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

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

3. Frank Cowell: Microeconomics Ex 2.9(1): Long-run costs Production function is homogeneous of degree 1 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 CRTS implies constant average cost  C(w, q) / q = A (a constant)  so C(w, q) = Aq  differentiating: C q (w, q) = A So LRMC = LRAC = constant So LRMC = LRAC = constant  Their graphs will be an identical straight line

4. Frank Cowell: Microeconomics Ex 2.9(2): Question method: Standard Lagrangean approach Standard Lagrangean approach

5. Frank Cowell: Microeconomics Ex 2.9(2): short-run Lagrangean In the short run amount of good 3 is fixed In the short run amount of good 3 is fixed  z 3 =  z 3 Could write the Lagrangean as Could write the Lagrangean as But it is more convenient to transform the problem thus But it is more convenient to transform the problem thus where where

6. Frank Cowell: Microeconomics z2z2 z1z1 Ex 2.9(2): Isoquants Sketch the isoquant map Sketch the isoquant map Isoquants do not touch the axes Isoquants do not touch the axes So maximum problem must have an interior solution So maximum problem must have an interior solution

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

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

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

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

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

12. Frank Cowell: Microeconomics Ex 2.9(3): short-run supply curve   average cost curve   marginal cost curve   supply curve q p q p   minimum average cost

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

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