MICROECONOMICS Principles and Analysis Frank Cowell 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

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 z 1 2

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

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

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

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

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

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:

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

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