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Midterm 1 Solutions - MIT OpenCourseWare

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solution if the prices of the two goods are di erent if the prices are the same, .... L∗ = 18. The elasticity of demand at the equilibrium wage is. dLD w∗. 4. 16. 8.

��

��

�� u = x + y� ��

�� P = 5 − 12 Q� �� P = 20 − Q� ��

��

QD 1

� 10 − 2P = 0

�

, P ≤5 , P >5

P

5

15

30

Q

Image by MIT OpenCourseWare.

��

QD 2 = 20 − P

�� QD M

=

QD 1

+

Q2D

� 30 − 3P = 20 − P

, P ≤5 , P >5

�� 20 − Q P = 10 − 13 Q

Q ≤ 15 Q > 15

�� QS = 2 + P � ��

�� p = 5 QS = 7 < 15 = QD �� QS = QD �� 2 + P = 20 − P �� P ∗ = 9� �� Q∗ = 11� ��

�� 1 U (b, n) = 3 min(n, b) �� U (b, n) = 3 min(3n, b) 3

�

Bracelets

(1, 3)

Necklaces

Image by MIT OpenCourseWare

�� pn = 2 �� pb = 2� ��

�� b = 3n �� 2n + 2b = 32� �� 2n + 6n

=

32

n

=

3

�� n = 4� b = 12� u = 12� �� pn = 10� �� 10n + 6n = 32

�� n = 2� b = 6� u = 6�

��

�� 2 ∗ 10 + 6 ∗ 2 = 32� �� nb2 � �� b2 2nb b

= =

2 2 2n

�� 2(2n) + 2n = 32

�

�� n = 163 � b = 323 � �� u = 163 ( 323 )2 = 512 27 �� b2 2nb b

= =

10 2 10n

�� 2(10n) + 10n = 32 160 16 160 2 �� n = 16 15 � b = 15 � �� u = 15 ( 15 ) = ��

2560 225

� ��

16 ∗ 162 ∗ 4 16 ∗ 162 ∗ 100 16 ∗ 162 ∗ 4 − ) )/( 27 27 ∗ 53 27 16 ∗ 162 (4 ∗ 53 − 100) = 27 ∗ 53 = 485.4 = (

�� Bracelets

(1, 3)

Necklaces Image by MIT OpenCourseWare.

�� LD = 34 − 4w

�� w ��

�

��

LS = L + 2w

�� L �� L = 10� ��

�� LS = LD �� 34 − 4w = 10 + 2w� �� w∗ = 4� L∗ = 18� �� D =

dLD w∗ 4 16 8 = −4 =− =− ∗ dw L 18 18 9

�� w∗ � �� L = 16� �� L = 16� �� 34 − 4w = 16 + 2w� �� w∗ = 3� L∗ = 22� �� w = 4� �� w > w∗ �� LD = 18 �� LS = 24� �� LS − LD = 6� W LD

LS

18

24

Unemployment Image by MIT OpenCourseWare.

�

L

�� τ = 1 ��

� �� τ � �� LS (p+ 8 ∗ τ ) = LD �� 16 + 2(w + τ ) = 34 − 4w� �� w∗ = 16 6 = 3 � �� w + τ = 8 11 3 + 1 = 3 �

�� X �� Y �� U (X, Y ) = X 0.2 Y 0.8 � �� $�� X �� Y �� $�� $�� X �� Y � �� X �� Y �� M UX = 0.2(Y /X)0.8 �� M UY = 0.8(X/Y )0.2 M RSY

��

X

= −M UX /M UY = −Y /4X

�� X �� Y � �� X �� Y � �� Y �� X � �� (X ∗ , Y ∗ )� �� M RSY ��X = −PX /PY � �� −Y /4X = −10/20 =⇒ −Y /4X = −1/2 =⇒ 2Y = 4X =⇒ Y = 2X

�� I = PX X + PY Y =⇒ 100 = 10X + 20Y �� (X ∗ , Y ∗ )� 100 = 10X + 20(2X) =⇒ 100 = 50X =⇒ X ∗ = 2 Y = 2X = 2(2) =⇒ Y ∗ = 4

��

�� −PX /PY �� M RSY �� X = −PX /PY �� Y = 2X � �� 100 = 20X +40Y � �� (X ∗ , Y ∗ ) = (1, 2)� �� X �� Y � �� X ∗ �� Y ∗ �� PX , PY �� I � M RSY

��

X

= −PX /PY =⇒ −Y /4X = −PX /PY =⇒ PY Y = 4PX X

�� I = PX X + PY Y �� X ∗ �� Y � I = PX X + 4PX X = 5PX X =⇒ X ∗ (PX ; I) = I/(5PX )

�� $100 �� X ∗ (PX ; I = 100) = 20/PX �

�� Y ∗ �� X � PY Y = 4PX X =⇒ Y = (4PX X)/PY =⇒ Y = (4PX [I/(5PX )])/P Y Y ∗ (PY ; I) = 4I/(5PY )

�� $100 �� Y ∗ (PY ; I = 100) = 80/PY �� Y �� $�� $�� Y �� X ∗ �� PY �� X ∗ = 2� Y ∗ �� Y ∗ = 80/PY = 80/10 = 8� �� Y �� X �� Y �� BCsub �� BC1 � �� IC0 � �� Y ∗ � Y

10

Inc

Total effect

8

IC1

Sub

4

BC

BC

su

0

b

2

BC

IC0

1

10

X

Image by MIT OpenCourseWare.

��

�� $10� �� 8 �� Y �� 8($10) = $80� �

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14.01SC Principles of Microeconomics

Fall 2011

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