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University of Toronto, Department of Economics, ECO 204 2010 ‐ 2011 Ajaz Hussain TEST 3 SOLUTIONS 1 These solutions are purposely detailed for your convenience

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Thanks: Asad Priyo

Page 1 of 37
S. Ajaz Hussain, Dept. of Economics, University of Toronto

ECO 204, 2010-2011, Test 3 Solutions This test is copyright material and may not be used for commercial purposes without prior permission

Question 1 [60 Points. All parts worth 10 points each] This question is based on the HBS case: The Prestige Telephone Company. For your convenience here is exhibit 1 from the case (figures below are for the Prestige Data services company January – March 2003) Exhibit 1: Prestige Telephone Company January 2003 February 2003 Intercompany Hours 206 181 Commercial Hours 123 135 Total Revenue Hours 329 316 Service Hours 32 32 Available Hours 223 164 Total Hours 584 512 In the case, the “commercial” price is $800/hour and the “intercompany” price is

March 2003 223 138 361 40 143 544 $400/hour:

Prestige Telephone Company
D

“Inter company” a Pi = $400/hr. t a

Other Services

Prestige Data Services

“Commercial”
Data

PC = $800/hr

Commercial Customers

The Prestige Data Services’ cost function was estimated to be (here is “hours of data services”): 223,436 28

The Prestige Data Services’ commercial inverse demand function in March 2003 was estimated to be: 1,466 4.83 Page 2 of 37
S. Ajaz Hussain, Dept. of Economics, University of Toronto

ECO 204, 2010-2011, Test 3 Solutions This test is copyright material and may not be used for commercial purposes without prior permission

(a) Under the terms of a regulation ruling PDS’s intercompany billing are capped at an average of $82,000/month. What does this imply for the average intercompany hours that can be billed per month? Is Prestige Data Services abiding by or violating the terms of the ruling? Give a brief explanation. Answer: If intercompany revenues are capped at $82,000/month and prices are $400/hr then: 400 82,000 205 This implies that intercompany hours should be 205 hours/month on average. Is PDS billing 205 hours on average each month? Look at: Exhibit 1: Prestige Telephone Company January 2003 February 2003 Intercompany Hours 206 181 Commercial Hours 123 135 Total Revenue Hours 329 316 Service Hours 32 32 Available Hours 223 164 Total Hours 584 512 Notice that average intercompany hours from Jan – Mar 2003 were: 206 181 3 223 610 3 203.33

March 2003 223 138 361 40 143 544

Thus, PDS is abiding by the terms of the agreement. (b) What kind of returns does PDS have for its “variable” inputs? Give a brief explanation based on the figures above (not below). Answer: The Prestige Data Services’ cost function was estimated to be (here is “hours of data services”): 223,436 28

Since this is a linear cost function PDS must have constant returns. In particular, notice that: 28 28 Page 3 of 37
S. Ajaz Hussain, Dept. of Economics, University of Toronto

ECO 204, 2010-2011, Test 3 Solutions This test is copyright material and may not be used for commercial purposes without prior permission

Since

is constant, PDS has constant returns.

(c) Recall that PDS has never earned profits. Calculate the breakeven number of commercial hours and the equation of the demand curve in the month when PDS breaks even. Show all calculations. Answer PDS sells data services to intercompany and commercial customers. Now profits are: Π Π Π Π Breakeven commercial output is when: Π – – Breakeven Breakeven Breakeven Assuming 28 and Breakeven 205 we have: 400 28 205 800 28 191 –

Π

Π



0

223,436

We can now compute the demand curve in the month when PDS breaks even. Assuming the “other factors” besides price are pushing the demand curve right we see that in the breakeven month, the demand curve has the same slope as the March 2003 demand curve:

Page 4 of 37
S. Ajaz Hussain, Dept. of Economics, University of Toronto

ECO 204, 2010-2011, Test 3 Solutions This test is copyright material and may not be used for commercial purposes without prior permission

PDS Commercial Demand Curve in March 2003 P = 1,466 – 4.83Q
Commercial Price

$1,466

$800

J

F

M

?

