : ; xT ; u0; : : : ; uT ) and T + 1 constraints. By applying the Lagrange condition to this problem, recover the maximum principle for the control problem (necessary conditions).
3. Consider the problem
subject to the initial and terminal conditions x0 = a; xT = b. One may think of it as a control problem by setting ut = xt+1-xt. Find the minimum and the optimal x *0 ; : : : ; x*T in two ways: directly (eg by Lagrangian method); and by writing the fundamental equation of dynamic programming for and computing Js(x) by backwards induction.
4. Consider the dynamic programming problem with \extended memory":
subject to xt+1 = g(t; xt; xt-1; ut) (x0; x-1 are given). Rephrase as a standard dynamic programming problem (with twice as many state variables).
1. Calculate the annual interest that you will receive on the described bond-A $500 Treasury bond with a current yield of 4 .2% that is quoted at 106 points? 2. Compute the tota
Realtors estimate that 23% of homes purchased in 2004 were considered investment properties. If a sample of 800 homes sold in 2004 is obtained what is the probability that at most
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Define A*B where: A = | 3 -3 6 | B = | 6 1 | | 0 4 2 | | 0 -5 |
which one is greater -4 4/25 or -4.12?
The perimeter of a rectangular swimming pool is 60m. The length of the pool is 4 m more than the width. What is the width of the pool?
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I have a question that hurts my head to work out. It is really confusing for me. It sais " By the start of the 21st century, only 1 in 6 babies in America was born with blue eyes.
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