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Unformatted text preview: SHORTRUN vs LONGRUN COSTS 0 Meaning, Intuition and Main Results: (i) Costs are Greater (or at best No Lower) in the Short Run
(ii) Costs Rise Faster in the Short—Run o A Simple Model Interpretation 1: The ﬁxed input, K, is ‘Capital’.
Interpretation 2: K is the number of ‘Production Plants’ Interpretation 3: K is the number of ﬁrms With identical costs.
(Instead of K use 71 to denote the number of ﬁrms.) SHORTBUN vs; LONGBEN COSTS lNTEBPBETATIQN 2: K : NIIMBEB QF PBQDUQTIQN PLANTS Suppose the cost function of a ﬁrm with K plants is given by 2 TO(K,y) : y? + K Then
0 If K : 1, total cost is 2 TC(1,y)=yT+1=y2+1 and marginal and average cost are given respectively by l
MC(l,y) : 2y and AC(1,y): (74+; and so Ininiiniinuni efﬁcient scale is W :1 with minAC(1,y) : A0(1,yE) : 2 o If K : 2, total cost is 2 TC(2, y) : y— + 2
—2¥
and marginal and average cost are given respectively by
y 2
M09730 = y and ACQw) = g + —
y and so Ininirnirnuni efﬁcient scale is
yE : 2 with minAC(2,y) : AC(2,yE) : 2 NOTE:
IN ORDER TO MINIMIZE COSTS, THE FIRM MUST ALLOCATE OUTPUT
ACROSS PLANTS SUCH THAT MARGINAL COSTS ARE EQUALIZED. FORMALIZEZ
Let yl be the output produced in plant 1. The costs of plant 1 are: T01 :nyrl, M01 : 2y1, A01 :yi+1/?Ji Let yg be the output produced in plant 2. The costs of plant 2 are: T02 : 1, M02 : 23/2, A02 : 3/2 + The ﬁrm Wishes to produce a total amount of output, y: y=yl+y2 It must choose 3/1 and 3/2 such that the cost of producing y is minimized,
that is2 TCl(y1) + T02(y2) subject to y = y1 + W To solve this problem, set up the Lagrangean function: minC : T01(y1) + TO2(?J2> ‘l' ’Y(y — 91 — 92> y19y2 Fill}: dTO
1 — ’7 : 0 :> M01 : ’7 dyl clTC
2 — 7 : 0 :> M02 : 7 dy2 which gives
M01 : M02 (What about SOC. conditions? Is this procedure always valid? NO.
Only if at least one of the M C’s is increasing and the other is not de—
creasing...) Apply in the speciﬁc example: M 01 : 2m and M 02 : 2y2.
Therefore M012M022>2y1:2y2:>yf:y§ and since y : yik + y; y=2yl=>yl=y/2 and 93:31/2 The minimum cost is Tan/l) + TO2(y§) H
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\_/ So, the T C (2, 3/) has a cost minimization problem hidden inside, namely
that of allocating production across plants.
This is the case in general, that is, With K plants. o If K : 3, total cost is yZ TCBwﬁrg+3
and marginal and average cost are given respectively by
3
M0679) : 2y/3 and AC(3,y) : % 1Lg and so ininiiniinuni efﬁcient scale is yE : 3 With : AC<37yE) : 2 o If K : 4, total cost is 2 —4¥
and marginal and average cost are given respectively by
4
MOMMU) : y/2 and AC(4,y) : % + _
y and so ininiiniinuni efﬁcient scale is W : 4 With : 140(4) yE) : 2 O and Sun“ If the number of plants, K, is variable, how many should the ﬁrm choose to
minimize the cost of a given output amount 3/? Same as before... 2 n1inTC(K,y) : y— + K K K Which gives, K * : y and thus a long—run cost
LRC®>=2y o What is the ﬁrm’s ‘short—run supply’ if it has K plants? o What is the ﬁrm’s ‘long—run supply’ ? lNTEBPBETATIQN 3: a 7 47
0 c 7
O o n (identical) ﬁrms together : an industry with 71 ﬁrms Total Industry Cost with 71 ﬁrms: 2 TC(n,y) : 3% +71 Industry Marginal and Average Cost with 71 ﬁrms: M002, y) : 2y/n and A091, y) : % +3 0 What is the ‘Aggregate Supply’ of an industry with 71 ﬁrms (that is,
assuming new ﬁrms cannot enter and existing ﬁrms cannot exit)? o What is the ‘Aggregate Supply’ of an industry with free entry and exit,
that is, the ‘long—run supply of the industry’? CAN the LONGBUN SUPPLY BE UPWARILSLQBINQZ Yes, if the ‘scarcity of inputs’ is taken into account.