123

135

138

Break even = 192

Commercial Hours

Thus, in the breakeven month: 4.83 Since $800 and 192 we have: 4.83 800 800

4.83 192 4.83 192 1,727.36

The commercial demand curve in the breakeven month is: 1,727.36 (d) In general, is it easier for the “commercial division” to breakeven if PDS comprises of “commercial and intercompany divisions” versus if PDS comprises of just a “commercial division”? Assume the two scenarios have the same total fixed cost. Answer With two divisions we had: Page 5 of 37
S. Ajaz Hussain, Dept. of Economics, University of Toronto

4.83

ECO 204, 2010-2011, Test 3 Solutions This test is copyright material and may not be used for commercial purposes without prior permission

Breakeven

Let’s compare this expression with the case of a single “commercial” division: Breakeven

(i.e. the “other”, intercompany, division has positive contribution Notice that so long as margin), then the commercial breakeven quantity with another “profitable” division is smaller than the breakeven quantity if the commercial division was by itself. (e) [This part is independent of all other parts] Suppose that in March 2003, the government imposes a 10% excise tax on commercial data services. Assuming all commercial customers can be modeled by a single representative consumer with income $ and utility function , (where good 1 is commercial data services and good 2 is “everything else”) what is the marginal utility due to the excise tax on commercial data services? Please show all calculations and specifically state all assumptions. Answer: Before the excise has been imposed, the “representative” consumer, with an unknown income $ , consumes 138 hours of commercial data services and some unknown quantity of other goods. Let “everything else” be the base good so that 1 and assume the consumer has a quasi‐linear utility function of the type: , Here is any function of good 1 such that 0 the income is the consumer surplus of good 1: 0. In this case, the total utility of a bundle minus

Page 6 of 37
S. Ajaz Hussain, Dept. of Economics, University of Toronto

ECO 204, 2010-2011, Test 3 Solutions This test is copyright material and may not be used for commercial purposes without prior permission

1,466 800

Pc
U(Q1, Q2 ) – Y

A Commercial Demand 138 Q1 Commercial Hours

Q2 (Everything Else)

Y/1 Q2
A

U(Q1, Q2 ) – Y

138

Y/800

Q1 Commercial Hours

That is at bundle A: , 138, 1 1,466 2 800 138 $45,954

Page 7 of 37
S. Ajaz Hussain, Dept. of Economics, University of Toronto

ECO 204, 2010-2011, Test 3 Solutions This test is copyright material and may not be used for commercial purposes without prior permission

1,466 800

Pc
U(Q1, Q2 ) – Y = $45,954

A Commercial Demand 138 Q1 Commercial Hours

Q2 (Everything Else)

Y/1 Q2
A

U(Q1, Q2 ) – Y = $45,954

138

Y/800

Q1 Commercial Hours

A 10% excise tax raises the price of commercial services to $880 and reduces demand to: 1,466 880 4.83 1,466 1,466 4.83 4.83 880 121.33

1,466 880 4.83 Thus, at bundle B: , 121.33,

1 1,466 2

880 121.33

$35,550

Page 8 of 37
S. Ajaz Hussain, Dept. of Economics, University of Toronto

ECO 204, 2010-2011, Test 3 Solutions This test is copyright material and may not be used for commercial purposes without prior permission

1,466 880 800 Pc
U(Q’1, Q’2 ) – Y = $35,550

B A

Commercial Demand 121.33 138 Q2 (Everything Else) Q1 Commercial Hours

U(Q’1, Q’2 ) – Y = $35,550

Y/1 Q2 Q’2
B A

U(Q1, Q2 ) – Y = $45,954

121.33 138

Y/880

Y/800

Q1 Commercial Hours

Therefore, the change in utility due to a 10% excise tax on commercial data services is: of a 10% excise tax on of a 10% excise tax on , , , , $35,550 , $45,954 , $10,404

