ECON 201 - PROBLEMS (Autumn 1996)

ECON 201 - PROBLEMS (Autumn 1996)

Microeconomics involves a number of simple techniques. Familiarity with them will help you not only to analyse problems yourself but also to understand the literature and pass exams The following exercises are for you to test your understanding of some of these techniques. Some of them will be discussed by the lecturers at fortnightly exercise classes starting in the third week of term. You should prepare the exercise neatly before the class and be prepared to ask and answer questions on the exercise. The lecturers may ask for work to be handed in at the class.

More difficult questions are marked *

Exercise 1 - For Monday 7th OctoberMLT 5 - 6 pm Week 3

1.An individual has income of M. Draw their budget curves in (x, y) space [X and Y are the only two goods] when

(i)Px = Py = 1

(ii)Px = 1, Py = 0.5

(iii)Px = 1, Py = 0.5,and consumers are prevented from consuming more than 0.8M units of X.

(iv)Px = 1, Py = 0.5 and the government distributes 0.2x to the individual free of charge.

(v)Px = 1, Py = 0.5, free distribution of 0.2x, but there is a 50% tax on X consumption beyond 0.2.

(vi)Px = 1 and Py = 0.5 upto M units of Y, but Py = 0.75 thereafter.

2.Show that true indifference curves from the same family can not intersect.

3.Find the MRS for the utility functions:

(i)U = xa yb

(ii)U = kx2ay2b

(iii)U = alogx + blogy

Comment on the results. What does this suggest about the cardinality of utility?

4.Sketch the indifference curves implied by the following utility functions for the values U = 1 and U = 2.

(i)U = 0.7x + 0.3y

(ii)U = x½ y½

(iii)U = x¼ y3/4

(iv)U = 2x

(v)U = min (x, y)

Drawing a budget line with Px = Py = 1 illustrate the proportions of X and Y consumed if U = 1. Which utility function offers the greatest scope for substitution between X and Y.

5.By hand or using Excel plot the indifference curves for

U = X½ Y½

Consider values of U from 1 to 2 in intervals of 0.2, and of X from 0.1 to 1.0 in intervals of 0.1.

Hint: Note that U = X½ Y½ can be rewritten

y = U2

X

So for each value of X and U in which we are interested (eg U = 1, X = 1,2,3 etc) we can solve for each Y and then plot the pairs of x and y to trace out each indifference curve.

Along with all the values of U also include

U = 1.581138 [U2 will be equal to 2.5]

Define Y as the numeraire (Py = 1) and use the diagram to plot the budget constraint when Px = 10 and income = 10 [10 = 10X + 1Y] on the same graph and indicate the optimum consumption bundle.

Next consider the case where Px = 25 and income = 10. Using half an A4 size piece of paper print the graph and use the lower half of the page to show how the Marshallian demand curve can be derived by varying the price between Px = 10 and 25.

Returning to the computer:

Draw the budget constraint for Px = 25, Py = 1 and income =15.81138 in addition to the previous budget constraints. Print as before. What is the consumption of X now? Illustrate the Hicksian Demand Curve for X using this information.

(i)the Marshallian demand curve for X, with income 10 and Px between 10 and 20,

(ii)the Hicksian demand curve for U = 1.4 for the same prices, and

(iii)the Engel curve with Px = 10.

Exercise 2 - For Monday 21rd October MLT 5 - 6 pm Week 5

1.An individual has a utility function defined over two goods as

if the price of X is 4, the price of Y is 2, and the consumer's income is 24, find the optimal levels of consumption of X and Y.

2.A consumer has the following utility function defined over the only two goods

u (x,y) = xy

They have income M and face prices Px and Py. Write down the maximization problem faced by the consumer; derive the first-order conditions; derive the Marshallian demand curves for X and Y.

Calculate the price and income elasticities for X and Y and also the compensated elasticity of demand for X.

Show, by example or otherwise, that the utility functions

u(x,y) = x0.5y0.5

and

u(x,y) = x2y2

yield the same demand curves.

*3.Derive the Marshallian demand curves and the Engel curves for the utility function

u(x,y) = (x - A) y(1-)

4.An individual has a utility function defined over two goods as

a)Using the Lagrange Multiplier method derive the demand curves for x and y.

b)Show that the adding up constraint holds.

c)If M = 40, Px = 1, Py = 1 find the actual consumption of x and y, and the original level of utility associated with this consumption bundle.

d)If the price of x were to rise to 2, how much income would you need to be able to purchase the original consumption bundle.

e)Compare this level of income with that required to restore you to your original indifference curve if the price of X were to rise to 2.

5.(i)Is it possible to simplify the utility function

U = 2.783 XY½

Explain why we can do this and what it tells us about utility functions

(ii)Find the marginal rate of substitution for the utility function, explain what it represents and its special relationship with the market rate of substitution.

(iii)Using the Lagrange Multiplier method, derive fully the demand curves for X and Y when agents have a utility function like the one above.

