Econometrics questions
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group-assign-13--univariate-calculus-part-b---pdf-_.pdf
Group Assignments #13 Univariate Calculus (Part B)
1. Define the following
a. Concavity & Convexity
b. Marginal Function
c. Marginal Cost
d. Marginal Revenue
e. Marginal Demand
2. For the following function: a) find the successive derivatives; and b) evaluate it at x = 3
3 2
2 2
2 2
3 3
3 3
4 4
4 4
1) 3 9 7
a) b) At = 3,
y x x x
dy dy x
dx dx
d y d y
dx dx
d y d y
dx dx
d y d y
dx dx
= + + −
= =
= =
= =
= =
4
2 2
2 2
3 3
3 3
4 4
4 4
5 5
5 5
2) (5 )
a) b) At = 3,
y x
dy dy x
dx dx
d y d y
dx dx
d y d y
dx dx
d y d y
dx dx
d y d y
dx dx
= −
= =
= =
= =
= =
= =
2 2
2 2
3 3
3 3
3) (4 7)(9 2)
a) b) At = 3,
y x x
dy dy x
dx dx
d y d y
dx dx
d y d y
dx dx
= − +
= =
=
= =
= =
3. Perform the operation as indicated
2
2
2
2
1) 3 7 12
2) 13 78
avg
3) 12
4) 18 0.5 (demand function)
46 5) 1.5 4
160 6) 2 3 5
TC Q Q
TC dTC AC MC
Q dQ
Q Q
d
Q dQ
TR Q Q
TR dTR P AR MR
Q dQ
P Q
dTR TR PQ MR
dQ
AC Q Q
dTC TC AC Q MC
dQ
AC Q Q Q
TC
MC
π
π π π
= + +
= = = =
= − +
= = =
= −
= = = = =
= −
= = = =
= + +
= = = =
= − + +
=
=
i
4. If TR and AR are defined a function of Q as follows:
AR(Q) = P = f (Q) TR(Q) = P·Q = f (Q) ·Q
a. What does AR and TR represent graphically?
b. What is MR? (hint: use the product rule) ?
dTR MR
dQ = =
c. Demonstrate the slope of the MR curve (MR') is twice as steep as that of the demand curve (AR').
5. Consider a total cost function where Q is the single argument such that
TC(Q) = 10 + 2Q2 and Q ≥ 0.
a. Derive both the average and marginal cost curves .
2( ) 10 2
'( )
( )
TC Q Q
TC Q MC
AC Q
= +
= =
=
b. Calculate the quantity where AC = MC.
( ) and ( )AC Q MC Q= =
c. Assume the industry is perfectly competitive, what price corresponds to long run zero profit?
Graph P, MC, and AC.
Quantity
P ri c e
0 1 2 3 4 5 6 7 8 9 10 0
2
4
6
8
10
12
14
16
18
20
d. What does the graph suggest about the relationship between these three functions? What is the economic interpretation?
e. At what quantities is the marginal cost curve greater than or less than average curve?
