BUIL1149 PROPERTY ECONOMICS
PROBLEM SHEET # 3 (MICRO MODULE)
(SOLUTIONS TO ASTERISKED PROBLEMS)
Q1*
A large retail chain must install the least cost solar heating system for its stores throughout the country. A number of optional solar heating systems are available on the market. Essentially, each such solar system may be viewed as the combination of a solar panel and an energy storage unit.
From a production function perspective, the heat generated by the solar heating system - measured in British Thermal Units (Btu’s) - may be viewed as output. On the other hand, the input levels applied to this heat production process may be viewed as the area A of the solar panel along with the volume V of the storage unit.
The firm has obtained advice from a consulting environmental scientist on the appropriate amount of energy that should be generated for the typical store in the retail chain.
In the first two columns of the following table find all input combinations (A, V) that will generate this recommended level of BTU’s including the ratio of their marginal products MPA/MPV in the third column.
AArea of Solar Panel (ft2) / V
Volume of Storage Unit (ft3) / MPA/MPV
3600 / 20861 / 2.3179
3800 / 20415 / 2.1489
4000 / 20000 / 2.0000
4200 / 19613 / 1.8679
4400 / 19252 / 1.7502
Required:
(a)Determine the input combination of the solar heating system that would be optimal from a least cost perspective if the annual cost or “price” per square foot (cubic foot) of the solar panel (the storage unit) is $56 ($26.06)?
The least-costinput-combinationof producing a given output levelis the one for which the marginal product of a dollar’s worth ofthe ithinput (denoted MP$i)is equatedto that of any other inputj used in the production process. In the present problem there are only two inputs to consider – A and V. Consequently the task is to find that pair of values: (A, V) that ensure:
Recall however, that the marginal product of a dollar’s worth of an input i is taken as the ratio of its marginal product MPi to its price Pi. Hence, the previous condition may be re-written as:
which may be further re-arranged as:
Immediately below we compute the ratio of A’s input price PAto that of V’s input price PV.
Next we caneasily confirm that (A, V) = (3800, 20415) is the cost-minimizing input combination because:
(b)Is there another way of determining the least cost solar heating system that meets the environmental scientist’s suggested requirements?
Another way of finding the least cost combination is to compute the total production cost associated with each input combination and then observe which of these possesses the least cost. The expression for computing the total cost TC associated with input combination (A, V) is given by:
TC = PAA + PVV = 56.00A + 26.05V
This expression allows us to determine the cost of each of the competing input-combinationsso that we are in a position to identify which one of them yields the lowest cost:
(c)As the area of the solar panel increases (see column 1) explain why the ratio of the marginal products fall (in column 3).
Due to the Law of Diminishing Marginal Returns MPA (MPB) falls(rises) as A increases (B decreases). Consequently the ratio of MPA to MPV will decline as we move down through the table from input combination (A=3600, V=20861) through to (A=4400, V=19252).
Q3*
Robson’s Construction Shack manufactures and assembles a stylish award winning aluminium garage for residential use. The firm’s production function for these garages is reproduced tabularly directly below:
K: Capital Inputs / L: Labour Inputs (Person-Years)(Machine - Years) / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8
1 / 30 / 52 / 80 / 110 / 130 / 145 / 155 / 162
2 / 50 / 80 / 120 / 164 / 200 / 220 / 235 / 248
3 / 80 / 124 / 175 / 226 / 260 / 274 / 282 / 287
4 / 100 / 160 / 218 / 272 / 302 / 320 / 335 / 345
The cost of each unit of capital is $50,000 per machine year and the gross salary of each construction worker per man-year is $35,000.
Required:
(a)Derive the short-run average cost curve for each of the 4 plant sizes and plot them on the same chart.
To derive the short run average costs schedule ATCkfor each plant size k (=1, 2, 3, 4) one first has to obtain schedules for total fixed cost TFCk, total variable cost TVCk and Total Cost TCk (= TVCk + TFCk). Then from the TCkmeasures one may obtain average total cost ATCk by computing TCk/Q values at each output level Q. All schedules are reproduced below:
The tabular schedules obtained for ATC1, ATC2, ATC3 and ATC4 are graphed together in the following chart:
(b)By examining the chart what can you deduce about returns to scale?
The envelope curve (or long-run average cost curve LRAC), which passes through the locus of minimum average cost points has a U-shape. This envelope curve is marked with asterisks on our chart (see below):
The descending (ascending) section of the LRAC curve is associated with increasing (decreasing) returns to scale. Increasing (decreasing) returns to scale arise when output increases more (less) than proportionately than an equi-proportionate increase in all inputs including plant scale. Thephenomenon of a descending (ascending) section of the LRAC is referred to as internal economies (diseconomies) of scale. There are several reasons - apart from returns to scale – that account for the phenomenonof internal economies and diseconomies of scale. These reasons areoutlined in both your power-point overheads for Micro Topic 3 and the reading in the prescribed text.
(c)Which of the four plant scales should the firm select if demand for these garages will be (i) 100 units? (ii) 250 units? and (iii) somewhere in the range 300 to 350? Explain your conclusions fully.
-Plant 1 offers the lowest unit cost if output equals100 units
-Plant 3 offers the lowest unit cost if output equals 250 units
-Plant 4 offers the lowest unit cost if output falls somewhere in the range 300 to 350 units
(d)Identify the long run average cost curve faced by Robson’s Construction Shack.
Theenvelope curve (which is marked with asterisks on the chart above) is referred to as the long-run average cost curve LRAC.
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