Vectors and Optimization Exercises
1. Vectors. Let vector a = (1;2) and vector b = (1;0): Plot these vectors on (x1; x2) space. Calculate vectors c = a + b and d = b a: Plot these vectors on the same diagram. Are any of the vectors, a; b; c; d orthogonal to each other? Calculate the scalar products of the corresponding vectors to prove your answer.
2. Vectors. Let vector a = (1;2;3) and vector b = (2; 1;3): Calculate vectors c = a + b; and d = a b and scalar products of vectors
ab ac bc
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3. Vectors and matrices. Consider vector a = (1; 1) and matrix B = 2 1 : 31
Note that a0 = 1 : This operation is called transpose, the notation is a0; (or T 1
a ) turns a string vector into the column vector and a column vector into the string vector. Calculate a B (this should be a vector of size 1 2) and B a0 (this should be a vector of size 2 1). Calculate scalar products a a0 (this should be a number) and a0 a (this should be a 2 2 matrix).
4. Di§erentiate the following functions (a) f (x) = 3×2
(b) f(x)= 3 x2
(c) f (x) = aex (d) f (x) = 1 e x
(e) f (x) = aln(x) (f) f(x)=h(g(x))
(g) f (x; y) = 3×3 + 21 y2
5. Implicit functions 1. Take the budget equation
p1x1 + p2x2 = w
Find the slope of the implicit function x1(x2). Show your work.
6. Optimization 1. A consumer seeks to maximise her utility by choosing how much of commodities A and B to consume. Let xA and xB denote the quantities demanded, and (pA;pB) the prices. Our consumer has utility
u(xA ; xB ) = ln(1 + xA ) + ln(1 + xB ); 1
and she is subject to the budget constraint
xApA+xBpB =M Find the optimal bundle (xA;xB):
7. Optimization 2. You need to enclose a rectangular Öeld with a fence. You have 100 meters of fencing material. Determine the dimensions of the Öeld that will enclose the largest area. Set this up as a constrained optimization problem and approach this with Lagrangean. Hints: use all the information to determine the objective function and the constraint. It may help to draw. Recall: What is the area of a rectangle? Call the short side x and the long one y. What is the perimeter of such rectangle?
8. Optimization with inequality constraints. Find
maxf (x;y) = xy s.to.x+y2 2
Approach this formally via Lagrangean. Write all the Karush-Kuhn-Tucker condi- tions. Argue that the non-negativity constraints will not bind and that the x+y2 2 constraint will hold as equality. Solve the resulting system of equations.
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