Eco 507 Midterm
ECO 507
Midterm Test
1.(i.) ∝ =∆lnQ/∆lnP
∝ =P/Q* (∆Q/∆K) = Elasticity
The coefficients of double log model are the corresponding elasticities
Price elasticity = 1.247
Income elasticity = 1.905
(ii.)Price elasticity = -1.2
Income elasticity = 2
Cross price elasticity = 1.5
Current volume = 10 mil
Average income increase by 2.5%
New qty after increase in income =
Ie=2
2=%∆Q%∆I
2=%∆Q/2.5
%∆Q=5%
New Qty = 11.445 mil
To increase the sales volume only by 9.2% you would have to reduce the price.
%∆Q/%∆P=Pe
-5.25%∆P=-1.2
%∆P=4.375%
(iii). a. Maximize…Z = M + .5S + .5MS - S²
Subject to 30000S + 60000M = 1200000
Lagrangean…L=M+.5S+.MS-S2+λ1,200,000-30,000S-60,000M
∂L∂S=0.5+0.5M-2S-30,000λ
∂L∂M=1+0.5S-60,000λ
∂L∂λ=30,000S+60,000M
Equating λ, I get
1 + 0.5S/60000 = 0.5 + 0.5M – 2S
M = 4.5S
By substituting into budget constraint, I get
30000S + 60000 * 4.5S = 1200000
S = 4
M = 18
b. Cost function = 30000S + 60000M
Marginal cost of S = 30000
Marginal cost of M = 60000
Total marginal cost = 90000
c.
(iv.) a. Demand…Q = a – bP
E = (P/Q)*(∆Q/∆P)
E = -b (P/Q)
-0.4 = -b(4/2) b = 0.2 a = Q + bP
= 2 + 0.2 * 4 a = 2.08
Demand Equation…Q = 2.08 – 0.2P
2.(i) Q = LK
∂Q∂L = K
∂2Q∂L2 = 0
The second order derivative did not give a negative value, so it ignores the condition of diminishing marginal productivity of labor.
b. Q (L, K) = LK
Q (mL, mK) = m²LK
The output increases more than proportionally, there are increasing returns to scale.
c.
Q = LK
TC = wL + rK
L = wL + rK + λ (Q-LK)
∂L∂L = w + λ (K) =0
∂L∂K = r + λ (L) =0 w /r = K/L =RTS
In this equation, the firm should use K and L as given that ratio to minimize cost of production.
The Lagrangean Multiplier is marginal cost of any input to marginal benefit of any input should be same for any input. It explains if marginal cost –benefit ratio is greater for K than L, we have to substitute L for K to minimize cost.
d.
225 = LK
225 = 16L+144K
Midterm Test
1.(i.) ∝ =∆lnQ/∆lnP
∝ =P/Q* (∆Q/∆K) = Elasticity
The coefficients of double log model are the corresponding elasticities
Price elasticity = 1.247
Income elasticity = 1.905
(ii.)Price elasticity = -1.2
Income elasticity = 2
Cross price elasticity = 1.5
Current volume = 10 mil
Average income increase by 2.5%
New qty after increase in income =
Ie=2
2=%∆Q%∆I
2=%∆Q/2.5
%∆Q=5%
New Qty = 11.445 mil
To increase the sales volume only by 9.2% you would have to reduce the price.
%∆Q/%∆P=Pe
-5.25%∆P=-1.2
%∆P=4.375%
(iii). a. Maximize…Z = M + .5S + .5MS - S²
Subject to 30000S + 60000M = 1200000
Lagrangean…L=M+.5S+.MS-S2+λ1,200,000-30,000S-60,000M
∂L∂S=0.5+0.5M-2S-30,000λ
∂L∂M=1+0.5S-60,000λ
∂L∂λ=30,000S+60,000M
Equating λ, I get
1 + 0.5S/60000 = 0.5 + 0.5M – 2S
M = 4.5S
By substituting into budget constraint, I get
30000S + 60000 * 4.5S = 1200000
S = 4
M = 18
b. Cost function = 30000S + 60000M
Marginal cost of S = 30000
Marginal cost of M = 60000
Total marginal cost = 90000
c.
(iv.) a. Demand…Q = a – bP
E = (P/Q)*(∆Q/∆P)
E = -b (P/Q)
-0.4 = -b(4/2) b = 0.2 a = Q + bP
= 2 + 0.2 * 4 a = 2.08
Demand Equation…Q = 2.08 – 0.2P
2.(i) Q = LK
∂Q∂L = K
∂2Q∂L2 = 0
The second order derivative did not give a negative value, so it ignores the condition of diminishing marginal productivity of labor.
b. Q (L, K) = LK
Q (mL, mK) = m²LK
The output increases more than proportionally, there are increasing returns to scale.
c.
Q = LK
TC = wL + rK
L = wL + rK + λ (Q-LK)
∂L∂L = w + λ (K) =0
∂L∂K = r + λ (L) =0 w /r = K/L =RTS
In this equation, the firm should use K and L as given that ratio to minimize cost of production.
The Lagrangean Multiplier is marginal cost of any input to marginal benefit of any input should be same for any input. It explains if marginal cost –benefit ratio is greater for K than L, we have to substitute L for K to minimize cost.
d.
225 = LK
225 = 16L+144K