Econ 604. Problem Set #1
KEY
1. Statistical evidence indicates that equations defining the demand and supply functions
of 1 pound bags of Starbucks Coffee at The Ukrops near VCU to be
Qd
Qs
=
=
Where
Currnelty
1050
60
+
50P
20P
-
100Ps + .1I
40w
P
Ps
I
is the price of a 1 pound bag of coffee
is the price of maple walnut scones ABD
is mean per capita monthly disposable income of students.
w
is the mean hourly wage pad to coffee roasters.
I
Ps
w
=
=
=
$1,000
$1.50
$8
a. Write the demand and supply curves for Starbucks coffee
Qd
=
=
1050
1000
-
50P
50P
-
150
Qs
=
=
60
-260
+
+
20P
20P
-
40(8)
b.
Solve for the equilibrium price and quantity
Qd
1000 -
50P
1260
P
Q
=
=
=
=
=
-260
70P
18
100
Qs
+
20P
+ 100
c. Graph your solution. Inverse supply and demand are
Ps
=
(Qs
+
260)/20
=
(1000 - Qd)/50
and
Pd
24
22
Price
20
18
16
14
12
10
0
50
100
150
200
250
Quantity
Suppose that Starbucks employees negotiate a $2 per hour wage increase.
d. Find the new supply curve
Qs
=
60
+
20P
=
-340 +
20P
-
40(10)
e. Find the new equilibrium
Qd
1000 -
50P
1340
P
Q
=
=
=
=
=
Qs
-340 +
70P
19.14
42.86
20P
f. Graph the change on your chart. New inverse supply is Ps = Qs/20 +17. Plotting as the
dashed line,
Price
24
22
20
18
16
14
12
10
0
50
100
150
Quantity
200
250
g. Analyze the changes. When did a change in supply occur? When did a change in
quantity supplied occur?
The upward shift of the supply curve in response to the wage increase represents a
change in supply. Quantity supply adjusts with price movements from the initial
equilibrium to the new one.
2. Suppose that a society’s conditions of medical care and military goods is given by the
equation
3X2
+
2Y2
=
100
where X represents Bradley fighting vehicles and Y represents doctors trained.
a. Graph this relationship. In your chare illustrate the following
- Inefficient points and efficient by unattainable points
- Increasing marginal opportunity cost
Unattainable
8
7
6
Increasing Marginal
Opportunity Costs
5
4
3
2
Inefficent
1
0
0
2
4
6
8
b. Find the slope of the above expression. Use it to analytically illustrate the
notion of increasing marginal opportunity costs.
Y
=
dY
=
dY/dX =
((100 – 3X2)/2)1/2
(½)((100 – 3X2)/2)-1/2(-3X)dX
(-1.5X)/Y
When X = 1, Y = 6.956, so dY/dX = -.215
When X = 2, Y = 6.63, so dY/dX = -.452
When X = 3, Y = 6.04, so dY/dX = -.75
And so on. In general, constant increments of X require society to give up
increasing increments of Y.
c. Suppose that the right hand side of the above expression increased to 200. What
effect would that have on your production possibilities frontier? What economic
explanation might motivate such a change?
The entire curve would shift out. A technological development may motivate such
a change.
2.2. If we cut four congruent squares out of the corners of a square piece of cardboard 12
inches on a side, we can fold up the four remaining flaps to obtain a tray without a top.
What size square should be cut in order to maximize the volume of the tray?
Volume = lwh. Here we set h=l. Thus we maximize
h (12-2h)2
h(144 – 24h +4h2)
144h -24h2 + 4h3
Taking FONC
144 – 48h + 12h2 =0
Dividing
12 -4h +h2
(6-h)(2-h)
Roots are 6 and 2. But from second order condition
-48 + 24h
We see that the expression is negative only for h=2.
Area is 2(8)(8) = 128
2.7 Suppose a firm’s total revenues depend on the amount produced (q) according to the
function
TR = 70q – q2
Total costs also depend on q:
TC
=
q2 + 30q + 100
a. What level of output should the firm produce in order to maximize profits (TR-TC)?

=
=
TR
70q-q2
d/dq =
70-2q
Thus 40 – 4q
q=10
-
TC
(q2 + 30q + 100)
=
2q-30 =
0
0
b. Show that the second-order conditions for a maximum are satisfied at the output level
found in part (a)
d2/dq2= -4 <0. Thus a maximum
c. Does the solution calculated here obey the “marginal revenue equals marginal cost”
rule? Explain
Yes,
MR
MC
=
=
70-2q and
2q+30