Frank Cowell: Microeconomics Exercise 2.10 MICROECONOMICS Principles and Analysis Frank Cowell November 2006

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

Frank Cowell: Microeconomics 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 z 2

Frank Cowell: Microeconomics Ex 2.10(1): Cost minimisation The Lagrangean: Differentiate w.r.t. z i to find the FOCs Rearrange to get: (the Lagrange multiplier) is an unknown We need to eliminate it We need to eliminate it

Frank Cowell: Microeconomics Ex 2.10(1): Finding Use the production function Use the production function And substitute in for z i : And substitute in for z i :  where From this we find that From this we find that

Frank Cowell: Microeconomics Ex 2.10(1): The cost function can be simplified to can be simplified to Substitute into expression for z i ; get optimal input demands Substitute into expression for z i ; get optimal input demands So minimised costs expressed as a function of w and q are So minimised costs expressed as a function of w and q are This can be written as  Bq 1/  where This can be written as  Bq 1/  where Differentiating this w.r.t. q, MC is Differentiating this w.r.t. q, MC is So MC is increasing in q if  < 1 So MC is increasing in q if  < 1

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

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

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

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

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