🔬 Tutorial problems iota

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🔬 Tutorial problems iota#

ι.1#

Find the largest domain SR2 on which

f(x,y)=x2y2xyx3

is concave.

How about strictly concave?

It is useful to review the Hessian based conditions for concavity and the conditions for definiteness of a Hessian of R2R functions.

ι.2#

Show that the function f(x)=|x| from R to R is concave.

Because the function is not differentiable everywhere in its domain, using the definition of concavity could be an easier way.

ι.3#

Consider the function f from R to R defined by

f(x)=(cx)2+z

Give a necessary and sufficient (if and only if) condition on c under which f has a unique minimizer.

ι.4#

Let C be an N×K matrix, let zR and consider the function f from RK to R defined by

f(x)=xCCx+z

Show that f has a unique minimizer on RK if and only if C has linearly independent columns.

Obviously, you should draw intuition from the preceding question.

Also, what does linear independence of the columns of C say about the vector Cx for different choices of x?

ι.5#

Consider the maximization problem

maxc1,c2(c1+βc2)

subject to c10, c20 and p1c1+p2c2m. Here p1,p2 and m are nonnegative constants, and β(0,1).

Show that this problem has a solution if and only if p1 and p2 are both strictly positive.

Solve the problem by substitution and using the tangency (relative slope) condition. Discuss, which solution approach is easier.

To answer the first part of the question, review facts of existence of optima.