Frank Cowell Microeconomics November 2006 Exercise 2 10

Frank Cowell: Microeconomics November 2006 Exercise 2. 10 MICROECONOMICS Principles and Analysis Frank Cowell

Ex 2. 10: Question Frank Cowell: Microeconomics n n 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 Frank Cowell: Microeconomics n Put the production function in a more manageable form n A quick check on the isoquant for m = 2: n Clearly isoquants do not touch the axes u z 2 Solution cannot be at a corner z 1

Ex 2. 10(1): Cost minimisation Frank Cowell: Microeconomics n The Lagrangean: n Differentiate w. r. t. zi to find the FOCs n Rearrange to get: n l (the Lagrange multiplier) is an unknown n We need to eliminate it

Ex 2. 10(1): Finding l Frank Cowell: Microeconomics n Use the production function n And substitute in for zi: u where n From this we find that

Ex 2. 10(1): The cost function Frank Cowell: Microeconomics n l can be simplified to n Substitute into expression for zi; get optimal input demands n So minimised costs expressed as a function of w and q are n This can be written as g. Bq 1/g where n Differentiating this w. r. t. q, MC is n So MC is increasing in q if g < 1

Ex 2. 10(2): Preliminary Frank Cowell: Microeconomics n In the “short run” the amounts of inputs k+1, …, m are fixed n So, define the term u (constant in the short run) n The production function can be written: n This is the only part that is variable in the short run. n We see that the problem has exactly the same structure as before u n but with different parameters. Therefore the solution has the same structure as before u but with different parameters.

Ex 2. 10(2): Short-run input demand Frank Cowell: Microeconomics n We can proceed by analogy with the long-run case Cost-minimising input demands must be: n where we have defined n Multiplying each input demand by wi and summing will give short-run variable costs n

Ex 2. 10(2): Short-run costs Frank Cowell: Microeconomics n Define short-run fixed costs u the amounts of inputs k+1, …, m are fixed n Then short-run total costs are given by n Substituting in for zi* costs in the short run are: n Clearly this expression has the form: n Differentiate costs w. r. t. q and we find short-run MC:

Ex 2. 10(3): short run supply Frank Cowell: Microeconomics n From the SRMC we get the short-run supply curve The condition “MC = price” gives n Solving this for q the supply function is n The elasticity of supply is n Clearly the elasticity falls if gk falls n u By definition of gk it must fall if k is reduced

Ex 2. 10: Points to remember Frank Cowell: Microeconomics n n 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