ajaz_204_2013_2014_HW_1

# ajaz_204_2013_2014_HW_1 - University of Toronto Department...

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University of Toronto, Department of Economics, ECO 204, 2013 - 20143 1 ECO 204 2013 2014 HW 1 For feedback, comments and suggestions, please e-mail __________________________________________________________________________________________________ Question 1 Plot the following functions in Wolfram Alpha : (a) from (b) from (c) from (d) ( ) ( ) from . By the way, what do you notice about the 2-D contour plots of parts (c) and (d)? Question 2 From Math 133 recall the following definitions (for uni-variate functions): Definition: A function ( ) is concave if and only if ( ) for all values of Definition: A function ( ) is strictly concave if ( ) for all values of (however, the reverse need not be true; that is, if a function is strict ly concave then it doesn’t necessarily mean that ( ) everywhere see question 3 below) Definition: A function ( ) is convex if and only if ( ) for all values of Definition: A function ( ) is strictly convex if ( ) for all values of (the reverse need not be true; that is, if a function is strictly convex then it doesn’t necessarily mean that ( ) everywhere see question 3 below) (a) Consider defined on ) When is this function concave? Convex? Strictly concave? Strictly convex? Both concave and convex? Neither concave nor convex? (b) Consider defined on ) and where . When is this function concave? Convex? Strictly concave? Strictly convex? Both concave and convex? Neither concave nor convex? (c) Write down a uni-variate function which is both concave and convex. (d) True or false: if the function ( ) is concave then the function ( ) is convex (and vice versa)?

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University of Toronto, Department of Economics, ECO 204, 2013 - 20143 2 (e) Suppose ( ) ( ) For example, ( ) ( ) Suppose the function ( ) is concave. Does this mean that ( ) ( ) is also concave? Question 3 Graph the function defined on . Does the function “look” strictly concave (decelerating and doesn’t have flat portions)? If so, is it true that strict concavity implies that ( ) for all values of (in particular )? Question 4 Convert each of the following minimization probl ems into a maximization problem (don’t solve the problems): (a) (b) (c) Question 5 Consider a perfectly competitive firm (i.e. price taker) with the cost function where is the total fixed cost and is a constant.
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