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DEPARTMENT OF ECONOMICS UNIVERSITY OF CANTERBURY

ECONOMICS 230W, 2010

Microeconomic Theory with Calculus

Mid-Year Exam: Model Answer

1. Memory-Dump Questions:

See the Memory-Dump Answers on Moodle for the answers to these questions.

2a) Cartman’s budget set is

244204(168)ISn;;！;

IS,0

158168！！n

where I is the number of units consumed of ice-cream, S is the number of units consumed of

sweets, and n is the number of hours of leisure (=168-the number of hours worked).

b) When I=5, the budget line is

44682.Sn;？

This gives a budget line with a slope of -1. The budget set is the shaded area below this line bounded by the vertical lines at n=158 and n=168. The key points on this trapezium-shaped

set are that S=2.5 when n=168 and S=12.5 when n=158.

3a) To show these preferences, divide your graph into four regions by drawing the vertical line at F=10, and the horizontal line at S=15. In the bottom-left quadrant of this graph, both

commodities are bads, so the indifference curves are downward sloping with the direction of increasing preferences being towards the origin. In the top-right quadrant, both commodities are goods, with the direction of increasing preferences being up and to the right. In the remaining two quadrants, one commodity is a good and one a bad, giving upward-sloping indifference curves, with, again, the direction of increasing preferences being away from the centre. To fully show the indifference curves, you should then connect up the indifference curves between each region. One way of doing this is to draw circles around Jack’s worst

possible point (where F=10 and S=15), but it is not the only way.

b) You are not given any information about whether the marginal rate of substitution is increasing or decreasing in each region, so it seems that you cannot tell whether Jack’s

preferences are convex or not. If you draw the indifference curves as circles, his preferences will be strictly concave, but you could draw a diminishing MRS in the top-right quadrant, an increasing MRS in the bottom-left quadrant, etc. so that the diagram looks like a spider web, and within each region preferences will be convex. The correct answer, however, is that preferences are definitely not convex. To see this, draw a straight line between a point on an indifference curve in the bottom-left quadrant to a point on the same indifference curve in the

ECON 230W: Microeconomic Theory with Calculus Mid-Year Exam, 2010: Model Answer

top-right quadrant. Unambiguously, this straight line must pass through lower indifference curves, and so Jack’s preferences are not convex.

4. Since her old bundle contained no tea, Emma can still just afford her old bundle after the price of tea falls. As a result, there is no need to shift the budget line back to remove the Slutzky-defined income effecti.e. there is no income effect for the Slutzky decomposition.

The fall in the price of tea will, however, make Emma better off, so a reduction in income is needed to get Emma back to her old indifference curve. With this reduced income at the new prices, Emma will consume only tea, but less than at her new bundle. There is therefore both an income and a substitution effect.

5a) First, we need to set the negative of the MRS equal to the ratio of prices:

MU1；？？;;MRSxx1, 12MU2

so we have

p1xx;;？1. (1) 12p2

We now need to combine this with the budget equation,

pxpxm;？, (2) 1122

xx.to solve for and Rearrange Equation (1) to give 12

pxpppx？；；(), 221221

and so the budget equation becomes

pxpppxm;；；？(), 111221

m(？；x1. (3) 1pp12

Put this back into Equation (1) to get

pm1x？；. (4) 2ppp212

xb) From Equation (3), we can see that the demand for will increase with income if 1

pp.ppxand will decrease with income if It would seem that whether is normal 12121

or inferior will then depend on prices. Note, however, that Equation (3) only describes the

xdemand for good 1 at interior solutions, and we can see that equation that can only be 1

pp.xpositive if It follows therefore that is normal at interior solutions. Similarly, 121

xfrom Equation (4) it is clear that must be inferior at interior solutions. Finally, 2

ECON 230W: Microeconomic Theory with Calculus Mid-Year Exam, 2010: Model Answer

p,xpirrespective of prices, decreases in and increases in and the reverse is true for 211

x. Both goods, therefore, ordinary and substitutes for the other. 2

6a) The indifference curves are L-shaped. Letting H be the number of hats and S the number of

hat stands, the indifference curve through the point (H,S)=(40,4) has its corner at that point,

and the indifference curve through (H,S)=(50,2) has its corner at (H,S)=(20,2).

b) To work out demands for perfect-complement preferences, ask how much a bundle at one of the corners would cost and then how many such bundles the consumer could afford. In this case, a bundle of 1 hat stand and 10 hats would cost Tomáš \$300. He can therefore afford

exactly 2 such bundles, which would imply buying 2 hat stands and 20 hats.

c) Now a bundle of 1 hat stand and 10 hats would cost Tomáš \$600 and so he can only afford

one such bundle. He thus will demand 1 hat stand and 10 hats. Since there is no substitution effect with perfect-complement preferences, the decline in demand for hat stands from 2 to 1 is entirely due to the income effect.

7a) Babe has Cobb-Douglas preferences, for which the demand functions are

mm0.2;0.8.AB？？ ppAB

(You can derive these, but it is easier to have memorised the formula for Cobb-Douglas demands from the memory dump testbank.) In a question where a consumer starts off with an endowment rather than monetary income, income, m, is just the market value of the

endowment:

mpp？;？1530750. AB

This gives demands of

AB？？15;30.

b) When the price of acorn changes, the budget line will pivot around the endowment point. From part a), you know that the endowment point is also is current optimal consumption. When the budget line tilts, therefore, the effect on his consumption will be the same as when isolating out the substitution effect. We know that the substitution effect will only be zero when the consumer has preferences that have a kink at the optimal point, and as these are Cobb-Douglas preferences, we know that the substitution effect must be negative. We also know that when there is a substitution effect, the substitution away from the original optimal consumption bundle will make the consumer better off. We can therefore be sure that Babe will be better off, even though we don’t know which way the price of acorns has moved.

ECON 230W: Microeconomic Theory with Calculus Mid-Year Exam, 2010: Model Answer

8a) Your graph should have swords on the horizontal axis and ploughshares on the vertical. The production function is an upward-sloping curve, starting from the origin with diminishing slope. The iso-profit line for profit = 0 is a straight line from the origin with a slope of 2/5. The iso-profit line for profit =1,000 is parallel to the profit=0 line, starting from P=200 on the

vertical axis. This line should cut through the production function, not be tangential to it or lie completely above it.

b) To maximise profit, Pierre will set the ratio of the input price to the output price equal to the marginal product. This gives

225wp？(？S625. 2pwS

Substituting the optimal S back into the production function gives

p P1250.w

ECON 230W: Microeconomic Theory with Calculus Mid-Year Exam, 2010: Model Answer

Summary Statistics:

Here are the median, average and maximum scores for each of the 9 questions on the test, and

for the overall grade as a percentage:

Question 1 2 3 4 5 6 7 8 9 Total (%)

Median

Average

Maximum

The number of students in each grade range was

Grade A+++ A++ A+ A A- B+ B B- C+ C C- D E

Bottom of Range 95 90 85 80 75 70 65 60 55 50 45 40 0

# Students

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