Notice this calculation does not require knowledge of the actual level of income, the exact utility function, or the amount of good 2 being consumed. Page 9 of 37
S. Ajaz Hussain, Dept. of Economics, University of Toronto

ECO 204, 2010-2011, Test 3 Solutions This test is copyright material and may not be used for commercial purposes without prior permission

(f) Recall that the PDS’s “variable” inputs were quasi‐variable power (denote by ) and quasi‐variable labor (denote by while its “fixed” inputs were quasi‐fixed power, quasi‐fixed labor and all other inputs. Denote all “fixed” inputs as capital . Suppose PDS’s production function is: Assume 1 and 0. Suppose the price of power is $ /hour, the price of quasi‐variable labor is $30.25/hour and the price of “capital” is . Given that $4/hour and $24/hour what is the price of quasi‐variable power ? Hint: Solve the CMP max
,

s. t. $4/hour and

and use the fact that Answer: We are told that:

$24/hour. Show all calculations below.

4 This means that: 4 To find we need to express the optimal demand for power in terms of the parameters and (hopefully) solve for . This means we have to solve the CMP. The total cost of hours of data services is: We could substitute values for some parameters now or we could work as long as possible in parametric form and then substitute numbers. We’ll do the latter so that you can see the algebra. The Cost Minimization Problem (CMP) is: min
,

s. t.

, ,

0

Now:

1 so that: min
,

s. t.

, ,

0

Page 10 of 37
S. Ajaz Hussain, Dept. of Economics, University of Toronto

ECO 204, 2010-2011, Test 3 Solutions This test is copyright material and may not be used for commercial purposes without prior permission

Since the production function is of the Cobb‐Douglas form we know that for 0 we must use some power and labor so that , 0. As such, we can drop the non‐negativity constraints: min
,

s. t.

Now: max
,

s. t.

Setup the Lagrangian: max
,

s. t.

max
,

s. t.
,

0

max L The FOCs are: L 7 L L The 1st FOC implies: L 7 7 7 The 2nd FOC implies: 0 0 6 7 0 0

Page 11 of 37
S. Ajaz Hussain, Dept. of Economics, University of Toronto

ECO 204, 2010-2011, Test 3 Solutions This test is copyright material and may not be used for commercial purposes without prior permission

L 6 7

6 7 7 6

0

Equating the ’s yields the familiar “the optimal input bundle is where the iso‐quant is tangent to the iso‐cost” result: 7 7 6 1 6 1 6 This allows us to isolate power (or for that matter labor) in terms of labor (power). For instance: 1 6 6 We can substitute this in the 3rd FOC: L 6 Page 12 of 37
S. Ajaz Hussain, Dept. of Economics, University of Toronto

0

ECO 204, 2010-2011, Test 3 Solutions This test is copyright material and may not be used for commercial purposes without prior permission

6 6 6 6 Now, we are told that $4 which means that:

4 Substitute the expression for to get: 4 4 4 6 1 6 1 6 4 6 Page 13 of 37
S. Ajaz Hussain, Dept. of Economics, University of Toronto

4 1

4

1

1

ECO 204, 2010-2011, Test 3 Solutions This test is copyright material and may not be used for commercial purposes without prior permission

4 Substitute $30.25 to get: 4 6

6

1

1 30.25 $1

0.9976

Page 14 of 37
S. Ajaz Hussain, Dept. of Economics, University of Toronto

ECO 204, 2010-2011, Test 3 Solutions This test is copyright material and may not be used for commercial purposes without prior permission