Consider the following story: oEachﬁrmM‘iarm’. a  7
O 0 Each farm can produce up to 1 unit of output using the ‘land unit7 and 0 Land productivity/ fertility varies. 0 Unit 1 (owned by farm 1) is the most productive, then comes Unit 2, andsmforth 0 Unit 1 requires 1 unit of labour to produce 1 unit of output. Unit 2 requires 2 units of labour to produce 1 unit of output
and so forth... Next answer the following questions: 1. If the price of labour is w : 1 and the price of ‘Unit 2" of land is 7‘1 (rent),
1 . ] E . E E .? 2. What is the supply of farm 2'? 3. What is the aggregate supply (assuming free entry/ exit)? 4. Does each farmer make positive proﬁt? 5. Is there ‘producer surplus’ here? What does it represent? Eormalize'
Let * Ln denote the labour employed by farm n * Kn the land occupied by farm 71. Then, the production function of farm 71: yn : min<Ln/n, Kn) Each farm occupies ONE land unit, ‘Unit 71’, so Kn : 1 for all n. Thus:
yn : mm(Ln/n, 1)
The higher is n, the lower is the productivity of the land unit of farm n:
Cost Min. Requires Ln/n : Kn : 1 :> Ln : n
So ‘Farm 71’ * requires 71 labour units to produce 1 unit of output. * Which means its labour (variable) cost of producing one unit is AVG” : um Where w is the wage rate. * is limited by the ‘one land unit7 to produce a maximum of 1 unit of ouput. From the above, the cost function of farm n is : ‘i’ Tm yn < 1 Where Tn is the rent (price) of ‘Unit 71’ (a ﬁxed cost).
Each ﬁrm is a ‘capacity constrained ﬁrm’, that is, the cost of producing yn > 1
is inﬁnite (it is impossible to do so). To simplify, assume w : 1, so the total cost of Farm n is given by : nyn ‘l— Tn: yn <1 and so (for yn < 1) marginal and average cost are MCn2n:AVCn and ACn:n+E
3m Note;
0 The more productive the land (lower n), the lower is M C . For example,
M01 : 1 (best : most productive/efﬁcient farm), M 02 : 2 (second best : half as productive or twice the variable cost)
M03 : 3 (third best), etc. On a graph, the supply of each farm 1 and 2 assuming the land rent is
sunk (already paid) and the aggregate supply are: 1 2 3
Farm 1 Farm 2 Aggregate Supply Next, suppose the price of the good is 19* : 2.5. How many farmers (ﬁrms) Will participate and how much Will each farm
supply? What will be the total output supplied? Farmers 1 and 2 (71* : 2); each supplies one unit : y; : 1)
so the total output supplied is 2 (3f : 2). If 19* is to be associated with LONG—RUN EQUILIBRIUM What must be true
about the proﬁt of each farm? 71'” : 0 :>p*(1)—n(1)—rn, for 71 :1,2 and
7t” : 0:>p*(0) —n(0) —7“n, for n: 3,4,5, So must be true about the EQUILIBRIUM RENTS in the long run 7T1:O¢rf:p*—l:2.5—1:1.5 W2:0?T3:p*—2:2.5—2:0.5 W3:0©r§:0—0:0
and likewise r::r’5k::rn:0 What is ‘total rent? 1.5 + .5 : 2. 10 NL IN: 0 What does ‘equilibirium rent’ reﬂect? The scarcity of land of a given productivity. 0 Why is ‘Unit 1’ more expensive that the others? Because it is the most productive land unit. 0 Does the landowner make ‘economic proﬁt’? No. She makes: ‘ECONOMIC RENT’ : ACTUAL RENT AMOUNT — MINIMUM AMOUNT REQUIRED TO
SUPPLY THE INPUT 0 If the price of output rises from 19* to p**, what will happen in the long
rnnZ More farms will participate (n** > More total output will be supplied (y** > Rents will rise (each farmer will make zero proﬁt) (7“:k Z 73:) 0 When we trace the points (p*, y*) and (p**, y**), what do we get?
The long—run behaviour of the industry: the LONG—RUN INDUSTRY SUP—
BLX o Is there ‘producer surplus’ in the long run?
The area below the price line and above the supply now represents the
economic rent received by the inputs whose price increased due to the
expansion of the industry. 11 We have simpliﬁed by leaving out things such as: 0 An explicit account of time: investment choices by the ﬁrm
(ADVANCED / GRADUATE MICRO, GROWTH THEORY) 0 An explicit account of uncertainty: ﬁrm behaviour toward risk given, e. g:
uncertain product/input prices (ADVANCED/GRADUATE MICRO, THE
ECONOMICS OF UNCERTAINTY AND INFORMATION)
0 An explicit account of ﬁnancial decisions (debt vs equity) (FINANCIAL ECONOMICS) 0 An explicit account of space/location (URBAN ECONOMICS) 0 An explicit account of the inner workings of the ﬁrm (e.g., separation between ownership and control) (CONTRACT THEORY, THE THEORY OF
THE FIRM, ORGANIZATIONAL ECONOMICS, ADVANCED / GRADUATE MI— CRO 12 ...
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This note was uploaded on 02/29/2012 for the course ECON 2350 taught by Professor Bardis during the Fall '12 term at York University.
 Fall '12
 Bardis

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