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Exercise 2.9

Exercise 2.9. MICROECONOMICS Principles and Analysis Frank Cowell . March 2007 . Ex 2.9(1): Question. purpose : demonstrate relationship between short and long run method : Lagrangean approach to cost minimisation. First part can be solved by a “trick”. Ex 2.9(1): Long-run costs.

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Exercise 2.9

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  1. Exercise 2.9 MICROECONOMICS Principles and Analysis Frank Cowell March 2007

  2. Ex 2.9(1): Question • purpose: demonstrate relationship between short and long run • method: Lagrangean approach to cost minimisation. First part can be solved by a “trick”

  3. Ex 2.9(1): Long-run costs • Production function is homogeneous of degree 1 • increase all inputs by a factor t > 0 (i.e. z→tz)… • …and output increases by the same factor (i.e. q→tq) • constant returns to scale in the long run • CRTS implies constant average cost • C(w, q) / q = A (a constant) • so C(w, q) = Aq • differentiating: Cq(w, q) = A • So LRMC = LRAC = constant • Their graphs will be an identical straight line

  4. Ex 2.9(2): Question method: • Standard Lagrangean approach

  5. Ex 2.9(2): short-run Lagrangean • In the short run amount of good 3 is fixed • z3 = `z3 • Could write the Lagrangean as • But it is more convenient to transform the problem thus • where

  6. z2 z1 Ex 2.9(2): Isoquants • Sketch the isoquant map • Isoquants do not touch the axes • So maximum problem must have an interior solution

  7. Ex 2.9(2): short-run FOCs • Differentiating Lagrangean, the FOCS are • This implies • To find conditional demand function must solve for l • use the above equations… • …and the production function

  8. Ex 2.9(2): short-run FOCs (more) • Using FOCs and the production function: • This implies • where • This will give us the short-run cost function

  9. Ex 2.9(2): short-run costs • By definition, short-run costs are: • This becomes • Substituting for k: • From this we get • SRAC: • SRMC:

  10. q Ex 2.9(2): short-run MC and AC marginal cost average cost

  11. Ex 2.9(3): Question method: • Draw the standard supply-curve diagram • Manipulate the relationship p = MC

  12. p q Ex 2.9(3): short-run supply curve • average cost curve • marginal cost curve • minimum average cost • supply curve p q

  13. Ex 2.9(3): short-run supply elasticity • Use the expression for marginal cost: • Set p = MC for p≥p • Rearrange to get supply curve • Differentiate last line to get supply elasticity

  14. Ex 2.9: Points to remember • Exploit CRTS to give you easy results • Try transforming the Lagrangean to make it easier to manipulate • Use MC curve to derive supply curve

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