Question 2 [40 Points. Parts (d) & (e) worth 5 points each, all other parts worth 10 points each] This question is based on the HBS case The Aluminum Industry in 1994. The following table contains the cost structure of the average CIS primary aluminum smelter, the average state primary aluminum smelter, and the average rational primary aluminum smelter (please note that cumulative capacity below is the total capacity of all smelters within a category (for example, the total cumulative capacity of all CIS smelters is 1.788 million tons per year): All cost figures are $/t Variable costs are in BOLD Average CIS Average State Smelter Smelter CIS CIS 243.73 148.62 382.13 63.69 4.51 76.92 39.57 17.80 68.76 67.11 1,788.07 (All CIS) 740.14 All over State 120.74 454.49 407.27 163.57 11.81 56.72 46.45 62.73 53.17 52.48 2,826.95 (All state) 1,135.25 Average Rational Smelter All over Rational 121.53 292.29 348.49 120.62 10.34 73.91 53.84 194.19 37.82 86.58 16,962.17 (All rational) 873.15

Smelter

Country Company Average Capacity (‘000s tpy) Total electricity cost Total alumina cost Other raw materials Plant power and fuel Consumables Maintenance Labor Freight General and administrative Cumulative capacity (‘000s of tons/year) (all smelters in a category) Total variable costs per ton ($/ton) =

Page 15 of 37
S. Ajaz Hussain, Dept. of Economics, University of Toronto

ECO 204, 2010-2011, Test 3 Solutions This test is copyright material and may not be used for commercial purposes without prior permission

(a) Solve the profit maximizing problem for the average rational primary aluminum smelter: max Π s. t. 0, Here price of aluminum/ton, Total fixed Cost, = Total variable cost and capacity. Assume 0. Show all calculations. Answer Since the case reports cost figures in “$ per ton” we assume that all smelter’s have constant returns so that: constant constant This will be useful below. Now, a smelter’s profit maximization problem (PMP) is: max Π s. t. 0, Note that smelters are price takers so that is a constant. Recall that all inequality constraints must be expressed in the form . Therefore: max Π s. t. 0, max Π Having expressed all constraints in terms of max L
, ,

s. t.

0,

, form the Lagrangian: 0

max L
, ,

The FOC is:

L The Kuhn‐Tucker conditions are: 0, , 0

0

Page 16 of 37
S. Ajaz Hussain, Dept. of Economics, University of Toronto

ECO 204, 2010-2011, Test 3 Solutions This test is copyright material and may not be used for commercial purposes without prior permission

0, 0, 0 Notice there are 4 possible cases that must be checked:

q = 0 (Check when λ2 ≥ 0)

λ2 = 0 (Check when q ≥ 0)

q = qc (Check when λ1 ≥ 0 ) Case A

λ1 = 0 (Check when q ≤ qc) Case B

q = qc (Check when λ1 ≥ 0 ) Case C

λ1 = 0 (Check when q ≤ qc) Case D

Case A 0, 0, 0. Need to check if > 0. Thus case A is impossible. Case B 0, 0 0, . Need to check if is automatically satisfied. Thus we need to check if

This requires that

0 and

Since 0 the KT condition with the FOC:

0. Start

L

0

Page 17 of 37
S. Ajaz Hussain, Dept. of Economics, University of Toronto

ECO 204, 2010-2011, Test 3 Solutions This test is copyright material and may not be used for commercial purposes without prior permission

0 Substitute 0 Thus, for 0 we need: 0 0 0 Case B is the solution when the price of aluminum is lower than 0 . This is because the marginal st cost of producing the 1 unit is greater than or equal to the price, or the , of the 1st unit. 0 0 and 0: 0 0

Constant Returns Technology

$ MC

P
B

MR qc Qty

Thus, anytime curve):

0 the competitive firm’s supply curve is

0 (that’s not the same as the MC

Page 18 of 37
S. Ajaz Hussain, Dept. of Economics, University of Toronto

ECO 204, 2010-2011, Test 3 Solutions This test is copyright material and may not be used for commercial purposes without prior permission

