a.Jane's MRS:
=
=
=
Sarah's MRS:
=
=
=
b.For equilibrium, each individual must equate MRS to
.
=
=
For Jane:
=
, Y =
X
For Sarah:
=
,
=
∙
Y =
X
To determine quantities substitute into each individuals budget constraint.
Jane's budget constraint: 600 = 10X + 20Y
Substitute Y = (1/2)X
600 = 10X + 20(1/2)X
600 = 10X + 10X
X = 30
600 = 10(30) + 20Y
300 = 20Y
Y = 15
Jane should consume 30 units of X and 15 units of Y.
Sarah's budget constraint: 700 = 10X + 20Y
Substitute Y = (3/4)X
700 = 10X + 20(3/4)X
700 = 25X
X = 28
700 = 10(28) + 20Y
420 = 20Y
Y = 21
Sarah should consume 28 units of X and 21 units of Y.
c.In equilibrium MRS
J should equal MRS
S.
=
Jane is consuming 15 units of Y and 30 units of X.
=
=
=
X
Sarah is consuming 21 units of Y and 28 units of X.
=
=
=
does equal
, and it is also True that the two individuals are consuming the available quantities of X and Y.
d.Equilibrium has been achieved. The equilibrium involves an equilibrium in resource markets determining Jane's and Sarah's incomes, in the product market allocating L and K between X and Y, and the condition for efficiency in output that matches production to Jane and Sarah's preferences.