1. EC202: Worked Example #3.3 Frank Cowell April 2004
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2. WX3.3: The Short Run

3. 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:

4. 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:

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

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

7. 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:

8. 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).

9. WX3.3: short run results
So we get:

10. 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

11. 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