Constant Returns Technology

$ MC

P qc

MR Qty

Case B when 0

0

0

Since

0 Case C , 0 Need to check if 0, 0 0 the KT condition 0 is automatically satisfied. Thus we need to check if

0:

L

0 0

Substitute Thus, for 0 we need: Page 19 of 37
S. Ajaz Hussain, Dept. of Economics, University of Toronto

and

0: 0

ECO 204, 2010-2011, Test 3 Solutions This test is copyright material and may not be used for commercial purposes without prior permission

0 Case C will be the solution if the aluminum price, the at full capacity: , is greater than ‐‐ the marginal cost

Constant Returns Technology

$ MC

P

MR

C

qc

Qty

the competitive firm’s supply curve is Thus, anytime competitive firm’s MC curve is not the supply curve:

. Again notice that the

Page 20 of 37
S. Ajaz Hussain, Dept. of Economics, University of Toronto

ECO 204, 2010-2011, Test 3 Solutions This test is copyright material and may not be used for commercial purposes without prior permission

Constant Returns Technology

$ MC

P qc

MR Qty

Case C when Case D 1, 0 Need to check if 0, Start with the FOC: 0

L

0 0

Substitute

0 0

Page 21 of 37
S. Ajaz Hussain, Dept. of Economics, University of Toronto

ECO 204, 2010-2011, Test 3 Solutions This test is copyright material and may not be used for commercial purposes without prior permission

This is the familiar ECO 100 result that a competitive firm produces where price equals marginal cost. The only problem is that we don’t know when case D will be a solution for sure. For that we need to the conditions under which 0 and . From: We have: Thus: 0 0 0 This gives us a condition for case D to be a solution and for the output supplied to be between 0 and full capacity. Intuitively, case D says that if the price of the product is between 0 and the firm will produce an output between zero and full capacity:

Constant Returns Technology

$ MC

P

MR

qc
Optimal output anywhere between 0 and full capacity

Qty

Page 22 of 37
S. Ajaz Hussain, Dept. of Economics, University of Toronto

ECO 204, 2010-2011, Test 3 Solutions This test is copyright material and may not be used for commercial purposes without prior permission

Thus, anytime

0

the competitive firm’s supply curve is also its MC curve:

Constant Returns Technology

$ MC

P

MR

qc

Qty

Case D when

0 0

0

Putting all cases together we have a competitive smelter’s supply curve:

Page 23 of 37
S. Ajaz Hussain, Dept. of Economics, University of Toronto

ECO 204, 2010-2011, Test 3 Solutions This test is copyright material and may not be used for commercial purposes without prior permission

Constant Returns Technology

$ Supply Curve

qc

Qty

Put another way, the quantity supplied is: 0, , , (b) Based on your answer to part (a) what is the impact on the average rational smelter’s optimal profits from, holding all else constant, a 1% increase in: • The price of aluminum? • Capacity? • The minimum output? • Fixed cost? Assume that $1,100/ton. Show all calculations. 0 0

Answer: We are being asked to investigate the impact on the average rational smelter’s profits due to a change in a parameter – the easiest way to solve this is by the envelope theorem. To do this, we must first find out the average rational smelter’s output. Recall that: Page 24 of 37
S. Ajaz Hussain, Dept. of Economics, University of Toronto

ECO 204, 2010-2011, Test 3 Solutions This test is copyright material and may not be used for commercial purposes without prior permission

All cost figures are $/t Variable costs are in BOLD Average CIS Average State Smelter Smelter CIS CIS 243.73 148.62 382.13 63.69 4.51 76.92 39.57 17.80 68.76 67.11 1,788.07 (All CIS) Since we assumed all smelters have constant returns, the average rational smelter’s 0, 121.53]: for 0, $873.15 740.14 All over State 120.74 454.49 407.27 163.57 11.81 56.72 46.45 62.73 53.17 52.48 2,826.95 (All state) 1,135.25 Average Rational Smelter All over Rational 121.53 292.29 348.49 120.62 10.34 73.91 53.84 194.19 37.82 86.58 16,962.17 (All rational) 873.15

