• MonashU
• ECC5650MicroTheory
• ProblemSet01

Try first to attempt the following problems without looking up a textbook.

Concavity, Convexity, Quasiconcavity, Quasiconvexity

1. (Chiang) Given the definition of a concave (convex) function, three theorems can be deduced:

   1. Theorem 1 (linear function) If

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      is a linear function, then it is a concave funcation as well as a convex function, but not strictly so.

   2. Theorem 2 (negative of a function) If

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      is a concave function, then

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      is a convex function, and vice versa. Similarly, if

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      is a strictly concave function, then

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      is a strictly convex function, and vice versa.

   3. Theorem III (sum of functions) If

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      and

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      are both concave (convex) functions, then

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      is also a concave (convex) function. If

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      and

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      are both concave (convex) and, in addition, either one or both of them are strictly concave (strictly convex), then

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      is strictly concave (strictly convex).

   => Prove these theorems.

2. Check

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   for concavity or convexity, and hence determine whether the function has a unique maxima or minima.

3. (Chiang) Check

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   for quasiconcavity and quasiconvexity.

4. (JR A1.48) Let

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   . Prover that f is quasiconcave.

Brouwer's Theorem

1. Prove that the empty set

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   and entire set

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   are both open and closed.

2. (JR, A1.38) Let

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   and suppose that

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   . Show that f has no fixed point even though it is a continous mapping from S to S. Does this contradict Brouwer's Theorem? Why, or why not?

Lagrange Multiplier Method

1. Find the extremum of

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   subject to

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   .

2. Find and characterise the critical points of

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   subect to

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   .

Kuhn-Tucker Conditions

1. Check the first and second-order conditions for the following problem:

   • Minimize

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   • subject to:

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Envelope Theorem

1. (Riley) To produce q units of output, a profit-maximising firm requires

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   units of the single input. The price of the output is p and the price of the input is r.

   1. Write down an expression for profit

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      .

   2. Solve for the profit maximizing output

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      and hence show that

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      and

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   3. Show also that maximized profit for different input and output prices is

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   4. Confirm that

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      and

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      .