Economic Principles Solutions to Problem Set 1

Transcription

1 Economic Principles Solutions to Problem Set 1 Question 1. Let < be represented b u : R n +! R. Prove that u (x) is strictl quasiconcave if and onl if < is strictl convex. If part: ( strict convexit of < ) strict quasiconcavit of u ). For x; 2 R n +;suppose < x and 6 x. Strict convexit of < implies that + (1 ) x x for all 2 (0; 1): Since u represents <, this means that u( + (1 Hence, u is strictl quasiconcave. ) x) > u(x) minfu(x); u()g: Onl if part: ( strict quasiconcavit of u ) strict convexit of < ). Suppose z < x; < x; with z 6 : We have to show that + (1 ) z x for all 2 (0; 1): Without loss of generalit, suppose z < ; i.e., u (z) u () : Strict quasiconcavit of u implies Hence, < is strictl convex. u( + (1 )z) > u () u (x) : Question 2. To prove that the two functions have the same indi erence curves, pick an arbitrar bundle ( ; x 2 ). For utilit function u(x) p x 2 the bundles indi erent to ( ; x 2 ) satisf the equalit p x 2 p 1 2 which b appling logarithms to both sides is equivalent to log + log x 2 log 1 + log 2 ; but this is the same indi erence condition we obtain if we use the utilit function v(x) log + log x 2. We conclude that the two utilit functions give the same set of bundles indi erent to x. Since x was arbitrar, the two utilit functions have the same indi erence curves. Since u 0 was chosen arbitraril, u and v have the same indi erence curves. For u; MRS MU 1 MU 2 p x2 2 p p x1 2 p x 2 x 2

2 For v; MRS MU x 2 x 2 u and v have the same indi erence curves and the same MRS because each utilit function is a strictl increasing transformation of the other. Speci call, v(:) 2 ln(u(:)): Question 3. Graph an indi erence curve, and compute the M RS and the Marshallian demand functions for the following utilit functions: a) Perfect substitutes: u ( ; x 2 ) + x 2 ; where > 0; > 0; x2 x1 Perfect Substitutes (slope ). MRS MU MU x2 It is possible to solve the problem graphicall. Here we do a little more algebraic solution. The problem we want to solve is max x1 ;x 2 + x 2 s.t. + x 2 0; x 2 0 The strateg we can use is to solve for x 2 in the budget constraint and substitute it in the objective function, turning it in a maximization problem in one variable. We need some care though, since we need to remember the nonnegativit constraints. From the budget constraint we can write: x 2 2

3 However we need to remember that x 2 0, that is must be contained in the interval [0; ]. So our original maximization problem is equivalent to the following: max ( ) + s.t. 2 [0; _ ] This is a ver eas maximization problem, since we are maximazing a straight line over an interval. The slope of the straight line is ( ): Therefore, if the slope is strictl positive (negative), the straight line is strictl increasing (decreasing), so the point of maximum is at the right (left) endpoint of the interval. If the objective function is constant (the slope is zero), the consumer will be indi erent tamong all the values in the interval. We can summarize these observations in our Marshallian demand function: (p; ) x 2 (p; ) 8 < : 8 < : 0 if < [0; ] if if > 0 if > (p; ) if if < b) The ke to solve this problem is the following observation. Suppose the solution satis es the budget constraint and has 6 x 2 Without loss of generalit suppose > x 2. The utilit of this bundle is x 2. This utilit can be increased if we decrease the consumption of b a small amount " > 0, use the mone we are saving ( ") to bu a little more of x 2 ( ") : The new bundle ( ",x 2 + ") still satis es the budget constraint with equalit, and we can choose " small enough so that it satis es the inequalit ( ") > (x 2 + "). The utilit of our new bundle is therefore (x 2 + ") > x 2 contradicting the hpothesis that our original bundle ( ; x 2 ) was optimal. We conclude that the solution to the maximization problem with goods that are perfect complements must satisf: a x 2. We can then solve the following sistem: a x 2 + x 2 Hence, the Marshallian demand functions are: ( ; ; ) x 2 ( ; ; ) + + An indi erence curve is shown in red in the following graph: 3