Smelter

Country Company Average Capacity (‘000s tpy) Total electricity cost Total alumina cost Other raw materials Plant power and fuel Consumables Maintenance Labor Freight General and administrative Cumulative capacity (‘000s of tons/year) (all smelters in a category) Total variable costs per ton ($/ton) =

Page 25 of 37
S. Ajaz Hussain, Dept. of Economics, University of Toronto

ECO 204, 2010-2011, Test 3 Solutions This test is copyright material and may not be used for commercial purposes without prior permission

Average Rational Smelter Supply Curve

$

MC = 873.25 Qty qc = 121.53

Currently, the price of aluminum is $1,100/ton and since: $1,100 $873.15

The average rational smelter will produce at full capacity (i.e. case “C”):

Average Rational Smelter Supply Curve

$

P = $1,100 MC = $873.25

MR

Qty qc = 121.53

Page 26 of 37
S. Ajaz Hussain, Dept. of Economics, University of Toronto

ECO 204, 2010-2011, Test 3 Solutions This test is copyright material and may not be used for commercial purposes without prior permission

Case C when Thus: 0 121.53 and: 1,100

0

873.25

226.75

Π Since 873.25 then 873.25 873.25 121.52 $106,117 . Moreover, adding up all average fixed cost items we have: $345 345 121.53 $41,928 Therefore: Π Π 1,100 121.52 41,928 106,117 133,672 41,928 106,117 $14,373 Π Notice that producing at full capacity maximizes profits in the sense that it minimizes loss. (As a study question, you should check that operating at full capacity is better than shutting down, confirming the rule that a rational company incurring losses should shut down when ). Now, what is the impact on the average rational smelter’s optimal profits from, holding all else constant, a 1% increase in: • The price of aluminum? • Capacity? • The minimum output? • Fixed cost? By the envelope theorem, the change in the objective (in this case profits) from a small change in a parameter is gotten by differentiating the Lagrangian with respect to the parameter, evaluated at the initial solution. _____________________________________________________________________________________ Optional note: Recall the Lagrangian was: Page 27 of 37
S. Ajaz Hussain, Dept. of Economics, University of Toronto

ECO 204, 2010-2011, Test 3 Solutions This test is copyright material and may not be used for commercial purposes without prior permission

max L
, ,

0

Any solution to this problem must satisfy the KT conditions: 0, , 0 0, 0, 0 The “product” terms in the KT conditions imply that at the optimum the following terms are zero: max L 0
, ,

This is why differentiating the optimal Lagrangian is equivalent to differentiating optimal profits with respect to the parameter. ____________________________________________________________________________________ Envelope Theorem: the marginal profit due to a 1% increase in aluminum price The Lagrangian was: max L
, ,

0

Differentiating the Lagrangian with respect to aluminum price: ΔΠ ΔP

L

This is the impact on profits from a $1 increase in aluminum price. To find the impact due to a 1% increase in aluminum price we have: %ΔΠ %ΔP Evaluate at optimal solution: %ΔΠ %ΔP %ΔΠ %ΔP 9.3%

L
L L Π

Π*

121.52 1,100 14,373 9.3% % Δ P 9.3% 1%

%ΔΠ %ΔΠ

Page 28 of 37
S. Ajaz Hussain, Dept. of Economics, University of Toronto

ECO 204, 2010-2011, Test 3 Solutions This test is copyright material and may not be used for commercial purposes without prior permission

%ΔΠ

9.3% 9.3% 0.093 0.093 Initial Π 0.093 Initial Π 0.093 $13,036 14,373

New Π Initial Π 100 Initial Π New Π Initial Π Initial Π New Π New Π New Π Initial Π Initial Π 14,373 New Π

