ECON 400 Homework Assignment 2 Answer Key. The Hicksian demand is the solution to the cost minimization problem.

Transcription

1 ECON 400 Homework Assignment Answer Key Question : Consider the following strictly quasi-concave utility function. u x ) = q + q a) 0 oints) Derive the Hicksian demand. Sorry for using x and x to denote goods and in the assignment. I m switching to our usual notation in the answer key. The Hicksian demand is the solution to the cost minimization roblem. min q + q + µ u q x,x,µ q ) µ q = 0 + µ q = 0 Using the tangency condition and then lugging exression for c into the utility constraint yields q = 4 q q + 4 q 4 + h, q h, b) 0 oints) Write down the cost function. c, ) = h, ) + h, ) = u + 4 u c, u 4 + c, ) =

2 c) 0 oints) Write down the indirect utility function. c, ) = u 4 + x = v, ) 4 + ) = x 4 + ) v, = x 4 + ) v, d) 0 oints) Write down the Marshallian demand. Use Roy s identity to get from the indirect utility function to the Marshallian demand. = x 4 + ) v, 4 4 x = x x 4 + = ) x 4 + x = g 4 + x 4 +, x = = x 4 + = x 4 + g, ) = x = 4x 4 + e) 0 oints) Verify the Slutsky equation for a change in the demand of good as the rice of good changes.

3 g = h g x g 4 = x 4 + ) 4 + ) ) 8 = x 4 + ) 8 = x 8 g = x h = v, 4 + g x g = x ) = x Hence the Slutsky equation is verified. 4x 4 + ) = 8x 4 + 4x Question : Consider the following two budget constraints. =,,4) =,/, / ) 0, 0 0 ), 4 + a) 0 oints) Draw the budget constraints. 4 + Good 4/ / 4 Good

4 b) 0 oints) Can you tell whether a consumer with strongly monotone references and a strictly quasi-concave utility function is better or worse off under the first budget constraint comared to the second? Give a detailed exlanation. Without further information we can t tell if the erson is better off or worse off under the old budget set. Neither budget set is a subset of the other one. So if the erson consumed a consumtion bundle under the old budget constraint that is no longer available under the new one, it is ossible that the erson is better off under the old budget set, worse off, or the same off. If we knew that the erson consumed a consumtion bundle on the old budget line that s below the new budget line or at the intersection with the new line, we could say that the erson is better off under the new budget set due to strong monotonicity in the first case and strict convexity in addition to strong monotonicity in the second. Similarly, if we knew that the erson consumed a consumtion bundle on the new budget line that s below the old budget line or at the intersection with the old line, we could say that the erson is better off under the old budget set due to strong monotonicity in the first case and strict convexity in addition to strong monotonicity in the second. But we don t have this information. c) 0 oints) Suose in addition to the assumtions in b) you re also told that the utility function is given by = q q u q,q from the first to the second budget constraint. This is a Cobb-Douglas utility function. Hence = x α v, c, ) α α α α α. Derive EV to evaluate the consumer s welfare change For EV we need to evaluate a erson s exenditure necessary to reach the new utility level at the old rices minus the erson s old fixed income. c 0 0 ) x 0 EV = c 0 EV = c 0 = u = v,/, / c 0 ) = / EV = 4 = / / α α / / / / / = / =

5 d) 5 oints) Exlain what the numerical value means that you obtained from your EV calculation. An EV of - means that we d have to take $ away from the erson s initial income if we want to make her as well off as she is after the rice change but stick with the old rices. This means that facing the new budget constraint has the same effect on the erson s welfare as if we d cut the initial income of the erson in half and left the rices at the original values. e) 0 oints) Derive CV to evaluate the consumer s welfare change from the first to the second budget constraint. CV = c c 0 ) CV = x c 0 = 4 / u 0 = v,,4 = c 0 / / / / / / / / = 4 = CV = 4 = f) 5 oints) Exlain what the numerical value means that you obtained from your CV calculation. The amount of -$/ tells us that the consumer would require $/ more dollars at the new level of income and rices in order to be as well off as with the old budget constraint, so twice as much income as what the consumer actually faces.