Exam 2: ECON207, 2024 Summer
Submission Instructions
Please submit your solutions before the deadline through the blackboard the Exam 01 folder. Late submissions will not be accepted and academic dishonesty will be punished.
On your answer sheet, write down following Honor Pledge with your signature:
“I affirm that I will not give or receive any unauthorized help on this exam, and that all work will be my own.” .
Please make sure that you must justify all your answers and show the steps you have taken. Also, when you draw graphs, please make sure to label the axis, functions, and points you specified. Missing labels will result in points losses.
Instruction for Reference
Since it is an open-book exam, I ask you to provide references for all of your steps to all of your answers (except for trivial ones). For instance, when you apply Lagrangian Ex- pression into a constraint optimization problem, you need to refer to one of the learning materials you’ve studied. (For instance, page of your textbook or page of lecture slide)
It will take a bit of your time, but I expect that if you actually studied, you can locate where you should refer in a very short time. Thus, the purpose of this practice is to check whether your answers came from your learning.
1. Furnace (45pt)
A furnace burns Fuel (coal or green) to turn Iron into steel.
(a) Suppose the firm needs 2kg of Fuel and 1kg of Iron to produce 1kg of steel. Find MRTS. (i.e. (i) 4kg of Fuel and 2kg of Iron turn into 2kg of steel, (ii) 5kg of Fuel and 2kg of Iron turn into 2kg of steel, and (iii) 3kg of Fuel and 1.7kg of Iron turn into 1.5kg of steel...)
(b) For (b)~(j), instead, assume that the unit of Steel produced and the unit of Iron and Fuel supplied to the furnace has the following relationship: Steel = 3 √Iron 3 √Fuel2 . What does it imply? Explain it as much as you can.
(c) Suppose the unit price of Iron is twice as expensive as that of Fuel. Draw corresponding isoquants and input costs. Put Iron on the horizontal axis and Fuel on the vertical axis.
(d) What can be inferred from (c)? Explain it as much as you can.
For (e)~(j), consider that a price-taking firm can use two types of Fuel: coal and green Fuel. Suppose their costs per unit are the same, but coal is twice as efficient as green Fuel. Then, suppose that the furnace has a capacity in its fuel tank, so the sum of coal and green Fuel that can be used in its production is fixed.
(e) Draw corresponding isoquants. (Put Iron on the horizontal axis and Green Fuel on the vertical axis.)
(f) What can be inferred from (e)? Explain it as much as you can.
(g) Suppose the firm can sell one Carbon emission allowance per one unit of green fuel usage. Using (e), find the price range of Carbon emission allowances that make the firm use green fuel.
(h) Suppose that the capacity has been doubled. Draw corresponding isoquants. (Again, put Iron on the horizontal axis and Green Fuel on the vertical axis.)
(i) Compare (e) and (h). Derive implications from comparison as much as you can.
(j) Based on your answers, briefly explain how the Carbon emission allowance works?
2. Local property (40pt)
Consider the following production schedule:
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Q
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0
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2
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3.414
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4.732
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6
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7.236
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8.449
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9.645
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10.82
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12
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13.16
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...
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X
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0
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1
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2
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3
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4
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5
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6
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7
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8
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9
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10
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...
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Y
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0
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1
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2
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3
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4
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5
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6
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7
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8
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9
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10
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...
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...
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1
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1.414
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2
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3.732
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5
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7.236
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8.449
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9.645
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10.82
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13.16
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14
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...
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...
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0
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2
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1
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3
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4
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5
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6
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7
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8
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10
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9
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...
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...
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1
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0
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1
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2
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3
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5
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6
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7
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8
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9
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11
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...
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(It is better to convert this table into a “familiar” functional form that you may have seen from the textbook or lecture : Q = F(inputs) =?, √2 = 1.414213..., √3 = 1.732050...)
(a) Let the term ‘local return to scale’ be return to scale at given initial inputs combi- nation. In which initial inputs combination, does production schedule exhibit locally con- stant/decreasing/increasing return to scale?
(b) Let the price of inputs (X, Y) is given by (x,y). Find the Long run cost function.
(c) In which Q, does Long run cost function exhibit locally linear/convex/concave shape with respect to Q? (in other words, ∂Q2/∂2CLR 0?)
(d) Consider Q = F2 (L, K) = Lα Kβ . Repeat parts (a), (b) and (c) with input prices (w, r). Explain implications of each cases: α + β > 1, α + β = 1 and α + β < 1.
(e) Give a Long run expansion path corresponding to F, and Long run expansion paths corresponding to F2 in each case, and find relationship between graphical representations you provide in this question and your answers to above questions.
(f) Suppose that a firm with F faces the market demand function QD (P) = 10 − 2P. What it implies? Explain the difference between input market and output market.
(g) Find market quantity supplied and long run supply function of the firm in (f). If you cannot find one or all of those, explain why and give a proof.
3. Short takes (20pt)
1) There are two petrochemical firms producing the same products. After an increase in oil price, one firm’s expenditure on oil has been increased but the other firm’s one has been de- creased. Explain how this situation can be happened? (Indeed, there are various approaches to explain it. To get the full point, examine a question from various angles.)
2) Let F(x,y) = α + βx2 + γ3y + 4 √xy3 be the firm’s production technology. Intuitively, can it exhibit decreasing scale to return? Explain why.
4. Production (30pt)
Suppose there is a firm which produces its commodity with 2 inputs. The price of input x1 is given by 8, and input x2 is given by 1/2. Firm’s production technology can be represented as follows: y = (min{4x1, 10x2 })1/2
(a) Suppose the firm’s input x2 is fixed at 160. Draw and label a graph that depicts a Short Run production function.
(b) Set the firm’s Short Run cost minimization problem. Derive the Short Run demand for x1 and Short Run cost function.
(c) Set the firm’s Short Run profit maximization problem. Draw the Short Run supply curve. (d) Set the firm’s Long Run cost minimization problem. Derive the Long Run demand for x1 and Short Run cost function.
(e) Set the firm’s Long Run profit maximization problem. Draw the Long Run supply curve. (f) Let p be price of the output. What is the firm’s output level and profit level(dependent on p) in Short Run? What about Long Run? Compare Short Run and Long Run results. Explain what your results implies.