A 1% increase in aluminum prices reduces the average rational smelter’s loss by 9.3% or from a loss of ($14,373) to a loss of ($13,036). Envelope Theorem: the marginal profit due to expanding capacity by 1% The Lagrangian was: max L
, ,

0

Differentiating the Lagrangian with respect to capacity: ΔL Δ

L

This is the impact on profits from a 1 unit increase in capacity. To find the impact due to a 1% increase in capacity we have: %ΔΠ %Δ Evaluate at optimal solution: %ΔΠ %Δ %ΔΠ %Δ 344.95 Π* 121.52 14,373 2.92% 1%

L
L L Π

%ΔΠ %ΔΠ

2.92%

2.92%

Page 29 of 37
S. Ajaz Hussain, Dept. of Economics, University of Toronto

ECO 204, 2010-2011, Test 3 Solutions This test is copyright material and may not be used for commercial purposes without prior permission

New Π Initial Π 100 Initial Π New Π Initial Π Initial Π New Π New Π New Π Initial Π Initial Π 14,373 New Π

2.92% 0.0292

0.0292 Initial Π 0.0292 Initial Π 0.0292 14,373

$13,953.31

A 1% increase in capacity reduces the average rational smelter’s loss by 2.92% or from a loss of ($14,373) to a loss of ($13,953.31). Envelope Theorem: the marginal profit due to raising the minimum output by 1% The Lagrangian was: max L
, ,

0 we have:

We had required that 0. Re‐writing this constraint as max L
, ,

Differentiating the Lagrangian with respect to ΔΠ Δ

:

L

This is the impact on profits from a 1 unit increase in minimum output. To find the impact due to a 1% increase in minimum output we have: %ΔΠ %Δ

L
L L

0

L

0

There is no impact on optimal profits from raising the minimum output requirement by 1%. Why? Because the optimal solution is for the smelter to produce well above zero, so that the minimum output constraint does not bind – as such, there is no value in relaxing the constraint. Envelope Theorem: the marginal profit of 1% higher fixed cost The Lagrangian was:

Page 30 of 37
S. Ajaz Hussain, Dept. of Economics, University of Toronto

ECO 204, 2010-2011, Test 3 Solutions This test is copyright material and may not be used for commercial purposes without prior permission

max L
, ,

0 :

Differentiating the Lagrangian with respect to ΔΠ Δ

L

1

This is the impact on profits from a $1 increase in TFC. To find the impact due to a 1% increase in TFC we have: %ΔΠ %Δ Evaluate at optimal solution: %ΔΠ %Δ %ΔΠ %Δ %ΔΠ %ΔΠ 1 1 41,928 14,373 2.92% Π* 2.92% 1%

L
L

1

L

1

Π

2.92% 2.92% 0.0292 0.0292 Initial Π 0.0292 Initial Π 0.0292 $14,793 14,373

New Π Initial Π 100 Initial Π New Π Initial Π Initial Π New Π New Π New Π Initial Π Initial Π 14,373 New Π

A 1% increase in TFC raises the average rational smelter’s loss by 2.92% or from a loss of ($14,373) to a loss of ($14,793). Page 31 of 37
S. Ajaz Hussain, Dept. of Economics, University of Toronto

ECO 204, 2010-2011, Test 3 Solutions This test is copyright material and may not be used for commercial purposes without prior permission

(c) Graph the primary aluminum industry supply curve as if all CIS smelters behave like the average CIS smelter, all state owned smelters behave like the average state smelter, and all rational smelters behave like the average rational smelter. Answer: From the case we know that CIS and state owned smelters most likely behave as irrational smelter’s, i.e. they produce aluminum even if the price falls below . Put another way, irrational smelters produce aluminum regardless of the price. Recall that: All cost figures are $/t Variable costs are in BOLD Average CIS Average State Smelter Smelter CIS CIS 243.73 148.62 382.13 63.69 4.51 76.92 39.57 17.80 68.76 67.11 1,788.07 (All CIS) Page 32 of 37
S. Ajaz Hussain, Dept. of Economics, University of Toronto

