Econ 401A: Economic Theory Mid-term. Answers

Transcription

1 . Labor suly Econ 40: Economic Theory Mid-term nswers (a) Let be labor suly. Then x 4 The key ste is setting u the budget constraint. x w w(4 x ) Thus the budget constraint can be rewritten as follows: wx x w4 value of consumtion = value of endowments Remark: t is not immediately intuitive to think of the wage as the rice of time and the day as your endowment of time. ut on the margin, if you decide to enjoy another hour of leisure, then your income goes down by w. Transform the utility function into the following concave function FOC u lnu ln x ln( a x ) mu w mu i.e. wx ( a x ) w4 a Therefore wx ( w4 a) Hence w w and so x [4 a)] a 4 x a w (b) s the rice of leisure (the wage rises, the substitution effect is to consumer more of the relatively cheaer commodity, x, and less leisure. Hence labor suly rises. The worker is better off as the wage rises. So income must be increased to move the consumer to her new otimal choice. n this homothetic model consumtion rises roortionally with income so the income effect is to increases both x and x. So the two effects are offsetting on leisure and hence on labor suly.

2 (c) a. This must be ositive. w. Walrasian Equilibrium (WE) in an exchange economy (a) f a utility function is homothetic, then the marginal rate of substitution is constant along a ray. (See below.) Let x be the consumtion choice of a consumer with income. Consider two consumers with incomes and. Let be the total. Then, for some, and ( k). Therefore s choice is s choice is deicted below. x kx and s choice is x ( k) x. k k t follows that x x kx ( k) x x Thus rather than solve for the two consumtion vectors and add them, it is equivalent to solve for the demand of one consumer with income. Remark: did not emhasie this intuitive exlanation strongly enough in my lecture on homothetic references. Next time will do better. For the reresentative consumer there can be no trade in a WE since there is no one to trade with. Thus x. (b) Consumer h, h,..., H has utility function x MRS( x) ( ) U( x, x ) ( x ) ( x ). h h h / h / h / and so is constant along a ray. h x (b) For the reresentative consumer there can be no trade in a WE since there is no one to trade with.

3 x (00, 400). FOC 400 MRS ( x) MRS ( ) ( ) 00 / (c) U( x, ) ( x ) ( x ) h h / h / For an efficient allocation x MRS ( x ) ( ) / = x MRS x / ( x ) ( ) x Therefore x x x x x x x x 4 Thus the PE allocations lie on the diagonal of the Edgeworth ox. (d) ETHER: WE is Pareto Efficient. Therefore risks are shared roortionally. OR MRS x MRS x ( ) ( ) Therefore a WE allocation is a PE allocation.. 3. Multi-roduct monooly (a) and f f and g are concave then for any f x f x f x 0 ( ) ( ) ( ) ( ) 0 g( x ) ( ) g( x ) g( x ). 0 x, x and convex combination x, Summing these inequalities, f x g x f x g x f x g x 0 0 ( ) ( ) ( )( ( ) ( )) ( ( ) ( )) 3

4 Define the sum h f g. Then h x h x h x 0 ( ) ( ) ( ) ( ) Remark: f you considered only functions of one commodity you had 0.5 oints deducted. (b) ( q) ( q ) q ( q ) q C( q) 60q q 90 q q ( q q ) q () Using the second derivative condition is fine for all terms excet ( q q). q q concave function since it is the sum of two linear (and hence concave) functions. n increasing concave function of a concave function is concave. Remark: f you missed this you had oint deducted. is a (c) Since ( q) is concave, the FOC are both necessary and sufficient 60 q ( q q) 60 4q q 0 q 90 4q ( q q) 4q 90 q 0q 0 q The two FOC can be written as follows: 4q q 60 () q 0q 90 () Double the second equation 4q 0q 80 () 4q q 60 () Subtract the bottom equation from the to q 0 so q Then q 90 0q 90 so q 3 3 (c) Substitute into () to obtain the maximied rofit ( q )

5 Remark: feel bad. The rofit maximiing oututs were intended to be integers to make this easy. (d) s the constant K ( q ) K 0 increases, the new rofit is ( q ) K. So roduce as long as 4. Walrasian Equilibrium in a three commodity model with roduction (a) ll of inut is used in roduction so Then lso Then 6 q 8 (6) 3 / / / x 6 U ln(6 ) ln3 ln(6 ) ln3 ln The derivative is / U( ) 0 6 for a maximum. Solving, (b) 8 and so q We have already argued that / 6 5

6 (c) (, ) q 8 / / 3 3 FOC (, ) 4 4( ) 0 / / / 3 3 (, ) 4 4( ) 0. / / / 3 3 To suort the otimum, these conditions must hold at (8,6) (, ) / 4 (, ) 3 4( ) Therefore if it follows that and 3. (d) Profit is 3q ( )(64 )

7 Remark: The roduction function exhibits constant returns to scale. Thus if yields a rofit yields twice as much. So WE rofit must be ero. (e) f you drew the diagram it was enough to oint out that the maximum rofit level set (the heavy green line) is also the boundary of the budget set for the reresentative consumer. From the figure, solves the consumer s maximiation roblem. ( x, x ) 3 Here is a more formal treatment. For any feasible roduction lan rofit cannot exceed maximied rofit so 3q 0 Note that x,, q x. Therefore 3 x ( x ). 3 3 Rearranging this exression, x 3x3. Thus the maximum rofit level set is the budget constraint for the reresentative consumer. From the figure, the red marker is a WE since demand is equal to suly in all markets. 7