4 x2 x1 Perfect complements MRS 1 when x 2 > ; MRS 0 when x 2 < ; and MRS is not well de ned when x 2 : Question 4. (JR 1.21). We have noted that u (x) is invariant to positive monotonic transformation. One common transformation is the logarithmic transform, ln (u (x)) : Take the logarithmic transform of the Cobb-Douglas utilit function; then using that as the utilit function, derive the Marshallian demand functions and verif that the are identical to those derived in class. Cobb-Douglas utilit function: Taking the logarithmic transformation, u( ; x 2 ) x 1 x 2 v( ; x 2 ) ln(u( ; x 2 )) ln + ln x 2 To nd the Marshallian demand functions, we solve the problem: The Lagrangian for this problem is: The F.O.C. are: max ln + ln x 2 s.t. + x 2 L ln + ln x 2 + ( x 2 ) x 2 + x 2 4

5 Taking the ratio of the rst two equations gives: x 2 Together with the budget constraint, we can solve for the optimal choice of and x 2. ( ) ) Substitute this into the budget constraint, we get x Hence, the Marshallian demand functions are the same as those we derived in class. Question 5. (JR 1.27). A consumer of two goods faces positive prices and has a positive income. Her utilit function is u ( ; x 2 ) max fa ; ax 2 g + min f ; x 2 g ; where 0 < a < 1: Derive the Marshallian demand functions. We can express the utilit function in the following wa: u( ; x 2 ) maxfa ; ax 2 g + minf ; x 2 g ax1 + x 2 if x 2 + ax 2 if x 2 Graphicall, an indi erence curve looks like this: x2 Indi erence Curve. Let s solve the maximization problem. We can distinguish several cases, depending on the relationship between the price ratio (i.e. the slope of the budget constraint) and the MRS (i.e. the slope of indi erence curves). 5 x1

6 < a < 1 a : In this case the budget constraint is atter than both the MRS above and below the 45-degree line. So the consumer will bu onl good 1. (p; ) x 2 (p; ) 0 a < 1 a. In this case the consumer is indi erent among all the bundles on the budget set and below the 45-degree line. (p; ) 2 [ + ; ] x 2 (p; ) (p; ) a < < 1 a. In this case the slope of the budget constraint is between the two MRS and the maximal point is at the kink, that is where x 2 B solving keeping in mind that the solution must satisf the budget constraint we get (; ; ) + x 2 (; ; ) + a < 1 a. In this case the consumer is indi erent between all the bundles that are on the budget line and above the 45-degree line. (p; ) 2 [0; + ] x 2 (p; ) (p; ) a < 1 a < In this case the budget constraint is steeper than both MRS, therefore the consumer will consume onl good 2. (p; ) 0 x 2 (p; ) Question 6 Consider the following monotonic transformation of the u(:): v(:) (u (:)) 2 + 2x 2 + 3x 3 : The three goods are perfect substitutes for each other and the consumer will choose the good that gives the highest MU i p i. 6

7 The Marshallian demand functions are as follows: if < minf 2 ; p 3 if 2 < minf; p 3 if p 3 < minfp 3 1; if g; x 3 1(p; ) ; x 2 (p; ) 0; x 3 (p; ) 0; g; x 3 1(p; ) 0; x 2 (p; ) ; x 3 (p; ) 0; g; x 2 1(p; ) 0; x 2 (p; ) 0; x 3 (p; ) p 3 ; < p 3 ; x 2 3 3(p; ) 0; (p; ) 0, x 2 (p; ) and (p; ) + x 2 (p; ) ; if p 3 < ; x 3 2 2(p; ) 0; (p; ) 0, x 3 (p; ) and (p; ) + p 3 x 3 (p; ) ; if p 3 < 3 1; (p; ) 0; x 2 (p; ) 0, x 3 (p; ) and x 2 (p; ) + p 3 x 3 (p; ) ; if p 3 ; x 2 3 1(p; ) 0; x 2 (p; ) 0, x 3 (p; ) and (p; ) + x 2 (p; ) + p 3 x 3 (p; ) : When 2, 3, p 3 5, we have < minfp 2 1 ; p 3 g and therefore x 3 1(p; ) 0; x 2 (p; ) ; x 3 3(p; ) 0. To achieve utilit level 6, she needs income such that r ) 54 7