Cost-Benefit Analysis
Principles of Economics: Long-run Competitive Equilibrium LONG-RUN COST Recall that all factors, particularly capital, are variable in the long-run. There are two ways (plus combinations of the two ways) that Capital can change in the long-run. 1.
Change in the size of individual firms through additional or improved equipment and buildings or through reduction or depreciation of equipment and buildings.
2.
Entry into or exit from the industry by firms of the same size as existing firms.
Economies of Scale: We analyze the effect of changes in the size of individual firm capital through the concept of economies of scale. Definition. Economies of Scale (Increasing Returns to Scale) occur when a % increase in all factor inputs causes a greater % increase in output. E.g. Suppose that 5 units of Labour and 3 units of Capital produce 100 units of output but that 10 units of Labour and 6 units of Capital produce 250 units of output. => 100% increase in factor inputs produces a 150% increase in output. Why does the concept of Economies of Scale not contradict Eventually Diminishing Returns (MP)? Definition: Diseconomies of Scale (Decreasing Returns to Scale) occurs when a % increase in all factors causes a smaller % increase in output. Definition: Constant Returns to Scale occurs when a % increase in all factors causes the same % increase in output. Average Cost decreases with Economies of Scale The key element of economies of scale is a decrease in Average Cost with increases in output.
-1-
Principles of Economics: Long-run Competitive Equilibrium Proof: Recall that Average Cost = ACo = (wL + rK)/q Define ‘a’ as the % change in inputs and ‘b’ as the % change in outputs. => AC1 = (waL + raK)/bq = a/b(wL + rK)/q = a/b * ACo a < b => Economies of Scale (Increasing Returns to Scale)
=> Average Cost decreases
a = b => Constant Returns to Scale
=> no change in Average Cost
a > b => Diseconomies of Scale (Decreasing Returns to Scale) => Average Cost increases Derivation of Long-run Average Cost from Short-run Average Cost Curves Since each quantity of capital gives specific short-run cost functions, we can derive long-run average cost from the short-run average cost function of each capital. Quantity ACo (Ko) AC1(K1>Ko) AC2(K2>K1) AC3(K3>K2)
20 $20 $30 $40 $60
Cost/unit AC2
40 $14 $20 $25 $45
60 $10 $12 $15 $32
80 $15 $8 $10 $20
100 $22 $10 $6 $14
120 $30 $14 $8 $10
Long-run Average Cost AC3
AC1 ACo
LRAC
Quantity
Average Cost falls to minimum and then rises for each Capital. Given that fixed cost is larger for larger capitals, average cost for small output is greater for larger capitals relative to smaller
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Principles of Economics: Long-run Competitive Equilibrium capitals. Average Cost is eventually lower for larger capitals relative to smaller capitals if there is economies of scale but minimum average cost will rise if there are diseconomies of scale. Note that falling long-run average cost implies economies of scale and rising long-run average cost implies diseconomies of scale. Long-run Average Cost tends to initially decrease due to increasing returns to scale with increased capital and labour and eventually increase due to decreasing returns to scale with increased capital and labour. There is only one minimum average cost attainable though it may occur with different amounts of capital and labour. The smallest capital that gives this minimum average cost is called Minimum Efficient Scale. Long-run Average Cost (LRAC) Long-run Average Cost shows the lowest average cost for each output in the long-run. This defines the size of capital for lowest average cost for each output. If capital is infinitely divisible, Long-run Average Cost is a smooth curve with each point from the short-run average cost of each capital. Average Cost Increasing Returns
Decreasing Returns
Minimum Efficient Scale
q
Profit Maximization in the long-run implies that firms change capital and labour until they find the capital with the Short-run Average Cost function that gives minimum Long-run Average Cost for the desired output.
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Principles of Economics: Long-run Competitive Equilibrium The diagram below represents the Long-run Average Cost curve for an industry and three sizes of capital in the industry with their attendant Short-run Average Cost curves. Provided that there is no technological change (i.e., no new minimum efficient scale), examination of the diagram shows that in competitive conditions price will fall from Po to P* as new firms enter the industry due to economic profits at Po for all capitals and above P* for the minimum efficient scale capital. In the long-run, no firm could obtain the average return on capital unless the firm was the size (capital) that permitted Minimum Efficient Scale.
