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Intermediate Microeconomics 301
First Mid-Term
KEY ANSWERS
Exercise 1 [30]: Demand is QD = 600 − 2p and supply is QS = 300 + 4p.
1. Supply = demand, so 600 − 2p = 300 + 4p, and thus equilibrium price and quantity are p∗ = 50
and Q∗ = 500.
2. Tax: 3$ per unit. The supply shifts. The new supply is QtS = 300 + 4(p − 3) and thus
QtS = 288 + 4p. The new equilibrium is determined by taking QtS = QD , and thus 288 + 4p =
600 − 2p. The new equilibrium price an quantity are p∗d = 52 and Qt = 496. Consumers now
pay p∗d = 52, and the suppliers get p∗s = 52 − 3 = 49. Thus consumers bear more the burden
of the tax compared to the suppliers.
Exercise 2 [30]: Linear demand curve, Q = 350 − 7P .
1. The inverse demand curve is P = − 17 Q + 50.
2. Give the definition of the elasticity (DEFINITION). Determine the elasticity for any P .
p
P 0
ε= Q
Q (P ) = 350−7P
(−7)
3. The price elasticity of demand at P = $25 is ε = −1, unit elasticity.
Exercise 3 [40]: U (x1 , x2 ) = x21 x2 .
1. The level of utility that corresponds to this indifference curve that passes through the point
(1, 4) is U (1, 4) = 12 × 4 = 4. The equation of the indifference curve is x2 = 4/x21 . The level
of utility that corresponds to this indifference curve that passes through the point (2, 6) is
U (2, 6) = 22 × 6 = 24. The equation of the indifference curve is x2 = 24/x21 .
2. GRAPH. The set of commodity bundles that Mary prefers to the bundle (1, 4) is on the right
of the IC of equation x2 = 4/x21 . The set of all commodity bundles (x1 , x2 ) such that Mary
prefers (2, 6) to these bundles is on the left of the IC of equation x2 = 24/x21 .
2
1
3. M RS(x1 , x2 ). DEFINITION. M RS(x1 , x2 ) = − MU
MU2 . M U1 = 2x1 x2 and M U2 = x1 , thus
2x1 x2
2x2
1
− MU
MU2 = − x2 = − x1 .
1
(a) The slope of Mary’s indifference curve at the point (1, 4) is M RS(1, 4) = − 2×4
1 = −8.
(b) The slope of her indifference curve at the point (2, 6) is M RS(2, 6) = − 2×6
2 = −6.
(c) The slope of his indifference curve at the point (2, 1) is M RS(2, 1) = − 2×1
2 = −1 and at
2×24
the point (1, 24) is M RS(1, 24) = − 1 = −48.
(d) The indifference curves you have drawn for Mary exhibit diminishing marginal rate of
substitution as along the same IC of equation x2 = 4/x21 , the M RS is decreasing: at the
point (1, 4) it is −8 and at (2, 1) it is −1. Same along the IC x2 = 24/x21 : the M RS is
decreasing: at (1, 24) it is −48 and at the point (2, 6) it is −6.
4. p1 = p2 = 2 and m = 20. The budget line is 2x1 + 2x2 = 20 or equivalently x2 = −x1 + 10.
1
2
2
5. At the optimal bundle, the M RS(x∗1 , x∗2 ) = − pp12 . So − 2x
x1 = − 2 which is equivalent to
x2 = 12 x1 . Plug that in the BC, 12 x1 = −x1 + 10, and thus x∗1 = 20
3 . Use x2 = −x1 + 10
.
Thus
the
level
of utility of Mary is
to define the optimal quantity of good 2, x∗2 = 10
3
20 2 10
4000
∗
∗
U (x1 , x2 ) = ( 3 ) 3 = 27 = 148.15.
2