Smelter

Average Rational Smelter All over Rational 121.53 292.29 348.49 120.62 10.34 73.91 53.84 194.19 37.82 86.58 16,962.17 (All rational) 873.15

Country Company Average Capacity (‘000s tpy) Total electricity cost Total alumina cost Other raw materials Plant power and fuel Consumables Maintenance Labor Freight General and administrative Cumulative capacity (‘000s of tons/year) (all smelters in a category) Total variable costs per ton ($/ton) =

All over State 120.74 454.49 407.27 163.57 11.81 56.72 46.45 62.73 53.17 52.48 2,826.95 (All state) 1,135.25

740.14

ECO 204, 2010-2011, Test 3 Solutions This test is copyright material and may not be used for commercial purposes without prior permission

Assuming constant returns and that smelters produce at full capacity (for cases C and D) we see that together, the state and CIS smelters will produce a cumulative output of 1,788.07 + 2,826.95 = 4,615.02 ‘000s tons at any price while the rational smelters will produce 16,962.17 ‘000s of tons so long as 873.15:

Primary Aluminum Industry $ Supply Curve

873.25

Rational smelters CIS & State smelters
4.61m 21.588m Qty (m tons)

(d) [This part is independent of all parts below] Graph the primary aluminum industry supply curve from part (c) below and then show the impact of all rational smelters experiencing “learning by doing” (assume learning by doing has a small effect). Answer With learning by doing there will be a decrease in every smelters’ . However, since the state and CIS smelters produce at any price, there will be no change in the their “supply” curve whilst learning by doing pushes the rational smelters’ supply curve down:

Page 33 of 37
S. Ajaz Hussain, Dept. of Economics, University of Toronto

ECO 204, 2010-2011, Test 3 Solutions This test is copyright material and may not be used for commercial purposes without prior permission

Primary Aluminum Industry $ Supply Curve

Rational smelters
873.25

CIS & State smelters
4.61m 21.588m Qty (m tons)

(e) Use the primary aluminum supply model in part (c) to predict the demand for primary aluminum for the case for the cases when (i) $800/ton, (ii) $1,100/ton. Please show all graphs below and explain your reasoning. Answer: Case (i):

$800/

. In this case:

Page 34 of 37
S. Ajaz Hussain, Dept. of Economics, University of Toronto

ECO 204, 2010-2011, Test 3 Solutions This test is copyright material and may not be used for commercial purposes without prior permission

Primary Aluminum Industry $ Supply Curve

873.25 P = 800

Rational smelters CIS & State smelters
4.61m 21.588m Qty (m tons)

Notice that all CIS and state smelters will produce aluminum while all rational smelters will shut down. The total output will be 4.61m tons. Assuming no inventory buildups or rundowns, demand for primary aluminum will be 4.61m tons. Case (ii): $1,100/ . In this case:

Primary Aluminum Industry $ Supply Curve

P =1,100 873.25

Rational smelters CIS & State smelters
4.61m 21.588m Qty (m tons)

Page 35 of 37
S. Ajaz Hussain, Dept. of Economics, University of Toronto

ECO 204, 2010-2011, Test 3 Solutions This test is copyright material and may not be used for commercial purposes without prior permission

Notice that all smelters will produce aluminum. The total output will be 21.588m tons. Assuming no inventory buildups or rundowns, demand for primary aluminum will be 21.588m tons. . The End ☺ WORKSHEETS

Page 36 of 37
S. Ajaz Hussain, Dept. of Economics, University of Toronto

ECO 204, 2010-2011, Test 3 Solutions This test is copyright material and may not be used for commercial purposes without prior permission

Page 37 of 37
S. Ajaz Hussain, Dept. of Economics, University of Toronto

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