AC, P
Po MES P* q This derivation shows that ‘perfect competition’ – all firms are the same size with similar cost functions – is an evolutionary result of a competitive market provided that there is no technological change. The assumption of perfect competition means that we do not need to draw the Long-run Average Cost function since we know that its minimum is minimum Short-run Average Cost of the firm. Technological change alters the minimum efficient scale, usually increasing it but sometimes decreasing it drastically. Electricity generation was dominated by $Billion plus plants (Hydro, Nuclear, or Coal) up until the 1970s since these huge plants produced the cheapest electricity. The development of Co-generation Gas Turbines in the 1970s introduced small jet engines that could produce electricity competitively because their small size and clean combustion of natural gas -4-
Principles of Economics: Long-run Competitive Equilibrium meant that they could be located in buildings to take advantage of the heat generated in combustion. The low natural gas prices of the 1980s and 1990s made these co-generators a formidable competitive of the huge plants, sparking the collapse of monopoly pricing in various jurisdictions, especially California. LONG-RUN COMPETITIVE EQUILIBRIUM We will assume perfect competition without technological change in this course so we can ignore the possibility of changes in the size of firm capital. Changes in capital can therefore only occur through the entry into or exit from the industry of perfectly competitive firms. Long-run Equilibrium => No change in Capital despite the possibility of Capital Change => No firms enter or exit the industry => 0 Economic Profit => P = min AC (Price = minimum Average Cost) We express long-run equilibrium graphically through the firm/industry diagram Demand and Supply intersecting at minimum Average Cost as in the following diagram. Long-Run Equilibrium Px
Cost/unit
So
MC AC P0
AC0
Do
q0
qx
Q0
Qx
E.g. Suppose that a perfectly competitive firm has the following cost functions. VC = 2q2 + 30q => AVC = VC/q = 2q + 30
FC = 1800
MC = 4q + 30
AC = TC/q = 2q + 30 + 1800/q
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Principles of Economics: Long-run Competitive Equilibrium 1.
What is firm Supply? Firm Supply = MC > min AVC => P = 4q + 30
2.
What is industry Supply if there are n firms? Industry Supply => P = 4(Q/n) + 30
3.
What is long-run equilibrium output of the firm? => q at MC = min AC=> 4q + 30 = 2q + 30 + 1800/q => 2q = 1800/q => q = 30
4.
What is long-run equilibrium price for the industry (and firm)? => P = min AC = 2*30 + 30 + 1800/30 = $150
5.
What is industry output if market demand is P = 690 – 0.9Q? => 150 = 690 – 0.9Q => Q = 600
6.
What is the number of firms in this industry at long-run equilibrium? Each firm produces 30 units and industry output is 600 units => q = Q/n => 30 = 600/n, => n = 20
Long-run Supply (Minimum Average Cost for each output) Definition: Long-run Supply is the set of price/quantity supplied combinations that give long-run equilibrium. => The assumption of perfect competition with entry and exit of optimally sized firms with no change in the size of firms means that Long-run Supply is the set of minimum Average Cost for each output not Long-run Marginal Cost. Constant Costs Definition: ‘Constant Costs’ assumes that changes in output of the industry do not affect the factor prices of the industry.
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Principles of Economics: Long-run Competitive Equilibrium We will assume ‘constant costs’ in our discussion of long-run equilibrium. This does not mean that factor prices (e.g., wages or the price of raw materials or fuel) cannot change exogenously but that factor prices do not change despite increased or decreased demand for factors due to increases or decreases in industry output. We know analyze the effect of changes in demand, changes in variable costs, and changes in fixed costs on long-run equilibrium given the assumption that factor prices are not affected by the size of an industry (constant costs) 1.
