MICROECONOMICS Principles and Analysis Frank Cowell 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 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

Ex 2.9(2): Question method: Standard Lagrangean approach

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

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

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

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

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

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

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

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

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

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