| 1. | Demand and marginal
revenue: |
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| Given: TR = P·Q, where price is a function of output: P=P(Q) | ||||
| Let u and v be two functions: (1) u=P(Q) => du/dQ = dP/dQ (2) v=Q => dv/dQ = dQ/dQ = 1 |
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| MR = dTR/dQ = u·dv + v·du = P + Q·dP/dQ | ||||
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Monday, April 01, 2013 |
Linking price, marginal revenue and elasticity
| 1. | Demand and marginal
revenue: |
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| Given: TR = P·Q, where price is a function of output: P=P(Q) | ||||
| Let u and v be two functions: (1) u=P(Q) => du/dQ = dP/dQ (2) v=Q => dv/dQ = dQ/dQ = 1 |
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| MR = dTR/dQ = u·dv + v·du = P + Q·dP/dQ | ||||
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| 2. | Marginal revenue and elasticity: |
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| Recall: (1) MR = P + Q·dP/dQ (2) |e| = -dQ/dP·P/Q, since dQ/dP < 0 |
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| Rewriting (1) and (2) slightly: (1) => MR = P·(1 + dP/dQ·Q/P) (2) => dP/dQ·Q/P = -1/|e| |
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| Substitutiting -1/|e| into equation (1) yields: MR = P·(1 - 1/|e|) => |
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