a) Y= ln(3x2+5x+10)
a) MR, MC, AR and AC
b) Average cost maximizing output
Average cost is independent of the price of the commodities. However, it depends on the elasticity of demand and supply. Average cost curve varies with the produced output in the short run. However, in the long run, the average cost curve is U shaped depicting increasing returns to scale. The average cost of a firm is related to the marginal cost. while the average cost declines as the quantity of products are increased, the marginal cost of the firm is less than the average cost. the two are equal whenever the average cost is neither increasing nor decreasing. However, the marginal cost is higher than the average cost of an organization whenever the average cost is increasing.
a) Profit function
Qb = 600-1.33Pb
TR=[Pa(350-0.5Pa)+Pb(600-1.33P)-$10,000 – 50(350-0.5Pa) – 150(600-1.33Pb)]
TR = 350Pa-0.5Pa2+600Pb-1.33Pb2-10,000-17500-25Pa-90000-199.5Pb
b) Profit maximizing quantity
=-400.5Pb-Pa-1.33P b2-117175 = 0
=Pa = -400.5Pb-1.33P b2-117175 substitute in the below equation
Pb= 122.18(-400.5Pb-1.33Pb2-117175)-0.18 (-400.5Pb-1.33P b2-117175)2– 117900.5
Pb= -48933.08b-162.49 Pb2-14316441+70.09+0.2394 Pb2+21091.5
Max U= 10x0.4y0.7 at 400 = 5x+7.50y
dx= 10x0.4y0.7 – µ(5x+7.50y-400)
=4 y0.7/ x-0.6 =5µ
= µ = 4y0.7/x0.3*1/5
µ = 4y0.7/5 x0.3
400= 5x+7.5 (7/6x)
Y = 7/6 (29.09)
Max U=10(29)0.4(3y)0.7=453.9358063 =454
c) MRTS = MPL/MPK
= 4x0.6y0.7/7x0.4y0.3 –(4y/7x) =-0.6695
The MRTS indicates the level at which a unit of good x can be exchanged for a unit of good y. A MRTS of -0.7 indicates that a substitution of a unit of good x for -0.7 of good y. the effect is negative indicating that no substitution is possible.
Question 5: Cobb-Douglas production function
a) The function represents the relationship between inputs and outputs during production. The above function has the following components:
Q- total production
L – Inputs in terms of labor
K – inputs in terms of capital
20 – is a total factor productivity component
0.6 and 0.5 – these are elasticity of labor and capital respectively.
The responsiveness of the total output to the changes in labor and capital inputs is measured by the elasticity of labor and capital. The above production function represents increasing returns to scale since the sum of the elasticity of labor and capital is greater than 1 (1.1).
b) MPL = 12L-0.4 K0.5
MPK = 10L0.6 K-0.5
MRTS = MPL/MPK = K/L
L/K= 50/100 = ½
MRTS = 12L-0.4 K0.5 / 10L0.6 K-0.5 = ½ therefore, L: K = 1:2
Therefore, K=2L substituting for K in the production function,
L= 176.77 = 177 units of labor
K = 2L = 2*177 = 354 units of capital
c) percent of cost spent on Labor
The firm spends 20% of its production cost on labor while the rest of the cost is spent on capital. It is evident that capital is expensive for the company. In addition, the business environment is capital intensive. However, the firm can substitute capital for labor at the rate of 1:2 in order to minimize on the cost of production.
Question 6: Demand and Supply
P=50-0.5X2 and P=2X2-10
Demand =supply, 50-0.5X2 = 2X2-10
24 = X2
X = 4,898 units
50-0.5X2 = 50-0.5*4.89892
P = $38,000
Therefore, equilibrium quantity is 24,000 while equilibrium price is $ 38,000.
Subsidy =$ 4 per unit produced
Supply equation = 2X2-10 – $4
50-0.5X2 = 2X2-10 – 4
64/2.5 = 2.5X2/2.5 = X2 = 25.6
X = 5,058 units
P = 50 – 0.5X2
P = 50 – 12.8
P = $37,2 00
Therefore, the new equilibrium quantity and price are 5,058 units and $ 37,200 respectively.
c) Sharing of subsidy between producers and consumers
The introduction of a subsidy by the government will enable the supply curve to shift to the right by the margin of the introduced subsidy. This implies an increase in the quantity supplied that will have an effect of reducing the price paid by consumers for the product.
d) Consumer and Producer Surplus
Equilibrium price is -$5
50-0.5X2-5 = 2X2-10 – 4
59/2.5 = 2.5 X2/2.5
X = 4.857 *1000
P = 50-0.5X2-5
0.5X2 = 45
P = 9.486*1000
P = $9,486
Supply curve p = 2X2-10 – 4
Intercept = -14*1000
Producer surplus= (14000+9486)*0.5*4857
Demand curve= 50-0.5X2-5 = 45 -0.5 X2
Intercept = 45*1000
A tariff is a trade barrier imposed by the government to restrict trade. The impact of a tariff is to increase in the import price of the commodity making consumers to reduce its consumption hence reduce the imported product. The consumer surplus would reduce while the producer surplus would also reduce. However, the government revenue would increase (Moyer, McGuigan & Kretlow, 2001).
Question 7: Saving for Retirement
a) The favorable interest rate for the person is that of 6.25% per annum. The interest rate is favorable for the person because its annuity factor is higher than that of 6.35% indicating that the savings of the person will earn a higher interest per year.
b) Between 25-29 – saves $1500 per year
Savings between ages 25-29
= n -5 while r= 1500 per year
Between 30-44 saves $ 2500
R =2500 and n=15
S = 69024.054 ((1+0.064508)/1)20=240982
The total value at retirement of the account is
c) Only save between 45-64
N=20, i= 0.0645080
R = 240982 (0.0645080/1.0645080-1)
R = 6239.99
Question 8: Net Present value
The net present values for each option is negative. The buy option has a higher negative value of -$27,620.4 compared to the lease that has a less negative value of -$27202.3. this indicates that the lease option is the better option that should be chosen since the cost to be incurred is lower than the cost to be incurred in a buy option.
Question 9: Bonds
$10,000 bond redeemable in 15.5 years at a monthly coupon of 5.5%
a) Purchase price today given a yield market interest of 6%
The bond price today can be given by the formula:
Where C – coupon payments, n – number of payments, i- interest rate and M – value at maturity.
Coupon payments= (12*15)+6 = 186
Value of coupon payments = coupon is paid monthly. Therefore, divide the coupon rate by 12. Coupon payments = 0.055/12 *10000 = $45.83
Monthly yield = 0.06/12 = 0.005
Bond price = $45.83*[1-[1/(1+0.005)186]]/0.005+10,000/(1+0.005)186
=$45.83*120.9 + 3955.38
b) Premium discount
A bond that sells at a given coupon rate is a bond that sells at par value. However, sometimes the bond can sell at a price that is either higher or lower than the par value. A premium is a bond that is sold for a price that is higher than the par value. On the contrary, a discount is a bond sold for a price that is less than t the par value.
c) Holding bond to maturity with monthly interest earnings saved in annuity. Earnings at maturity:
C = Cash flow per period
i = interest rate
n = number of payments
Value of coupon payments = 0.055/12 *10000 = $45.83
Monthly yield = 0.062/12 = 0.0052
Moyer, C., McGuigan, R. & Kretlow, J. (2001) Contemporary Financial Management, 8th Edition. OH: Cincinnati: South-Western College Publishing.