Ollscoil na hÉireann, Gaillimh GX_____ National University of Ireland, Galway
Semester 1 Examinations 2008/2009 Exam Code(s) 2BA5 Exam(s) 2nd B.A. (Economic & Social Studies) Module Code(s) EC222 Module(s) Techniques of Analysis Paper No. 1 Repeat Paper External Examiner(s) Professor Robert Wright Internal Examiner(s) Professor Eamon O’Shea
Mr. Stephen McNena
There are two sections in this exam. Please answer any 6 questions from section A and 6 questions from section B.
Duration 3 hours No. of Pages 6 pages, including this cover sheet Department(s) Economics Course Co-ordinator(s) Breda Lally, St. Angela’s College, Sligo Requirements: MCQ No Handout Handout of formulae to be distributed at St. Angela’s
College, Sligo Statistical Tables Yes, to be distributed at St. Angela’s College, Sligo Graph Paper Yes, if students request it Log Graph Paper No Other Material No
SECTION A MATHS
Answer any 6 of the 8 questions. 15 minutes per question. 1. (a) Consider the demand function represented by the equation: Qd = 50 – 5P
(i) Express Total Revenue (TR) as a function of Qd. (ii) What is the slope of the demand curve? What is the value of its intercept? (iii) Sketch the demand curve. (iv) At P = 6, calculate the quantity and the total revenue.
(b) Consider the following production function: Y = A f (K, L) = 10AK0.5L0.5 where Y is national output, A is technological knowledge, K is the capital stock, and L is labour.
Transform this Cobb-Douglas production function into a linear model using logarithms.
2. (a) A firm that makes paint sells their product for €7. Their cost function is
represented by the equation: TC = 200 + 5Q. (i) Calculate the breakeven quantity. (ii) Determine the Total Revenue, Fixed Cost, Variable Cost and Total Costs. (iii) Sketch the Total Cost and Total Revenue curves.
(b) Consider the following demand and supply equations:
Demand: P = 100 – 3Q Supply: P = 10 + 2Q
(i) Solve for the equilibrium price and quantity. (ii) What are the slopes of the two curves? 3. Consider an open economy described by the following equations: C = 300 + 0.9Yd Govt spending = 600 I (autonomous) = 400 Tax rate = 20% of incomes
Exports = 800 Imports = 27.5% of disposable income Note that C, Y and M are endogenous, and I, G, t and X are exogenous constants.
(a) Solve for the equilibrium national income, Y. (b) Then calculate disposable income, consumption and imports.
(c) Determine the Marginal Propensity to Consume and the Marginal Propensity to Save.
(d) Calculate the expenditure multiplier. (e) If an injection of extra autonomous expenditure causes national income to rise by
400, calculate the size of the extra expenditure.
4. Consider the demand function represented by the equation:
Qd = 30 - ⅓ P
(a) Express Total Revenue (TR) as a function of Q. (b) Determine the equation for Marginal Revenue. (c) Using differentiation, calculate the quantity where TR is maximised. (d) Calculate the price where TR is maximised. (e) Calculate the maximum Total Revenue.
5. Assume a Total Cost function of the form: TC = 30Q - 15Q² + 3Q3
(a) Derive a function for Average Cost (AC). (b) Derive a function for Marginal Cost (MC). (c) Find the levels of output that minimise AC and MC. (d) Calculate the minimum AC and the minimum MC. (e) Using algebra, confirm that AC = MC when AC is at a minimum.
6. A firm faces the following demand and Total Cost functions:
P = 125 – Q TC = 500 + 5Q + 0.5Q²
(a) Determine the equations for Total Revenue, Marginal Revenue, Average Cost, Marginal Cost and Profit.
(b) Derive the profit-maximising level of output. (c) Calculate the maximum profit.
(d) Derive the price and Marginal Revenue at this level of output. 7. A monopolist produces a single good (X) but sells it in two separate markets. The cost function is TC = 120 + 8Q. The demand function in each market is
P1 = 50 – 4q1 P2 = 80 – 3q2 where P1 and q1, P2 and q2 are the prices and quantity in markets 1 and 2. Find the prices and quantities in each market that maximise overall profits. 8. A firm has a Cobb-Douglas production function as follows: Q = L0.3K0.7
The price of labour is 3 and the price of capital is 15. The firm faces a total cost constraint of 150. Determine the values of L and K for which production is maximised, subject to the cost constraint.
SECTION B STATISTICS
Answer any 6 of the 8 questions. 15 minutes per question. 1. (a) Your sell newspapers at €1.80 each, breakfast rolls at €3.50 each and textbooks at €40 per book. During one day you sell 150 newspapers, 25 breakfast rolls and 10 textbooks. What is the weighted mean selling price?
