Math Review Solutions Set for Econ 321

Transcription

1 Math Review Solutions Set for Econ 3. Practice Problems: dπ. dq Q 4 Q + 6 Q8. PP-. :. MR- Le Axis - Right Axis

2 dπ. dq Q 4Q 6Q Q PP-. : MR- Le/ Axis - Right Axis Practice Problems: dx 4x + dx x + dx x dx αaxα βcx dx x 5 x + x 5 x+ x dx 5 x 6 x + 5 x 3 + x 3

3 3 7. dx β x α x α + β + β x α α x α α dx x x + dx x 3x + x 6x / 3x + x3 x 6 x 3/ + 3 x + dx Aαxα x +β Ax α x Axα α x +β x +β Axα x x +β dx x dx 4 3 x dx x+3 x x+3 x 4 x 4 3 dx α + Axα βx α Aαx α β 4.3 Practice Problems: dx 5 e5 x dx e3 x dx e x dx α x + b eα x +bx+c dx 5 x dx c x dx 5 x+ x+ dx x3 ln x + 5 x 3 9. V e $7.83; V 5 e 5 $, 8.5; V e $4, dv t dt 5e t

4 4. lnv t ln + t PP4.3:. 6 LN Vtle4 axis vs :me V e $4.33; V 5 e 5 $ V e $, dv t dt 5 e t t

5 5 lnv t ln + t PP4.3: LN Vtle4 axis vs :me Practice Problems: z z. x x + y; y x + y; dz x + ydx + x + y. z x y x x +y ; z xy x +y ; dz y x x +y dx z z x 3. x ; x +x +x3 x 3 z 3x ; x +x +x3 x x +x +x3 3 x dz dx x +x + 3x dx +x3 3 x +x + 3 +x3 3 x +x +x3 3 z 4. x ex z lny; y ex y ; dz ex lnydx + e x y 5. Y xy x +y ; dx 3 A Kα L α Y ; K αakα L α Y ; L αakα L α dy K α L α da + αak α L α dk + αak α L α dl Y 6. K α Kρ + β L ρ ρ ρ α K ρ Y ; L α Kρ + β L ρ ρ ρ β L ρ dy α K ρ + β L ρ ρ ρ α K ρ dk + α K ρ + β L ρ ρ ρ β L ρ dl 6. Practice Problems:. F x x, y 4; F y x, y ; dx 4

6 6. F x x, y 8y; F y x, y + 8x; dx 8y +8x 3. F x x, y y + 5; F y x, y y + xy ; dx y +5 y+xy 4. F x x, y 3x 4xy + 3y ; F y x, y x + 6xy; dx 3x 4xy+3y x +6xy 5. F x x, y y 4 + xy + x ; F yx, y y + 8xy 3 + 5x ; dx y4 +xy+ x x +y +8xy 3 +5x 7. Practice Problems:.. dx 4x; dx x; d y dx 4: convex d y dx : concave 3. dx x ; d y dx x : concave 4. dx 4ex ; d y dx 4e x : convex K αakα L α ; α αak α L α : With < α < as is standard, K the function is concave. With α > the function is convex. 5. Y Y 6. Take the first derivative with respect to either K or L, both will respond similarly. Y K α Kρ + β L ρ ρ ρ α K ρ > no matter what the ρ, as everything else is assumed to be positive. Now take the second derivative with respect to the same variable. ρ α α K ρ + β L ρ ρ ρ α K ρ α K ρ + β L ρ ρ ρ K ρ Y K When < ρ < the second derivative is negative and the function is concave: Y with Rho< Y with Rho<

7 7 When ρ the second derivative equals and the function is linear: Y with Rho Y with Rho When ρ > the second derivative is positive and the function is convex: Y with Rho> 5 5 Y with Rho> dt C dq 9Q dt C + 4Q ; dq will always be convex. 8Q + 4 : With Positive quantities of Q, the function 8. Practice Problems:. dπ dq Q 4 ; Q8; d π dq. dπ dq Q 4Q ; Q; d π dq dπ 3. dl AL 4 ; L dπ 4. dl ALα α w ; L w αa standard < α < 64 A d ; π α ; : Maximum 6: Maximum /4 A : Maximum dl L 3/ d π dl α αal α : Maximum with a

8 8 9.4 Practice Problems: Substitution:. du. du 3. du 4. du 5. du y + 3 y y + + y y W c + y ; x y 5 y ; x 9; y y ; x 33 3 y 66 3 β c ; + y y ; x y 5 c W +β c W β +β Lagrangian:. L x x λ.77.. L x 3 x λ L x x λ.3. λ ; λ ; λ ; L y L y L y y y y λ ; x y 5 λ ; x 9; y λ ; x 33 3 ; y L x x y λ.5. λ ; L y x y λ x y 5

9 9 L 5. c c λ +β W. λ ; L c β c λ ; c W +β c W β +β.3 Practice Problems:. A+B B+C A+C. A-B B-C A-C 3. Transpose A

10 Transpose B Transpose C *A *B -3*C α A A*B [ 3 [ α α α α α 3 α 5 α α 3 α 4 ] [ ] [ ] ] [ 4 7 [ ] ]

11 A*C B*C B*A C*A C*B Practice Problems:. DETA5 3 DETB-*9-3*-96 DETC*4-6*4 DETD*---6* : INVA INVB [..3.5 ] [ ]

12 INVC 3. [.5.5 ] [ ] [ ] [ ] [ ] [ ] [ [ ] [ ] [ ] ].7 Practice Problems: : a: x y b: x

13 3 c: y x / /3 x / d: Not Possible. : Singular matrix: the determinant is zero..9 Practice Problems: a β c c +r y + y +r c c +r b dc c + c +r c +r dc dc + dβ + d: dc c + c +r +r dc dc dβ dc + c dβ + c d + r + +r + c y +r d + r + +r + +r dc c dβ c d + r +r c y +r d + r +r +r

14 4 [ c +r c +r ] [ ] dc c c dc c y +r +r +r +r dβ d + r dc dβ c c c +r +r c +r < The higher is the discount factor β the more utility the consumer receives from consuming in the second period. In other words, the consumer discounts future spending less with a higher β and will therefore decrease this period s consumption, save more borrow less to increase future consumption. dc d+r c c c +r y +r +r +r c +r c +r? The result here is ambiguous. There are two effects at work. One is a wealth effect and the other is a substitution effect. With an increase in + r a net saver will experience a positive wealth effect, but will experience a negative substitution effect away from c towards c. dc c +r c +r c +r > As first period income rises there is a wealth effect and the agent will consume more in the first period. dc

15 5 c +r +r c +r c +r > As second period income rises, the agent can save less for future spending or borrow more for current spending. Either way, with higher income levels coming in the second period, the agent can increase his first period consumption. dc dβ c +r c c +r c +r > As the value the agent places on future consumption rises, the more he will shift current consumption to savings for future consumption. dc d+r c +r c c +r c +r y +r +r c +r? The result here is ambiguous. There are two effects at work. One is a wealth effect and the other is a substitution effect. With an increase in +r a net borrower will experience a negative wealth effect, but will experience a positive substitution effect away from c towards c. dc c +r c +r c +r >

16 6 As income rises in the first period, the agent consumes more in the first period, but also save some of the increased income for the second period in order to smooth consumption. dc c +r c +r +r c +r > As income rises in the second period, there will be a wealth effect and the consumer will increase his second period consumption.