Midsemester 1 - solutions

Midsemester 1 - solutions - ECON 200 Mid-semester Test 1...

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ECON 200 Mid-semester Test 1 Question Two (20 marks) – A Problem Suggested time: 20 minutes Consider the following Cobb-Douglas utility function for a consumer who must choose a bundle of goods 1 and 2: () 1 , 2 1 2 1 = + = β α x x x x U ( 1 ) (a) Very often in dealing with this type of utility function, we study the log-form: () 2 1 2 1 ln ln , ln x x x x β α + = = How do we know that this form of the utility function represents the same preference ordering as (1)? ( 2 marks ) Because the ln() function is an increasing monotonic transformation of U, it preserves the ranking of U, and so represents the same preference ordering. The commodity bundle that maximizes u will also maximize U. (b) The marginal utility with respect to good 1 and good 2 are given by: 2 2 1 1 x x u x β α = = What happens to marginal utility as we consume more of each commodity? ( 2 marks ) Student should note that the marginal utility of both goods is diminishing as we consume more of both. Answers which show that thought went into will be given one more mark than answers that just saying – they are both diminishing. = ↑⇒ = ↑⇒ 2 2 2 1 1 1 x x u x x x β α (c)
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ECON200 Midsemester 1 - Solutions 2007 indifference curves should look like. Your drawing need not be exact. (Hint: consider what happens to the MRS 21 as we change x 1 and x 2 in particular.) ( 5 marks ) MRS = -alpha x 2 / beta x 1`` (2 marks) Slope = negative (1 mark) Shape = convex – as for high x 2 – low x 1 , MRS is steep, and for high x 1 – low x 2 , MRS is relatively flatter. (1 mark) Drawing of convex, negatively sloped indifference curves. (1 mark) (d) Assume prices are given by (p 1 , p 2 ) and income is fixed amount ‘m’. State the consumer’s maximization problem. What is the equilibrium condition for a consumer with this utility function? Is it a corner or interior solution? How do you know? ( 4 marks ) Max U (x 1 ,x 2 ) s.t. p 1 x 1 + p 2 x 2 = m (1 mark) Equilibrium condition – MRS 21 = -p 1 /p 2 (2 marks) Interior solution –we know this because indifference curves are convex and budget line is regular. Another way to show this is to say that the MRS is undefined when the consumer has zero of either good (because MU is undefined), so that this consumer will never be at that point. (1 mark) (e) The ordinary demand functions for good 1 and 2 that satisfy this equilibrium condition are given by: 2 2 1 1 p m x p m x + = + = β α What influence do ‘alpha’ and ‘beta’ have on our demand for these commodities? Derive the own price elasticity of demand for good 1 and interpret it.(
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Midsemester 1 - solutions - ECON 200 Mid-semester Test 1...

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