Mary has preferences over two goods x and y described by the utility function U(x, y) = x^2y. She has a budget of $3000 and must satisfy her budget constraint 3000 = Px*x + Py*y, where Px and Py are prices of x and y respectively. 1. Assume that Px is $10 and Py is $5. Write down Mary’s constraint optimization problem and derive the first order conditions using the Lagrange multiplier method. 2. Solve the first order conditions for critical point(s) and apply second order sufficient conditions to classify it (them). 3. Eliminate the constraint by substituting it into the objective function and rewrite Mary’s problem as a one variable unconstrained optimization problem.
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Step 1
Step 1: Mary's constraint optimization problem can be written as: Maximize U(x, y) = x^2y Subject to: 3000 = 10x + 5y The Lagrangian for this problem is: L(x, y, λ) = x^2y + λ(3000 - 10x - 5y) The first order conditions are: ∂L/∂x = 2xy - 10λ = 0 ∂L/∂y = x^2 - Show more…
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1. State any theorems/conditions that you use in determining your solutions. a) State the necessary and sufficient conditions for a function f(x,y) to have a stationary point subject to the constraint condition g(x,y) = c where c is a constant. b) Find the critical values for minimizing the costs of a firm producing two goods x and y when the total cost function is c(x,y) = 8x^2 - xy + 12y^2 and the firm is bound by contract to produce a minimum combination of 42 goods i.e. x + y = 42. You must use the Lagrange multiplier. c) By calculating the Bordered Hessian show that you have obtained a minimum value in part (b) above. d) Given a budget constraint of £108 optimize the generalised Cobb-Douglas production function: q(K,L) = K^0.4 L^0.5 the constraint being expressed as: 3K + 4L = 108 where K represents capital and L represents labour. By using the Lagrange multiplier and vector calculus (the gradient of a scalar field) only optimize q(K,L) subject to this constraint. e) A cubical construction is being made with volume, V and surface area A. Using vector calculus only, find the critical value of V(x,y,z) = xyz Subject to A = 2xz + 2zy + xy = 12 Hint: You do not need to determine the Bordered Hessian for the critical values; you simply have to locate it.
Adi S.
Use a CAS to perform the following steps implementing the method of Lagrange multipliers for finding constrained extrema: a. Form the function $h=f-\lambda_{1} g_{1}-\lambda_{2} g_{2},$ where $f$ is the function to optimize subject to the constraints $g_{1}=0$ and $g_{2}=0 .$ b. Determine all the first partial derivatives of $h$ , including the partials with respect to $\lambda_{1}$ and $\lambda_{2},$ and set them equal to $0 .$ c. Solve the system of equations found in part (b) for all the unknowns, including $\lambda_{1}$ and $\lambda_{2} .$ d. Evaluate $f$ at each of the solution points found in part (c) and select the extreme value subject to the constraints asked for in the exercise. Minimize $f(x, y, z, w)=x^{2}+y^{2}+z^{2}+w^{2}$ subject to the constraints $2 x-y+z-w-1=0$ and $x+y-z+w-1=0$
Partial Derivatives
Lagrange Multipliers
Use a CAS to perform the following steps implementing the method of Lagrange multipliers for finding constrained extrema: a. Form the function $h=f-\lambda_{1} g_{1}-\lambda_{2} g_{2},$ where $f$ is the function to optimize subject to the constraints $g_{1}=0$ and $g_{2}=0.$ b. Determine all the first partial derivatives of $h$, including the partials with respect to $\lambda_{1}$ and $\lambda_{2},$ and set them equal to 0 . c. Solve the system of equations found in part (b) for all the unknowns, including $\lambda_{1}$ and $\lambda_{2}$. d. Evaluate $f$ at each of the solution points found in part (c) and select the extreme value subject to the constraints asked for in the exercise. Maximize $f(x, y, z)=x^{2}+y^{2}+z^{2}$ subject to the constraints $2 y+4 z-5=0$ and $4 x^{2}+4 y^{2}-z^{2}=0.$
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