Review Questions 5
econ 11a
ams 11a
Solutions
1. The demand equation for a monopolist’s product is p = 250 − 0.2q.
a. Find the price-elasticity of demand (as a function of q).
250 − 0.2q
p
250 − 0.2q
q
q
=
.
=
ηq/p = dp
−0.2
−0.2q
dq
b. What is the price elasticity of demand when p = $50? Is demand elastic, inelastic, or
does demand have unit elasticity at this point?
To answer this, we first need to find the value of q when p = 50:
p = 50 =⇒ 50 = 250 − 0.2q =⇒ 0.2q = 200 =⇒ q = 1000.
250 − 200
1
= − . Since |η| < 1, it follows that demand is inelastic
−200
4
at that point on the demand curve.
Now, compute η =
c. Suppose that the price is lowered (from $50) to $49.25. Use your answer to part b. to
estimate the percentage change in demand.
We use the approximation, %∆q ≈ η · (%∆p). First, we need to compute the percentage change in price,
%∆p =
pnew − pold
49.25 − 50
· 100% = −1.5%.
· 100% =
pold
50
Now, we can compute the (approximate) percentage change in demand,
%∆q ≈ (−1/4)(−1.5%) = 0.375%.
I.e., demand will increase by approximately
3
8
of 1 percent.
d. What effect will this change in price have on the firm’s revenue? Be as precise as you
can, and explain your answer.
Recall the relationship between elasticity and marginal revenue:
dr
1
=p 1+
.
dq
η
In the example above, when q = 1000 and p = 50, η = −0.25, so
dr
= 50(1 − 1/0.25) = 50(1 − 4) = −150 < 0.
dq
1
Since marginal revenue is negative when q = 1000, an increase in q brings about a
decrease in revenue, and since lowering the price raises demand, the effect on revenue
of lowering the price will be to lower revenue.
2. The (short-run) production function for ACME Widgets is given by Q = 50K0 (L−10)2/3 ,
where Q is the weekly output of widgets, L is the weekly labor input, measured in $1000s,
and K0 is the fixed level of capital input.†
a. Compute the labor-elasticity of output, ηQ/L , as a function of L.
ηQ/L =
L
2
2L
dQ L
· = · 50K0 (L − 10)−1/3 ·
.
=
2/3
dL Q
3
50K0 (L − 10)
3(L − 10)
b. What is the labor-elasticity of output when labor input is $45000 a week?
6
90
= .
ηQ/L =
105
7
L=45
c. Suppose that ACME hires two additional widget polishers, at a combined cost of
$1500 a week. Use your answer to part b. to estimate the resulting percentage change
in output.
This is done using the approximation formula %∆Q ≈ ηQ/L (%∆L). First, compute
∆L
· 100% ≈ 3.33%. So,
the percentage change in labor input, %∆L =
Lold
%∆Q ≈
6
· 3.33% ≈ 2.85%.
7
d. Can the answers above be used to estimate the change in ACME’s weekly revenue? If
so, what is the resulting change in revenue? If not, explain why not.
No. Revenue depends on the output and the price of widgets. The production
function gives no information about the demand for widgets, and so is no help with
the price or the revenue.
3. The demand equation for the SlugWorks Calculator company is q = 500(p + 50)e−0.04p .
a. Compute the price-elasticity of demand, as a function of p.
dq
First,
= 500 (e−0.04p − 0.04e−0.04p (p + 50)) = −500e−0.04p (0.04p + 1), so
dp
ηq/p =
dq p
p
p(0.04p + 1)
· = −500e−0.04p (0.04p + 1) ·
=−
.
−0.04p
dp q
500(p + 50)e
p + 50
†
The difference between short-run and long-run production functions is that in the short run, capital
input is assumed to be fixed, and in the long run, both capital and labor inputs are allowed to vary.
2
b. Find the price, p0 , where demand has unit elasticity.
Unit price-elasticity of demand means η = −1, so we solve the equation
p(1 + 0.04p)
= −1 =⇒ p + 0.04p2 = p + 50 =⇒ p2 = 1250.
p + 50
√
Since the price must be positive, we conclude that p0 = 1250 (≈ 35.36) is the price
for which demand has unit elasticity.
η = −1 =⇒ −
c. How many calculators can SlugWorks’ sell at the price you found in part b.? What
will their revenue be?
√
When the price is p0 = 1250, Slugworks can sell
√
√
√
1250 + 50 · e−0.04· 1250 ≈ 10376
q0 = q( 1250) = 500 ·
calculators, rounding to the nearest whole calculator.
The firm’s revenue will be r = p0 · q0 ≈ $366, 847.
d. What will happen to their revenue if they raise the price above p0 (that you found in
part b.)? What will happen to revenue if they lower the price? Explain your answers.
To answer this, we return to the relationship between revenue and price elasticity that
we used in problem 1d., and we use the precise form of η that we found in part a. of
this problem:
1
p + 50
0.04p2 − 50
0.04p2 − 50
dr
=p 1+
=p 1−
=p·
=
.
dq
η
p(1 + 0.04p)
p(1 + 0.04p)
1 + 0.04p
From this we see that
√
dr
> 0 (since 0.04p2 −50 > 0 in
dq
this case), while on the other hand q decreases, (price increases =⇒ demand decreases).
If demand decreases when marginal
revenue is positive then revenue decreases! Thus,
√
raising the price above p0 = 1250 will lower the revenue.
√
dr
< 0 (since 0.04p2 − 50 < 0
(b) if p decreases below p0 = 1250, then on the one hand
dq
in this case), while on the other hand q increases. If q goes up and marginal
√ revenue
is negative, then revenue will decrease. So, lowering the price below p0 = 1250, also
lowers revenue.
(a) if p increases above p0 =
1250, then on the one hand,
Conclusion: changing
√ the price in either direction lowers revenue, so revenue is maximized when p = 1250.
Comments: In problems 1 and 3 we computed price elasticity of demand in two different
ways, because in problem 1 the demand equation had the form p = f (q), and in problem
3
3 the demand equation had the form q = h(p). The price elasticity of demand, however,
has the exact same interpretation in both cases, and it has the exact same relationship
with marginal revenue. We took advantage of this relationship with marginal revenue in
both problems.
4. The supply curve is given by qs = 0.01p2 + 0.4p − 2. (I’m using qs to denote the supplied
output.)
p(0.02p + 0.4)
0.02p2 + 0.4p
dqs p
·
=
=
. So, when p0 = 10 the price
a. ηqs /p =
dp qs
0.01p2 + 0.4p − 2
0.01p2 + 0.4p − 2
elasticity of supply is
6
ηqs /p = = 2.
3
p0 =10
b. According to the approximation formula,
%∆qs ≈ ηqs /p · %∆p = 2 ·
10.5 − 10
· 100% = 2 · (5%) = 10%.
10
4