Cost identities, averages and marginal curves
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| marginal cost/ˈmɑːdʒɪnl kɒst/ | 边际成本 | biān jì chéng běn |
| average fixed cost/ˈævrɪdʒ fɪkst kɒst/ | 平均固定成本 | píng jūn gù dìng chéng běn |
A decision you can investigate
- A higher total bill need not mean a higher cost per unit. An additional unit below the existing average pulls the average down.
- Define the denominator and compare the right cost measures.
Build the explanation
- TC=TFC+TVC. At positive Q, ATC=TC/Q, AFC=TFC/Q and AVC=TVC/Q, so ATC=AFC+AVC. MC=ΔTC/ΔQ; with unchanged fixed cost it also equalsΔTVC/ΔQ. Fixed costs stay unchanged over the relevant short-run range, so AFC declines as positive output rises. Variable costs depend on activity; they need not be proportional to output.
- An MC below an average cost pulls that average down; MC above it pushes the average up. In a smooth interior model MC crosses AVC and ATC at their respective minima. Do not interchange total and average curves. Short-run curves hold some capacity fixed, whereas LRAC describes the lowest achievable average cost when the firm can choose its input scale. A short-run operating point need not be the cheapest long-run scale.
Work through the evidence
- A fictional firm has TFC60. At Q10, TVC140 gives TC200, AVC14, AFC6 and ATC20. At Q12, TVC168 gives TC228, AVC14, AFC5 and ATC19. The two-unit interval MC=(228−200)/(12−10)=14. ATC falls because this interval cost14 is below the starting average20; AVC stays14 because variable cost here is locally proportional.
- A separate smooth illustration uses TC=60+20Q−3Q²+0.2Q³. Then MC=20−6Q+0.6Q², AVC=20−3Q+0.2Q², AFC=60/Q and ATC=AVC+60/Q for Q>0. At Q10, MC20, AVC10, AFC6 and ATC16. AVC reaches8.75 at Q7.5 where MC also8.75. These smooth values are a different model from the earlier finite table, not data to splice together.
What is ATC at Q12 in the finite example?
TC228 divided by12 gives19.
What is the two-unit interval MC?
Cost rises28 across two units, giving14 per unit.
Average total cost is defined at zero output by dividing fixed cost by zero.
Average cost requires positive output; the quotient at zero is undefined.
Test the limits
- At Q0, average costs are undefined because division by zero is not allowed; total fixed cost may still exist. Fixed means fixed over a specified period/output range, not unavoidable forever. A rent can become avoidable when a contract ends.
- Step-fixed capacity costs can jump; marginal costs then require the actual interval rather than a smooth derivative. MC intersects an average at a minimum only under appropriate smooth conditions; discontinuous data may straddle it. LRAC is not the same as a single SRATC curve. Compare feasible scales, input prices, technology and demand before interpreting low average cost as high profit.
What is AVC at Q7.5 in the smooth illustration?
20−3×7.5+0.2×7.5²=8.75.
Apply and explain your answer
- Why does ATC fall while AVC remains14 in the finite example?
- Fixed cost60 is spread over more output: AFC falls6→5, while variable cost per unit remains14.
Match the terms to their meanings.
Use each term for its stated economic relationship.
Use the terms precisely
- average fixed cost 平均固定成本: Fixed cost divided by positive output.
- marginal cost 边际成本: Change in total cost per additional output unit over a stated interval, or its derivative in a smooth model.
A fictional firm has TFC60. At Q10, TVC140 gives TC200, AVC14, AFC6 and ATC20. At Q12, TVC168 gives TC228, AVC14, AFC5 and ATC19. The two-unit interval MC=(228−200)/(12−10)=14. ATC falls because this interval cost14 is below the starting average20; AVC stays14 because variable cost here is locally proportional. A separate smooth illustration uses TC=60+20Q−3Q²+0.2Q³. Then MC=20−6Q+0.6Q², AVC=20−3Q+0.2Q², AFC=60/Q and ATC=AVC+60/Q for Q>0. At Q10, MC20, AVC10, AFC6 and ATC16. AVC reaches8.75 at Q7.5 where MC also8.75. These smooth values are a different model from the earlier finite table, not data to splice together.
At Q0, average costs are undefined because division by zero is not allowed; total fixed cost may still exist. Fixed means fixed over a specified period/output range, not unavoidable forever. A rent can become avoidable when a contract ends. Step-fixed capacity costs can jump; marginal costs then require the actual interval rather than a smooth derivative. MC intersects an average at a minimum only under appropriate smooth conditions; discontinuous data may straddle it. LRAC is not the same as a single SRATC curve. Compare feasible scales, input prices, technology and demand before interpreting low average cost as high profit.
TC=TFC+TVC. At positive Q, ATC=TC/Q, AFC=TFC/Q and AVC=TVC/Q, so ATC=AFC+AVC. MC=ΔTC/ΔQ; with unchanged fixed cost it also equalsΔTVC/ΔQ. Fixed costs stay unchanged over the relevant short-run range, so AFC declines as positive output rises. Variable costs depend on activity; they need not be proportional to output.