(iv)Derive the unique property regarding the share of X and Y in expenditure when the utility function has the Cobb-Douglas form.

What, therefore, is the elasticity of demand for X as income rises?

6.If agents have the utility function

U = X¼Y¾

and the budget constraint is

PxX + PyY = I

(i)Find the marginal rate of substitution for the utility function and the market rate of substitution.

(ii)Find the demand function for x and outline its principal properties. What is the demand for X when Px = 5, Py = 7.5 and I = 1000?

(iii)Without solving explicitly use your knowledge of demand and utility functions to explain why a utility function

U = ¼ log X + ¾ log Y

and a budget constraint of

10 X + 15 Y = 2000

would yield identical demand functions to (b).

7.Faced with vectors of prices P1 and P2, a consumer buys bundles given by the vectors Q1 and Q2 where

What can we say about the relative ranking of Q1 and Q2? Sketch a diagram showing the relevant budget lines to answer this question and add an indifference curve map which would be consistent with the observed behaviour.

Exercise 3 - For Monday 11th November MLT 5 - 6 pm Week 8

1.In the figure above Jo is in equilibrium with M = 300, Px = 4 and Py = 10.

(a)How much X does Jo consume?

(b)If Px were 2.5 how much X would Jo consume?

(c)How much income must be removed to compensate for the price fall?

(d)Identify the income effect

(e)Identify the substitution effect

(f)Is X inferior?

2.Let U = xy, M = 40, Px = 1, Py = 2.

(a)Find consumption of X and Y. Sketch the budget curve and locate the equilibrium

(b)Let Py fall to 1. What income would have purchased the original consuption bundle? At this income and prices what would actually be consumed. Draw the new budget curve and equilibrium.

(c)Identify the substitution effect on Y based on the compensation in (b) [with Slutsky rather than Hicksian compensation]

(d)Now assuming no compensation, identify the equilibrium and draw the budget curve at the new prices.

(e)Identify the income effect in terms of Y, and in money terms

(f)Identify the income and substitution effects on X.

3.We have three observations A, B, C, of Peter's purchasing behaviour across three goods

A B C

goodprice quantity price quantity price quantity

1 10 2 20 1 8 3

2 40 3 1 4 8 2

3 3 2 1 6 2 2

Write out a (3 x 3) matrix showing the cost of each bundle at each set of prices.Put prices down side and bundles across the top, thus;

Bundle

ABC

A

PricesB

C

Identify with a ring in cell (i, j) - row i column j - if bundle i is directly preferred to bundle j. Do the data satisfy WARP?

Now identify cases of indirect preference. Do the data satisfy SARP?

4.Illustrate the income and substitution effects of halving the price of X when

(i)indifference curves are L-shaped

(ii)indifference curves are linear

Exercise 4 - For Monday 25th November MLT 5 - 6 pm Week 10

6Jeffrey is a wealthy student at the University of Birmingham and has been given £2000 by Mummy and Daddy to spend as he chooses throughout the year. Jeremy also has a nice new car which he drives the 400 yards from Selly Oak to campus each day. Jeremy can park legally in the student car park for nothing or in the closer staff car park illegally from which he gets a higher utility. To park illegally costs Jeremy £40 for a staff parking card on the student black market plus he runs a risk that his car will be discovered and vandalised by Dr X, who resents wealthy students getting a free education and parking in her parking space. There is a 50% chance of being detected by Dr X and Jeremy always parks legally if he is indifferent between the two options.

(i)If Jeremy has a utility function of 2 log x if he parks illegally and log x if he parks legally, how much damage will Dr X have to inflict to persuade Jeremy to stick to the student car park?

(ii)The university is distressed by the level of vandalism on students' cars but is unable to catch Dr X, who is doing an average of £1000 worth of damage. It can take action to reduce the number of parking cards on the black market. To what price P would the black market price of the parking card have to rise to persuade Jeremy park legally? (You do not have to solve for a precise number for P).

(iii)Alternatively the university could improve its surveillance. If it patrolled the car parks constantly Dr X would stay away. However, they would then confiscate Jeremy's card if he was caught using the staff car park. What would the detection rate have to be to make Jeremy use the student car park if the black market price of the card was £200?

Production and Cost Functions

7(i)A perfectly competitive industry produces a good which sells for £100 per unit. If the cost function for the industry is wL + rk and the production function is Q = KL, find the long-run total cost function for the firm when  and  = ½.

(ii)In the short-run the quantity of capital employed by the firm cannot be varied. By setting K = and solving for L, show that the short run cost function is

If K = 1000 and W = 5 find the quantity produced by the firm in the short-run.

Continued....

(iii)If P were to rise to £120 per unit in the short-run what would happen to output?

(iv)With particular reference to the value of  +  in the production function, indicate what factors are likely to influence the equilibrium price in the long run.

CR1.1/SR/Econ9a.rev 11.10.95