Changes in Demand
E.g. Decrease in Demand
Long-Run Equilibrium Decrease in Demand Cost/unit
Px
S1
MC
So
AC P1 P0
ACS MCS
Economic Loss
Ps
Do D1 qS
qx
q0 q1
Q1 QS Q0
Qx
Demand decreases, causing a fall in Price (PS) and Quantity (QS) in the short-run. The firm produces at qS from MC at PS for an Economic Loss of qS(ACS – PS). In the long-run, the economic loss causes some firms to leave the industry. Industry Supply decreases as firms leave until price rises to minimum Average Variable Costs where economic profits are zero. No more -7-
Principles of Economics: Long-run Competitive Equilibrium firms leave and the industry is at equilibrium. Since Average Cost has not changed, the new longrun equilibrium Price (P1) equals the original long-run equilibrium price. The output and the zero economic profit/loss of the firm also do not change in the long-run. Industry output falls relative to Qo and QS due to the decline in firms. The assumption of constant costs results in a perfectly elastic Long-run Supply function for changes in Demand. 2.
Changes in Variable Costs
E.g. Decrease in Variable Costs (e.g., decrease in labour or raw material costs or per unit taxes)
Cost/unit
Long-Run Equilibrium Decrease in Variable Costs Px MCo ACo MC1
S0 SS S1
AC1 P0 MCS ACS
Economic Profit
Ps P1 P0-dVC
Do qx
q0 qS q1
Q0 QS Q1
Qx
The diagram above depicts a decrease in variable cost of exactly the same amount per unit. Since the variable cost per unit decreases by the same amount, the average cost of every output decreases by the decrease in variable cost per unit. Average Cost shifts down parallel to the original Average Cost function with minimum average cost as q0 as before. Marginal Cost, the cost of each additional unit, also shifts down by the decrease in Variable Cost per unit to intersect
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Principles of Economics: Long-run Competitive Equilibrium the new Average Cost at its minimum at q0. (Average Variable Cost also shifts down by the same amount per unit, which we see here as the shift down of the shut-down price as minimum Marginal Cost) Since Industry Supply is Marginal Cost (above minimum Average Variable Cost) times the number of firms (in a perfectly competitive market), Supply decreases by amount of the decrease in variable costs per unit. We show this by shifting Supply down at q0 to pass through P0-dVC. Shortrun equilibrium Price (PS) and Quantity (Q) is from the intersection of this new Supply (SS) and the original Demand function. Price does not fall as much as the fall in variable costs because fall in price induces movement along demand and an increase in quantity demanded. The firm produces qS in the short-run where PS equals Marginal Cost. Average Cost ACS if from the Average Cost function at qS. This gives an Economic Profit of qS (PS - ACS) because Price did not fall by as much as Variable Costs. In the long-run, the economic profit draws firms into the industry. Supply increases (shifts right) until price falls to the new minimum Average Cost. Economic profits are zero and no new firms enter the industry. Long-run equilibrium in the industry occurs at P1 and Q1 > QS > Qo. The individual firm decrease production as Price falls to q1 at minimum Average Cost where there is zero Economic Profit. There is therefore no change in industry output or economic profit or loss in the long-run. Price falls by the amount of the fall in per unit costs and industry output increases through the entry of firms into the industry. 3.
Changes in Fixed Costs
E.g. Decrease in Fixed Costs (e.g., decrease in property tax or licensing fee)
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Principles of Economics: Long-run Competitive Equilibrium A change in Fixed Costs does not change Marginal Cost or Supply. Industry price and quantity and firm quantity do not change in the short-run therefore. The decrease in Fixed Cost does cause a non-parallel decrease in Average Cost, with a minimum (= MC) at less output and lower price than before the decrease in Fixed Cost, and an economic profit (equal to the decrease in Fixed Costs) at qS = q0 in the short-run. Firms enter the industry in the long-run increasing supply until supply intersects demand at price P1 equal to minimum Average Cost and industry quantity Q1 > Q0. Firm output q0 is at minimum average cost with 0 economic profit. Note that industry output increases in the long-run since firms enter but firm output decreases in the long-run because fixed costs are lower.
