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Dynamic Optimization 03.378 Dynamic Optimization BP/LP (nur für das Wahlpflichtfach Umweltökonomie) 2st Di 10-12, Geoma

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## Dynamic Optimization 03.378 Dynamic Optimization BP/LP (nur für das Wahlpflichtfach Umweltökonomie) 2st Di 10-12, Geoma

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**Dynamic Optimization03.378 Dynamic Optimization BP/LP(nur**für das Wahlpflichtfach Umweltökonomie) 2st Di 10-12, Geomatikum Uwe A. Schneider www.fnu.zmaw.de**Topics**• Review Integration, Simple Differential Equations • Calculus of Variation • Optimal Control Theory • Phase Diagrams / Stability Analysis • Dynamic Programming**Integration and Dynamic Optimization**• Integration often needed for solving systems of differential equations • Euler's Equation of COV • Hamiltonian of Optimal Control • more difficult than differentiation • computers and look-up tables make life easier**Simple Integration Rules**• Sums: • Powers: • Constant coefficients: • Log: • Exponentials:**Variable Separation**• Derived from chain rule of differentiation**Integration by Parts**• Derived from product rule of differentiation**Integration by Substitution**• Derived from chain rule of differentiation**Integrating Factor**• A function by which an ordinary differential equation is multiplied in order to make it integrable • Used to aid solving • linear first and higher order differential equations • bernoulli equations • non-exact equations • several others • Makes a differential equation look like a known antiderivative • A given differential equation may have zero, 1, or more integrating factors**Differential Equations, 1**• Equations that involve dependent variables and their derivatives with respect to the independent variables are called differential equations • Ordinary Differential Equation: Differential equations that involve only ONE independent variable**Differential Equations, 2**• Order: The order of a differential equation is the highest derivative that appears in the differential equation • Degree: The degree of a differential equation is the power of the highest derivative term. … Second order of degree 3**Differential Equations, 2**• Linear … if there are no multiplications among dependent variables and their derivatives … all coefficients are functions of independent variables • Non-linear … do not satisfy the linear condition • Quasi-linear … if there are no multiplications among all dependent variables and their derivatives in the highest derivative term**Differential Equations, 3**• Homogeneous … if every single term contains the dependent variables or their derivatives. • Non-homogeneous … Otherwise • Autonomous … if independent variable does not appear in the equation • Exact vs. nonexact (see later slides)**Linear First Order Differential Equations (FODE)**• Constant coefficients: • Variable coefficients:**General Solution to Linear FODE**Integrating Factor**General Solution to Linear FODE**Integrating Factor**Linear Second Order Differential Equations (FODE)**• Constant coefficients: • Variable coefficients:**Bernoulli Equation**• Fundamental equation of motion for neo-classical growth models with Cobb-Douglas technology • Use integrating factor: (1-a)y-a • Substitute k by z = y(1-a) yielding: • Solve this linear first order differential equation • Substitute z by y = z1/(1-a)**Exact Equations, 3**• Integrate to find (y) • Note that (y) is a function of y only. Therefore, in the expression giving '(y) the variable, x, should disappear. • Write down the function F(x,y) • All solutions given by F(x,y) = k • Alternatively, one can integrate over y and use (x)**Calculus of Variation**• Find path x(t), which optimizes • F(.) is twice differentiable • Starting and end points are known**Derivation of First Order Conditions**• Decompose x(t) into optimal path x*(t) + deviation a*h(t) • Setup • Evaluate g’(0) = 0 (Optimum is where g' = 0 and a = 0)**First Order Necessary Condition**= Euler Equation**Second Order Necessary Conditions**= Legendre condition • Maximization: • Minimization:**Sufficient Conditions**• F is jointly concave in x and x' for a maximization • F is jointly convex in x and x' for a minimization**Production and Inventory Planning Example**• Need to deliver B units at T • Production cost rise linearly with production rate • Cost of holding inventory is constant per unit of time • Zero inventory at beginning • Minimize total cost**Problem Extensions**• Fixed starting point - free end value • Fixed starting point - free horizon • Transversality conditions • Salvage value • Several Functions**Free Starting Value, Solution**Free Starting and End Value, Solution**Limitations**• Functional constraints • Continuity • Differentiability • Integrability (to solve for x(t)) • Set-up • Continuous decisions • Simple functions**Optimal Control (OC)**• Generalization of Calculus of Variation • Pontryagin, L.S. et al. late 1950's • State variable(s) … x(t) • Control variable(s) … u(t) • can deal with corner points • can have binary variables (0,1)**OC – Simplest Problem**Objective function } State equation Boundary Conditions**Relationship to COV**• letting u(t) = x'(t) • yields calculus of variation problem with free end value