# ECON/AMS 11B WINTER 2014 Final Exam Instructions

Question

ECON/AMS 11B

WINTER 2014

Final Exam

Instructions

• Please turn oﬀ all phones and other electronic devices.

• There are 6 questions worth a total of 54 points. 100%=50 points.

• No notes or books. A table of integration formulas is provided.

• You may use a simple scientiﬁc calculator. No graphing or programmable calculators.

• Read the questions carefully and check your answers.

•

For full credit—show all your work.

Good Luck!!!

NAME:

Problem

1

2

3

4

5

6

Total

Score

/8

/8

/8

/10

/10

/10

/50

1

ECON/AMS 11B

FINAL EXAM

Selected Integration Formulas

Basic rules.

uk+1

+ C,

k+1

1.

uk du =

2.

1

du = ln |u| + C.

u

3.

eu du = eu + C.

4.

f (u) ± g(u) du =

5.

c · f (u) du = c ·

k = −1.

f (u) du ±

g(u) du.

f (u) du.

Rational forms containing (a + bu).

6.

1

du

= ln |a + bu| + C.

a + bu

b

7.

u du

u

a

= − 2 ln |a + bu| + C.

a + bu

b

b

8.

u2 du

u2 au a2

=

− 2 + 3 ln |a + bu| + C.

a + bu

2b

b

b

9.

u2 du

u

a2

2a

= 2− 3

− 3 ln |a + bu| + C.

2

(a + bu)

b

b (a + bu)

b

Forms containing

10.

11.

12.

√

a + bu.

√

2(3bu − 2a)(a + bu)3/2

u a + bu du =

+ C.

15b2

√

u du

2(bu − 2a) a + bu

√

=

+ C.

3b2

a + bu

√

2(3b2 u2 − 4abu + 8a2 ) a + bu

u2 du

√

=

+ C.

15b3

a + bu

Exponential and logarithmic forms.

eau

+ C.

a

13.

eau du =

14.

ueau du =

15.

un eau du =

16.

un ln u du =

eau

(au − 1) + C.

a2

un eau n

−

a

a

un−1 eau du .

un+1 ln u

un+1

−

+ C,

n+1

(n + 1)2

n = −1.

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WINTER 2014

ECON/AMS 11B

FINAL EXAM

1. (8 pts) A ﬁrm’s marginal cost function is given by

dc

5q + 12

=

dq

q+2

and the ﬁrm’s ﬁxed cost is $3000. Find the ﬁrm’s cost function.

3

WINTER 2014

ECON/AMS 11B

FINAL EXAM

WINTER 2014

2. (8 pts) Find the Consumers’ surplus and Producers’ surplus at equilibrium for the market whose supply

and demand equations are given below.

• Supply: p = 10 + 0.25q,

• Demand: p = 180 − 0.1q 2 .

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ECON/AMS 11B

FINAL EXAM

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ECON/AMS 11B

FINAL EXAM

WINTER 2014

3. The average monthly demand for a ACME Widgets’ product is given by the relation

Q = 4 ln(5Y 2 − 3p),

where Q is the monthly demand, measured in 1000s of widgets; p is price of an ACME Widget, measured

in dollars; and Y is the average monthly household income in the market for the ﬁrm’s product, measured

in $1000s.

(a) (5 pts) Find Q, Qp and QY when p = 25 and Y = 4.

(b) (3 pts) Find the price-elasticity of demand when p = 25 and Y = 4.

Round your answers to 2 decimal places.

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ECON/AMS 11B

FINAL EXAM

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ECON/AMS 11B

FINAL EXAM

WINTER 2014

4. (10 pts) A monopolistic ﬁrm produces two types of widgets, type A and type B widgets. It costs the

ﬁrm $8 per unit to produce type A widgets and $4 per unit to produce type B widgets, and their weekly

ﬁxed cost is $2000. The demand functions for these products are given by

QA = 400(PB − 2PA + 14)

QB = 400(PA − PB ),

where PA and PB are the selling prices per unit of type A and type B widgets, and QA and QB are the

numbers of type A and type B widgets that the ﬁrm sells in a week.

Find the prices that the ﬁrm should set to maximize their weekly proﬁt. Use the second derivative test to

justify your claim that the ﬁrm’s proﬁt is maximized at the prices you found.

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ECON/AMS 11B

FINAL EXAM

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WINTER 2014

ECON/AMS 11B

FINAL EXAM

WINTER 2014

5. ACME Widget’s (AW) production function is given by

Q = 10K 0.6 L0.4 ,

where Q is the ﬁrm’s annual output, measured in widgets, K is the ﬁrm’s annual capital input and L is

the ﬁrm’s annual labor input. The price per unit of capital is pK = $5, 000 and the price per unit of labor

is pL = $2, 000.

(a) (6 pts) Find the levels of capital and labor input that AW should use to minimize the cost of

producing Q0 = 4, 000 widgets. What is the minimum cost?

(b) (2 pts) What is AW’s marginal cost? Explain your answer.

(c) (2 pts) Use the envelope theorem and linear approximation to estimate the change in AW’s (minimum)

cost of producing 4,000 widgets, if the price per unit of capital increases to $5,200 (assuming that all

else stays the same).

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ECON/AMS 11B

FINAL EXAM

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ECON/AMS 11B

FINAL EXAM

WINTER 2014

6. The Smith family’s utility function is given by

U (x, y, z) = 12 ln x + 7 ln y + 5 ln z,

where x, y and z are the quantities of X-goods, Y-goods and Z-goods that they consume per month. The

average prices of these goods are px = $15, py = $10 and pz = $5, respectively.

(a) (7 pts) Find the quantities of X-goods, Y-goods and Z-goods that the Smith family should consume

each month to maximize their utility, given that their monthly XYZ-budget is B = $3600. What is

their maximum utility?

(b) (3 pts) By approximately how much will the Smiths need to increase their XYZ-budget if they want

to increase their monthly utility by 3 utils? Explain your answer.

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FINAL EXAM

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FINAL EXAM

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