Long-Run Equilibrium Decrease in Fixed Cost
Cost/unit
Px MCo
S0
S1
ACo AC1 MCS ACS
Ps = P0
Economic Profit
P1
Do q1
q0 qS
qx
Q0 QS
- 10 -
Q1
Qx
Change in the size of individual firms through additional or improved equipment and buildings or through reduction or depreciation of equipment and buildings.
2.
Entry into or exit from the industry by firms of the same size as existing firms.
Economies of Scale: We analyze the effect of changes in the size of individual firm capital through the concept of economies of scale. Definition. Economies of Scale (Increasing Returns to Scale) occur when a % increase in all factor inputs causes a greater % increase in output. E.g. Suppose that 5 units of Labour and 3 units of Capital produce 100 units of output but that 10 units of Labour and 6 units of Capital produce 250 units of output. => 100% increase in factor inputs produces a 150% increase in output. Why does the concept of Economies of Scale not contradict Eventually Diminishing Returns (MP)? Definition: Diseconomies of Scale (Decreasing Returns to Scale) occurs when a % increase in all factors causes a smaller % increase in output. Definition: Constant Returns to Scale occurs when a % increase in all factors causes the same % increase in output. Average Cost decreases with Economies of Scale The key element of economies of scale is a decrease in Average Cost with increases in output.
-1-
Principles of Economics: Long-run Competitive Equilibrium Proof: Recall that Average Cost = ACo = (wL + rK)/q Define ‘a’ as the % change in inputs and ‘b’ as the % change in outputs. => AC1 = (waL + raK)/bq = a/b(wL + rK)/q = a/b * ACo a < b => Economies of Scale (Increasing Returns to Scale)
=> Average Cost decreases
a = b => Constant Returns to Scale
=> no change in Average Cost
a > b => Diseconomies of Scale (Decreasing Returns to Scale) => Average Cost increases Derivation of Long-run Average Cost from Short-run Average Cost Curves Since each quantity of capital gives specific short-run cost functions, we can derive long-run average cost from the short-run average cost function of each capital. Quantity ACo (Ko) AC1(K1>Ko) AC2(K2>K1) AC3(K3>K2)
20 $20 $30 $40 $60
Cost/unit AC2
40 $14 $20 $25 $45
60 $10 $12 $15 $32
80 $15 $8 $10 $20
100 $22 $10 $6 $14
120 $30 $14 $8 $10
Long-run Average Cost AC3
AC1 ACo
LRAC
Quantity
Average Cost falls to minimum and then rises for each Capital. Given that fixed cost is larger for larger capitals, average cost for small output is greater for larger capitals relative to smaller
-2-
Principles of Economics: Long-run Competitive Equilibrium capitals. Average Cost is eventually lower for larger capitals relative to smaller capitals if there is economies of scale but minimum average cost will rise if there are diseconomies of scale. Note that falling long-run average cost implies economies of scale and rising long-run average cost implies diseconomies of scale. Long-run Average Cost tends to initially decrease due to increasing returns to scale with increased capital and labour and eventually increase due to decreasing returns to scale with increased capital and labour. There is only one minimum average cost attainable though it may occur with different amounts of capital and labour. The smallest capital that gives this minimum average cost is called Minimum Efficient Scale. Long-run Average Cost (LRAC) Long-run Average Cost shows the lowest average cost for each output in the long-run. This defines the size of capital for lowest average cost for each output. If capital is infinitely divisible, Long-run Average Cost is a smooth curve with each point from the short-run average cost of each capital. Average Cost Increasing Returns
Decreasing Returns
Minimum Efficient Scale
q
Profit Maximization in the long-run implies that firms change capital and labour until they find the capital with the Short-run Average Cost function that gives minimum Long-run Average Cost for the desired output.
-3-
Principles of Economics: Long-run Competitive Equilibrium The diagram below represents the Long-run Average Cost curve for an industry and three sizes of capital in the industry with their attendant Short-run Average Cost curves. Provided that there is no technological change (i.e., no new minimum efficient scale), examination of the diagram shows that in competitive conditions price will fall from Po to P* as new firms enter the industry due to economic profits at Po for all capitals and above P* for the minimum efficient scale capital. In the long-run, no firm could obtain the average return on capital unless the firm was the size (capital) that permitted Minimum Efficient Scale.
