EC202: Worked Example #3.3 Frank Cowell April 2004 This presentation covers exactly the material set out in the file WorkedExamples.pdf, but with the addition of a few graphics and comments To start the presentation select Slideshow\View Show or click on icon below left. Mouse click or [Enter] to advance through slide show

WX3.3: The Short Run

Solution cannot be at a corner WX3.3: Preliminary Solution cannot be at a corner Put the production function in a more manageable form: z 1 2 A quick check on the isoquant: Write down the Lagrangean:

The l is an unknown: we need to eliminate it WX3.3: maximisation Differentiate the Lagrangean to get the FOCs: The l is an unknown: we need to eliminate it And rearrange:

WX3.3: Manipulating the FOCs From the FOC: Use the constraint (the production function): Rearrange to get l:

WX3.3: Deriving the conditional input demand functions Substitute this into this to get this:

Increasing with Q if g<1 WX3.3: the cost function multiply these: by wi and then add to get this: Increasing with Q if g<1 Differentiate to get marginal cost:

WX3.3: short run formulation Again put the production function into a convenient form: Constant in the short run So this is the only bit that is variable in the short run. But this means that the problem has exactly the same structure (but with different parameters). Therefore the solution has the same structure (but with different parameters).

WX3.3: short run results So we get:

WX3.3: short run MC and supply Differentiating the cost function we get marginal cost: This forms the short-run supply curve for the firm Clearly the elasticity falls if gk falls. gk falls if k is reduced Same applies for conditional input demand

WX3.3: 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