Assignment 7

Assignment7

1. It is known that quantity demanded decreases by two units for each $1 increase in price. At a price of $5, quantity demanded is ten units.

a. What will be the quantity demanded if price is zero?
b. Write an equation for quantity demanded as a function of price.
c. Write an equation that expresses price as a function of quantity.
d. Write an equation for total revenue.


Ans : Let us assume that the demand function is given by Q = a + bP

It is given that quantity demanded decreases (Q) by 2 units when Price increases (P) by 1 unit

We know that Slope = Q/P = -2 (negative sign is because if Q decreases , P increases)

Substituting the value of slope in the above equation , we get

Q=a + (-2)P
Q= a – 2P

At P = $5, Q=10

So, 10 = a-2*5
a = 20
Q = 20 – 2P

a. At price=0, Q=20
b. Q=20-2P
c. P=10-0.5Q
d. TR = PQ=10Q-0.5Q2

2. A market consists of three people, A, B, and C, whose individual demand equations are as follows:


A.: P=35-0.5QA
B: P=50- 0.25QB
C: P=40-2.00 QC

The industry supply equation is given by QS = 40 +3.5 P.

a. Determine the equilibrium price and quantity.
b. Determine the amount that will be purchased by each individual.

Ans: Market demand is the horizontal summation of individual demand.
So, first find our indidual demand
QA=70-2P
QB=200-4P
QC=20-0.5P

QD= QA+ QB+ QC=290-6.5P


a. QS =QD (Since at equilibrium quantity demanded = quantity supplied)
Equating the above equation , we get P=25,Q=127.5
b. QA=70-2P = 70-2*25 = 20
QB=200-4P = 200-4*25 = 100
QC=20-0.5P = 20-0.5P = 7.5

3. The demand equation faced by Du Mont Electronics for its personal computers is given by P = 10,000- 4Q

a. Write the marginal revenue equation.
b. At what price and quantity will marginal revenue be zero?
c. At what price and quantity will total revenue be maximized?
d. If price is increased from $6,000 to $7,000, what will the effect be on total revenue? What does this imply about price elasticity?
Ans:
P=10000 – 4Q
a. MR = dPQ/dQ = 10000-8Q
b. MR= 0 = 10000-8Q
So, Q = 1250, P=10000-5000 = 5000
c. TR is maximum when MR =0 , So, TR will be maximized when P=5000 and
Q=1250
d. At price $6000, Q=1000, TR=PQ=6000000
At price $7000, Q=750, TR=PQ=5250000
Since TR decreases, demand is elastic.

4. Write a demand equation for which the price elasticity of demand is zero for all prices.

Ans: QD = a
5. The price elasticity for rice is estimated to be -0.4 and the income elasticity is 0.8. At a price of $0.40 per pound and a per capita income of $ 20,000, the demand for rice is 50 million tons per year.

a. Is rice an inferior good, a necessity, or a luxury? Explain.
b. If rice per-capita income increases to $20,500, what will be the quantity demanded of rice?
c. If the price of rice increases to $0.41 per pound and income per capita remains at $20,000, what will be the quantity demanded?
Ans:
a. Inferior goods are those where quantity demanded decreases when Income increases. For luxury goods price elasticity is > 1, For necessity price elasticity is less than than 0
In this case income elasticity is 0.8 which is greater than 0 and hence it is not inferior
Price elasticity is -04. which is less than 0 and hence it is a necessity.
b. %change in Q/%change in I = 0.8
Q2-Q1
---------
Q1
--------- = 0.8
I2-I1
---------
I1

Q2-50/50/20500-20000/20000 = 0.8
Q2 = 51
c. Q2-50/50/0.41-0.40/0.40 = -0.4
Q2 = 49.5 million

6. The McNight Company is a major producer of steel. Management estimates that the demand for the company’s steel is given by the equation.

Qs =5,000 – 1,000 Ps +0.1I + 100 Pa

where QS is steel demand in thousands of tons per year, PS is the price of steel in dollars per pound, I is income per capita, and Pa is the price of aluminum in dollars per pound. Initially, the price of steel is $1 per pound, income per capita is $20,000, and the price of aluminum is $0.80 per pound.

a. How much steel will be demanded at the initial prices and income?
b. What is the point income elasticity at the initial values?
c. What is the point cross elasticity between steel and aluminum? Are steel and aluminum substitutes or complements?
d. If the objective is to maintain the quantity of steel demanded as computed in part (a), what reduction in steel prices will be necessary to compensate for a $20 reduction in the price of aluminum?

a. Qs = 5000-1000*1+0.1*20000+100*.80 = 6080
b. Point income elasticity = dQd/dI*I/Qd = 0.1*20000/6080 = 0.33
c. Point cross elasticity = dQs/dPa*Pa/Qs = 100*0.80/6080 = 0.01
d. 6080 = 5000-1000*Ps+0.1*20000+100*0.6
1000Ps = 60+2000+5000-6080
1000Ps = 980
Ps = 0.98
Reduction required = 1-0.98/1*100 = 2%

7. Consider the following five data points:

X -1.0 0.0 1.0 2.0 3.0
Y -1.0 1.0 1.0 2.5 3.5

a. Use regression analysis to calculate by hand the estimated coefficients of the equation Y = B + aX.
b. Compute the coefficient of determination.
c. What is the predicted value of Y for X = 1.0? For X = 3.5?

8. Annual prices and beef consumption per capita in six cities are as follows:

City Price per Pound Consumption per Capita
1 $2.00 55
2 1.90 60
3 2.10 50
4 1.80 70
5 2.30 45
6 2.20 48

If demand equation is to be estimated using these data, would the linear form (i.e., Q = B + aP) or the multiplicative form (i.e., Q = BPa) be more appropriate? Explain.

Ans:
R2(linear model) = 0.92
R2(multiplicative model) = 0.96 Hence multiplicative model is more appropiate

Click below to get the calculations
Click Here

9. The MacWend Drive- In has determined that demand for hamburgers is given by the following equation:

Q = 205.2 + 23.0A – 200.PM + 100.PC + 0.51I

where Q is the number of hamburgers sold per month (in 1,000s), A is the advertising expenditures during the previous month (in $1,000s), PM is the price of MacWend burgers (dollars), PC is the price of hamburgers of the company’s major competitor (dollars), and I is income per capita in the surrounding community (in $1,000s).
a. Are the signs of the individual coefficients consistent with predictions from economic theory? Explain.
b. If A = $5,000, PM = $1, PC = $1.20, and I = $20,000, how many hamburgers will be demanded?
c. What is the advertising elasticity at A = $5,000?

10. Motorland Recreational Vehicles estimates the monthly demand (Q) for its product is given by the equation

log Q = 1.00 – 1.50log P + 3.00 log I R2 = 0.21

where P is price and I is income per capita in thousands. The t-statistics are shown in parentheses and logarithms to the base 10 were used to transform the equation. Assume that estimates are generated by sample of 400 observations.

a. Rewrite the expression as a multiplicative demand equation.
b. Based on the equation, is the product an inferior good, a necessity, or a luxury good?
c. Is the equation likely to be useful in predicting demand for Motorland’s product? Why or why not?