AC, P
Po MES P* q This derivation shows that ‘perfect competition’ – all firms are the same size with similar cost functions – is an evolutionary result of a competitive market provided that there is no technological change. The assumption of perfect competition means that we do not need to draw the Long-run Average Cost function since we know that its minimum is minimum Short-run Average Cost of the firm. Technological change alters the minimum efficient scale, usually increasing it but sometimes decreasing it drastically. Electricity generation was dominated by $Billion plus plants (Hydro, Nuclear, or Coal) up until the 1970s since these huge plants produced the cheapest electricity. The development of Co-generation Gas Turbines in the 1970s introduced small jet engines that could produce electricity competitively because their small size and clean combustion of natural gas -4-
Principles of Economics: Long-run Competitive Equilibrium meant that they could be located in buildings to take advantage of the heat generated in combustion. The low natural gas prices of the 1980s and 1990s made these co-generators a formidable competitive of the huge plants, sparking the collapse of monopoly pricing in various jurisdictions, especially California. LONG-RUN COMPETITIVE EQUILIBRIUM We will assume perfect competition without technological change in this course so we can ignore the possibility of changes in the size of firm capital. Changes in capital can therefore only occur through the entry into or exit from the industry of perfectly competitive firms. Long-run Equilibrium => No change in Capital despite the possibility of Capital Change => No firms enter or exit the industry => 0 Economic Profit => P = min AC (Price = minimum Average Cost) We express long-run equilibrium graphically through the firm/industry diagram Demand and Supply intersecting at minimum Average Cost as in the following diagram. Long-Run Equilibrium Px
Cost/unit
So
MC AC P0
AC0
Do
q0
qx
Q0
Qx
E.g. Suppose that a perfectly competitive firm has the following cost functions. VC = 2q2 + 30q => AVC = VC/q = 2q + 30
FC = 1800
MC = 4q + 30
AC = TC/q = 2q + 30 + 1800/q
-5-
Principles of Economics: Long-run Competitive Equilibrium 1.
What is firm Supply? Firm Supply = MC > min AVC => P = 4q + 30
2.
What is industry Supply if there are n firms? Industry Supply => P = 4(Q/n) + 30
3.
What is long-run equilibrium output of the firm? => q at MC = min AC=> 4q + 30 = 2q + 30 + 1800/q => 2q = 1800/q => q = 30
4.
What is long-run equilibrium price for the industry (and firm)? => P = min AC = 2*30 + 30 + 1800/30 = $150
5.
What is industry output if market demand is P = 690 – 0.9Q? => 150 = 690 – 0.9Q => Q = 600
6.
What is the number of firms in this industry at long-run equilibrium? Each firm produces 30 units and industry output is 600 units => q = Q/n => 30 = 600/n, => n = 20
Long-run Supply (Minimum Average Cost for each output) Definition: Long-run Supply is the set of price/quantity supplied combinations that give long-run equilibrium. => The assumption of perfect competition with entry and exit of optimally sized firms with no change in the size of firms means that Long-run Supply is the set of minimum Average Cost for each output not Long-run Marginal Cost. Constant Costs Definition: ‘Constant Costs’ assumes that changes in output of the industry do not affect the factor prices of the industry.
-6-
Principles of Economics: Long-run Competitive Equilibrium We will assume ‘constant costs’ in our discussion of long-run equilibrium. This does not mean that factor prices (e.g., wages or the price of raw materials or fuel) cannot change exogenously but that factor prices do not change despite increased or decreased demand for factors due to increases or decreases in industry output. We know analyze the effect of changes in demand, changes in variable costs, and changes in fixed costs on long-run equilibrium given the assumption that factor prices are not affected by the size of an industry (constant costs) 1.
