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Exercise 2.10

Exercise 2.10. MICROECONOMICS Principles and Analysis Frank Cowell. November 2006 . Ex 2.10: Question. purpose : to derive and compare short-run and long-run responses. method : derive AC, MC, supply in original and modified models . z. 2. z. 1. Ex 2.10(1): Preliminary steps.

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Exercise 2.10

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  1. Exercise 2.10 MICROECONOMICS Principles and Analysis Frank Cowell November 2006

  2. Ex 2.10: Question • purpose: to derive and compare short-run and long-run responses. • method: derive AC, MC, supply in original and modified models

  3. z 2 z 1 Ex 2.10(1): Preliminary steps • Put the production function in a more manageable form • A quick check on the isoquant for m = 2: • Clearly isoquants do not touch the axes • Solution cannot be at a corner

  4. Ex 2.10(1): Cost minimisation • The Lagrangean: • Differentiate w.r.t. zi to find the FOCs • Rearrange to get: • l (the Lagrange multiplier) is an unknown • We need to eliminate it

  5. Ex 2.10(1): Finding l • Use the production function • And substitute in for zi: • where • From this we find that

  6. Ex 2.10(1): The cost function • l can be simplified to • Substitute into expression for zi; get optimal input demands • So minimised costs expressed as a function of w and q are • This can be written as gBq1/g where • Differentiating this w.r.t. q, MC is • So MC is increasing in q if g < 1

  7. Ex 2.10(2): Preliminary • In the “short run” the amounts of inputs k+1,…,m are fixed • So, define the term • (constant in the short run) • The production function can be written: • This is the only part that is variable in the short run. • We see that the problem has exactly the same structure as before • but with different parameters. • Therefore the solution has the same structure as before • but with different parameters.

  8. Ex 2.10(2): Short-run input demand • We can proceed by analogy with the long-run case • Cost-minimising input demands must be: • where we have defined • Multiplying each input demand by wi and summing will give short-run variable costs

  9. Ex 2.10(2): Short-run costs • Define short-run fixed costs • the amounts of inputs k+1,…,m are fixed • Then short-run total costs are given by • Substituting in for zi* costs in the short run are: • Clearly this expression has the form: • Differentiate costs w.r.t. q and we find short-run MC:

  10. Ex 2.10(3): short run supply • From the SRMC we get the short-run supply curve • The condition “MC = price” gives • Solving this for q the supply function is • The elasticity of supply is • Clearly the elasticity falls if gk falls • By definition of gk it must fall if k is reduced

  11. Ex 2.10: Points to remember • Get the constraint into a convenient form • Get a simple view of the problem by deriving ICs • Use a little cunning to simplify the FOCs • Re-use your solution for other problems that have the same structure

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