The supply function of a profit-maximizing pricetaking mining company

The price of a commodity such as copper, p, is determined in the global market.

A mining company’s profit is its revenue minus its cost.

Company’ s total revenue is price of copper p, times the quantity of output sold, q.
TR(q) = p.q,

Total cost of production, TC, is either the firm's short run cost function or its long run cost
function, depending on whether we are interested in short run or long run supply.

Thus, the firm's profit function is
π(q) = TR(q) - TC(q)
= pq - TC(q),

Theory: The firm chooses its output q to maximize its profit π(q), taking price as given.
The supply function of a profit-maximizing pricetaking mining company (Cont’d)
 If we solve the maximization problem for all values of p, we get a function
q(p). The resulting function would be the firm's supply function.
Max
π(q)
F.O.C
𝜕π /𝜕𝑞=0
S.O.C
𝜕 2 π /𝜕 2 q < 0
Profit maximization
 To maximize the profit function we have to take the first derivative of the
profit function, and setting it equal to zero, and solving for q.

Taking the first derivative of π(q) = p.q -TC(q), with respect to q
p - 𝜕𝑇𝐶/𝜕𝑞= 0,
F.O.C
 Noting
tahat
𝜕𝑇𝐶/𝜕𝑞 is marginal cost, MC,
p = MC(q*).
The supply function of a profit-maximizing pricetaking mining company (Cont’d)
 Note that at a maximum profit the second derivative of profit at q* must be
negative. Thus taking derivative of p –MC(q) =0, we get:
S. O.C
- 𝜕𝑀𝐶/𝜕𝑞 < 0,
 Or changing the sign from negative to positive (multiplying both side by -1):
𝜕𝑀𝐶/𝜕𝑞 > 0,
 the marginal cost curve must be upward sloping at a profit-maximizing
output.
 This gives us the supply curve.
The supply function of a profit-maximizing pricetaking mining company (Cont’d)
 A mining company will continue to produce as long the expected cost of
producing each extra unit of out put is less than its expected price.
 The term "expected" is important, because it implies that the company
estimates its costs and commodity prices at time t.
 In other words, the company does not know what the price of copper, zinc,
nickel, or gold would be at the time when its product reaches the market.
 The company also is estimating its costs because it does not know how the
costs of its inputs labour, transportation, insurance and inventories would
change over this time horizon.
Total Cost, Average Cost and Marginal cost

Average cost, AC (q) is total cost per unit
of output
AC(q)= TC(q) / q

Marginal cost, MC(q) is the first derivative
of total cost with respect to q
MC(q) = 𝜕𝑇𝐶/𝜕𝑞

Marginal cost shows how much total cost
change as a result of producing one extra
unit of output.
Total cost
$ cost/t
TC(q) = FC +VC(q)
Fixed cost
Total cost, TC, at each period of time is
the sum of fixed and average costs.
{
$ cost per unit of output

𝑞𝑎
𝑞𝑏
Marginal cost
𝑞𝑎
𝑞𝑏
Average cost
Output per unit of time
Why marginal cost is U-shaped?

Capital equipment typically operate optimally at certain level of
production capacity as the output increases towards that optimal
level the marginal cost at first would be declining. Because each
extra unit of output would utilize the idle capital, and the
economies of scale are realized.

But after the optimal utilization of capital the diseconomies of
scale emerge . These emanate mainly from three kinds of
discords;
(i). The principal-agent and control problem;
as monitoring the
productivity and the quality of output in a larger scale operation
becomes increasingly difficult when different stakeholders have
conflicting objectives .
(ii). Problems arising from lack of co-ordination and efficient
flows of information; as production processes become more
complicated.
(iii). HR Challenges; as workers productivity diminish because of
the impersonal working conditions.
Cost-based production capacity
•
Full capacity, is defined as the level of production
that corresponds to the minimum long run
average cost curve of a firm. Cassel (1937) ,
Hickman (1964)
•
This level corresponds to the concept of
Minimum Efficient Scale, MES, which is the
production quantity, or range of quantities, in
which the economies of scale have been
exhausted but diseconomies of scale have not yet
surfaced.
•
Over the long-run, a perfectly competitive firm
can adjust the amount it uses of all factor inputs,
including those that are fixed in the short-run, by
adjusting the physical size of its plant or engaging
in merger and acquisition activities.
•
In making such adjustments, the firm will seek to
minimize its long-run average cost
Cost-based production capacity
•
In the short run, production capacity is
typically constrained by prior capital
expenditures.
•
Klein argued that long run average
cost curve may not have a minimum,
and proposed that full capacity is at
the point where the short run average
cost curve is tangent to the long run
average cost curve
•
Berndt and Morrison have suggested
that the full capacity is at the
minimum point of the short run
average cost curve beyond which the
cost of producing additional output
rises sharply.
Cost-based production capacity and unceratinty
•
All these concepts of capacity are
important.
•
During the time of uncertainty
firms focus on short term capacity.
• They resort to intensive margin
strategy
•
During the more stable times firms
are more confident about future
and are ready to invest.
• Extensive margin strategy of
growth
Commodity prices and mines’ profitability
Global Demand and Supply
Mine 1
Mine 2
𝑀𝐶1
p
𝑃𝑒
𝐴𝐶1
Mine 3
𝑀𝐶2
𝐴𝐶2
p
p
𝑃𝑒
𝑃𝑒
Loss
𝑀𝐶3
𝐴𝐶3
S
D
p
𝑃𝑒
Profit
S
𝑞1
𝑞2
D
𝑄𝑒
𝑞3
2
Q
Commodity prices and mines’ profitability
Global Demand and Supply
Mine 1
Mine 2
𝑀𝐶1
p
𝑃𝑒
𝐴𝐶1
Mine 3
𝑀𝐶2
𝐴𝐶2
p
p
𝑃𝑒
𝑃𝑒
Loss
𝑀𝐶3
𝐴𝐶3
p
S
D
𝑃𝑒
Profit
S
𝑞1
𝑞2
𝑄𝑒
𝑞3
P
AC = TC/𝑞3
Total
Revenue
P . 𝑞3
𝑞3
D
Total
Cost
AC . 𝑞3
𝑞3
Q
Economic rent versus profit
Replacement cost for an ounce of gold, 2014
$1,800
$1,600
$1,400
Cash Operating Costs
[VALUE]
$1,200
$1,000
$800
$600
$400
$200
Sustainig Capital
[VALUE]
Construction Capital
[VALUE]
Discovery Costs
[VALUE]
Overhead [VALUE]
Acceptable profit
[VALUE]
Tax [VALUE]
Source: CIBC world Market
$0
1
GFMS metric: the full marginal cost of mining 2014
Thomson Reuters GFMS group provides an all-in cost metric as a way of measuring the full
marginal cost of gold mining.
 All-in cost – is designed to reflect the full marginal cost of gold mining.
 It is defined as total production cost plus ongoing capital expenditure, indirect costs and
overheads. Where
 Total production cost – is total cash cost, plus depreciation, amortisation and
reclamation cost provisions, and
 Total cash cost – comprises mine site cash expenses (mining, ore processing, onsite general and administrative costs), refining charges, royalties and production
taxes, net of by-product credits.


GFMS group estimates the All-In Cost of production at approximately $1,300/oz for the first
nine months of 2014.
GFMS = Gold Fields Mineral Services