Changes in Demand
E.g. Decrease in Demand
Long-Run Equilibrium Decrease in Demand Cost/unit
Px
S1
MC
So
AC P1 P0
ACS MCS
Economic Loss
Ps
Do D1 qS
qx
q0 q1
Q1 QS Q0
Qx
Demand decreases, causing a fall in Price (PS) and Quantity (QS) in the short-run. The firm produces at qS from MC at PS for an Economic Loss of qS(ACS – PS). In the long-run, the economic loss causes some firms to leave the industry. Industry Supply decreases as firms leave until price rises to minimum Average Variable Costs where economic profits are zero. No more -7-
Principles of Economics: Long-run Competitive Equilibrium firms leave and the industry is at equilibrium. Since Average Cost has not changed, the new longrun equilibrium Price (P1) equals the original long-run equilibrium price. The output and the zero economic profit/loss of the firm also do not change in the long-run. Industry output falls relative to Qo and QS due to the decline in firms. The assumption of constant costs results in a perfectly elastic Long-run Supply function for changes in Demand. 2.
Changes in Variable Costs
E.g. Decrease in Variable Costs (e.g., decrease in labour or raw material costs or per unit taxes)
Cost/unit
Long-Run Equilibrium Decrease in Variable Costs Px MCo ACo MC1
S0 SS S1
AC1 P0 MCS ACS
Economic Profit
Ps P1 P0-dVC
Do qx
q0 qS q1
Q0 QS Q1
Qx
The diagram above depicts a decrease in variable cost of exactly the same amount per unit. Since the variable cost per unit decreases by the same amount, the average cost of every output decreases by the decrease in variable cost per unit. Average Cost shifts down parallel to the original Average Cost function with minimum average cost as q0 as before. Marginal Cost, the cost of each additional unit, also shifts down by the decrease in Variable Cost per unit to intersect
-8-
Principles of Economics: Long-run Competitive Equilibrium the new Average Cost at its minimum at q0. (Average Variable Cost also shifts down by the same amount per unit, which we see here as the shift down of the shut-down price as minimum Marginal Cost) Since Industry Supply is Marginal Cost (above minimum Average Variable Cost) times the number of firms (in a perfectly competitive market), Supply decreases by amount of the decrease in variable costs per unit. We show this by shifting Supply down at q0 to pass through P0-dVC. Shortrun equilibrium Price (PS) and Quantity (Q) is from the intersection of this new Supply (SS) and the original Demand function. Price does not fall as much as the fall in variable costs because fall in price induces movement along demand and an increase in quantity demanded. The firm produces qS in the short-run where PS equals Marginal Cost. Average Cost ACS if from the Average Cost function at qS. This gives an Economic Profit of qS (PS - ACS) because Price did not fall by as much as Variable Costs. In the long-run, the economic profit draws firms into the industry. Supply increases (shifts right) until price falls to the new minimum Average Cost. Economic profits are zero and no new firms enter the industry. Long-run equilibrium in the industry occurs at P1 and Q1 > QS > Qo. The individual firm decrease production as Price falls to q1 at minimum Average Cost where there is zero Economic Profit. There is therefore no change in industry output or economic profit or loss in the long-run. Price falls by the amount of the fall in per unit costs and industry output increases through the entry of firms into the industry. 3.
Changes in Fixed Costs
E.g. Decrease in Fixed Costs (e.g., decrease in property tax or licensing fee)
-9-
Principles of Economics: Long-run Competitive Equilibrium A change in Fixed Costs does not change Marginal Cost or Supply. Industry price and quantity and firm quantity do not change in the short-run therefore. The decrease in Fixed Cost does cause a non-parallel decrease in Average Cost, with a minimum (= MC) at less output and lower price than before the decrease in Fixed Cost, and an economic profit (equal to the decrease in Fixed Costs) at qS = q0 in the short-run. Firms enter the industry in the long-run increasing supply until supply intersects demand at price P1 equal to minimum Average Cost and industry quantity Q1 > Q0. Firm output q0 is at minimum average cost with 0 economic profit. Note that industry output increases in the long-run since firms enter but firm output decreases in the long-run because fixed costs are lower.
Long-Run Equilibrium Decrease in Fixed Cost
Cost/unit
Px MCo
S0
S1
ACo AC1 MCS ACS
Ps = P0
Economic Profit
P1
Do q1
q0 qS
qx
Q0 QS
- 10 -
Q1
Qx