(b) You invest some of your savings in a managed fund. The annual percentage returns earned by the fund for the last 5 years are outlined in the table below. Calculate the geometric mean annual return earned by the fund.
9.4% 13.8% 1.7% 11.9% 4.7%
(c) (i) Calculate the mean, median and mode of the following set of numbers. 0, 9, 9, 6, 9, 7, 9, 3, 3, 2, 1 (ii) Explain carefully why the median is sometimes a preferable measure of central location than the mean. 2. (a) A student is taking two courses: maths and economics. The probability the
student will pass maths is 0.75, and the probability of passing economics is 0.90. The probability of passing both subjects is 0.70. What is the probability of passing at least one subject?
(b) A firm uses two types of machines, the S40 and the S60. The probability of the S40 machine being available is 0.90 and the probability of the S60 being available is 0.80. What is the probability that both types of machine are available? What is the probability that both types of machine are not available?
(c) If you ask three strangers on campus, what is the probability that:
(i) All were born on a Wednesday? (ii) All were born on different days of the week?
(d) Sarah is the owner of a convenience store. Sarah collects data on how many items
each customer purchases, as shown in the table below:
X = no. of items P(x) 0 1 2 3
0.10 0.40 0.30 0.20
(i) Compute the mean and standard deviation for the distribution of number of items. (ii) If a customer enters the shop, what is the chance that they buy at least 2 items?
3. (a) A factory manager knows that over the long-run, 80% of the shipments from the factory arrive at their customers’ premises on-time, defined as within five working days. This week 20 shipments are dispatched from the factory.
(i) What is the probability that all of this week’s shipments are delivered on- time?
(ii) What is the probability that just 12 of this week’s shipments are delivered on-time?
(iii) Compute the mean and standard deviation of this probability distribution.
(b) The mean starting salary for college graduates in 2005 was €22,000. Assume that the distribution of starting salaries follows the normal distribution with a standard deviation of €3,000. What percent of the graduates have starting salaries:
(i) Between €19,000 and €25,000? (ii) More than €30,000? (iii) Calculate the wage level beyond which the 10% top-earning graduates lie.
4. (a) Assume that the mean life of a mobile phone battery is 70 hours. The distribution of the battery lives follows a normal distribution with a standard deviation of 5.5 hours. As part of their testing process, the manufacturer tests samples of 25 batteries.
(i) What can you say about the shape of the distribution of sample means? (ii) What fraction of the samples will have a mean life of more than 66 hours? (iii) What is the probability that the sample mean life is less than 80 hours?
(b) Information from the Irish Insurance Federation indicates the mean amount of life
assurance per household in Ireland is €150,000. The distribution is positively skewed. The standard deviation of the population is unknown.
(i) Suppose you select sixty samples of households. What is the expected shape of the distribution of the sixty sample means?
A random sample of 50 households revealed a standard deviation of €40,000. (ii) What is the probability of selecting a sample with mean greater than
€170,000? (iii) What is the probability of selecting a sample with mean less than €100,000?
5. The manager of a bank branch wants to study certain characteristics of the branch’s
customers. Specifically, the average current account balance, and whether they have a mortgage or not. A sample of 90 customers provides the following data:
420!=x s = €60 47 customers have a mortgage (a) Set up a 90% confidence interval for the population mean current account balance.
(b) Set up a 95% confidence interval for the proportion of customers who have a mortgage. 6 (a) The Sligo Chamber of Commerce wants to estimate the average time workers
spend travelling to work. A sample of 15 workers reveals the following number of minutes spent commuting:
29 38 38 33 38 21 45 34 40 37 37 42 30 29 35
The standard deviation of the sample data is 6 minutes. Develop a 99% confidence interval for the population mean.
(b) Policymakers in the Government want an estimate of the proportion of the population that support carbon taxes. The estimate is required to be within 0.02 of the true proportion. Assume a 98% level of confidence. How large a sample is required? How large a sample would be required if the Government have some prior knowledge that the proportion of people that support a carbon tax is 0.40? 7. The tram service operator in a large city claims that at least 40% of tram passengers have switched from commuting by car. A sample of 100 passengers reveals that 28 of them have switched from commuting by car. Is the sample experience different from that claimed by the tram operator at the 98% confidence level? You must follow all five steps in your answer. 8. The following information is available:
Ho: µ = 50 H1: µ ≠ 50
The sample mean is 49, the sample size is 36, and the population standard deviation is 5. Use the 0.05 significance level. Test the null hypothesis, making sure to state the decision rule and calculate the test statistic.