Estimate the demand function – price elasticity of demand
Assignment instructions:
Answer all questions and show all your work. Whether the question requires mathematical calculations or descriptive analysis (deductive or inductive) to arrive at the final answer, you must you show all steps, calculations, and discussions needed to arrive at the final answer or conclusion. Answers should be written in easy to read and understand manner.
Note: You are required to apply the relevant methods learned in this course when answering the questions. Use the language of economics or terminology you have learned in the chapter when writing your answer or solving a problem.
Assignment:
1. Suppose the estimated demand equation of good X looks as follows:
QX = 10,000 – 2 PX + 3 PY – 4.5M
Where Px is the price of the product itself
PY is the price of another (related) good
M is buyers’ income.
Suppose currently PX = $100, PY = $50, and M = $2,000.
a. What is the price elasticity of demand of good X?
b. What is the cross-price elasticity of good x with respect to the price of good y?
c. what is the income elasticity of good X?
d. Estimate the demand function (representing the demand curve) of this product.
e. Suppose the local government decides to raise the sales tax on product X, causing the price to rise by 10%. Will sales of the product X rise or fall, and by what percentage amount?
f. If the seller of the product (X) wants to increase her sales quantity by 10% through a price- change, what should she do to price – increase? decrease? By what percentage amount?
2. A manufactures of a well-known brand of tooth paste is considering changing the way he prices his tooth paste. Currently, he is selling it at $5.00 per bottle. He hired a college professor of economics to estimate the demand function for the product. The demand function estimated by the professor is given below:
Qd = 3 – 0.25P
Where P represents the price per bottle.
Is the manufacturer maximizing his sales revenue by charging $5.00 for each bottle? If your answer is no, how much should he charge in order to maximize his sales revenue?
3. For 2006-2011, a company has collected the following data on quarterly sales of its product.
| Year |
Quarter |
t |
Sales (000) |
| 2005 |
q1 |
1 |
2270.00 |
|
q2 |
2 |
1800.00 |
|
|
q3 |
3 |
2000.00 |
|
|
q4 |
4 |
2050.00 |
|
| 2006 |
q1 |
5 |
2440.00 |
|
q2 |
6 |
2000.00 |
|
|
q3 |
7 |
2080.00 |
|
|
q4 |
8 |
2200.00 |
|
| 2007 |
q1 |
9 |
2410.00 |
|
q2 |
10 |
2200.00 |
|
|
q3 |
11 |
2450.00 |
|
|
q4 |
12 |
2600.00 |
|
| 2008 |
q1 |
13 |
2630.00 |
|
q2 |
14 |
2500.00 |
|
|
q3 |
15 |
2530.00 |
|
|
q4 |
16 |
2300.00 |
|
| 2009 |
q1 |
17 |
2309.00 |
|
q2 |
18 |
2150.00 |
|
|
q3 |
19 |
2380.00 |
|
|
q4 |
20 |
2632.00 |
|
| 2010 |
q1 |
21 |
2700.00 |
|
q2 |
22 |
2410.00 |
|
|
q3 |
23 |
2560.00 |
|
|
q4 |
24 |
2600.00 |
|
| 2011 |
q1 |
25 |
2900.00 |
|
q2 |
26 |
2800.00 |
|
|
q3 |
27 |
2710.00 |
|
|
q4 |
28 |
2680.00 |
a. The manager would like to know if there is an upward trend in sales of the product. Use the data above to estimate the quarterly trend in sales using a linear trend model of the form:
Qt = a + bt.
Where Qt = quarterly sales revenue
Does your statistical analysis indicate a trend? If so, is it an upward or downward trend and how great is it? Is it a statistically significant trend (use the 5 percent level of significance)?
b. Now adjust your statistical model to account for seasonal variation in sales. Estimate the following model of sales:
Qt = a + bt + c1 D1 + c2 D2 + c3 D3
Where D1, D2 and D3 are respectively dummy variables for quarters I, II, and III (e.g., D1 is equal to 1 in the first quarter of each year and 0 in other quarters).
Does the data indicate a statistically significant seasonal pattern (use the 5 percent level of significance)? If so, what is the seasonal pattern of sales of the product?
c. Comparing your estimates of the trend in sales in parts a and b, which estimate is likely to be more accurate? Why?
d. Using the estimated forecast equation from part b, forecast sales for quarter 1, 2, 3 and 4 of 2012; quarter 1, 2, 3, and 4 of quarter 13, and quarter 1 of 2014. What possible concerns do you have about this forecasts?
4. Paul’s Plumbing is a small owner operated business. Paul the owner and manager knows the fixed cost of the plant is $780 per day and includes the rent for building, tools and equipment. Variable costs included: labor, energy and supplies costs. He collected data on output and total variable cost over a ten day period.
Day Q AVC
1 10 $50.00
2 20 43.00
3 40 26.00
4 50 25.00
5 70 18.85
6 80 18.37
7 100 15.50
8 120 16.67
9 150 21.00
10 160 28.75
Where Q is total output per day.
a. Use the data to run the appropriate regression to estimate the parameters for the empirical cost function:
AVC = a + bQ + cQ2
b. Using a 10 percent significance level, discuss the suitability of the parameter estimates obtained in part a. Consider both the algebraic signs and statistical significance of the parameter estimates.
c. Present the estimated average variable cost, total variable cost, and short-run marginal cost functions.
d. At what level of output does AVC reach its minimum value? What is the minimum value of AVC at its minimum?
