Announcements Last lecture Tonight 5-6 ARTS Main LT Thursday 9-10 Muirhead Main LT: Fazeer Exercise Class on Production WorkSheet Marks on Test 2 hopefully by Friday CORRECT Version of Solution to Test 2 on Network (some minor 1 am errors in original)

Tests in Last Week 201 Test 3 on Production Worksheet –Monday 11th Dec 5-6 Vaughan Jeffries –see final previous years 201 TFU Final –Thursday 14th Dec 5-6 Vaughan Jeffries –see final previous years 203 Final (Two essays) –Wednesday 13th 12-1 Vaughan Jeffries Will meet in Semester 2 to review test procedure

So have 3 Distinct Problems 1) Maximise profits Max x1, x2  = P f (x 1, x 2 ) – w 1 x 1 – w 2 x 2 Gives factor demand functions X 1 = x 1 (w 1, w 2 ) X 2 = x 2 (w 1, w 2 ) May not be well defined if there are constant returns to scale Varian Appendix Ch 18

2) Maximise subject to a constraint 3) Minimise subject to a constraint Problem 3) is the Dual of 2) Called Duality Theory Essentially allows us to look at problems in reverse and can often give very important insights. Varian Appendix Ch19 Varian Appendix Ch 18

Take Cobb-Douglas example of Problem 2: e.g (1) (2) (3) =0

From 1) + 2)

Substitute into (3) This is a the factor-demand function for L

Strictly it is a Cost-constrained factor- demand function

So maximising output subject to a cost or finance constraint yields a solution for the use of the two factors: Similar to demand theory

So Production subject to a cost or a finance constraint is just like the usual consumption problem And a Cost-constrained factor-demand function is just a like normal Marshallian demand function in consumer theory …. with all its associated properties, adding- up, conditions, Cournot etc.

But what about the third problem… That of minimising cost subject to an output constraint? Again we will work through a Cobb-Douglas example to give you a flavour of the story.

Cobb-Douglas Example of problem 3 =0

From (1) + (2) Which is precisely the same condition as before, So both problems tell us that we should use K and L according to the same rule (4)

But now comes the different bit! The constraint (3) was and (4)

But if have CRS a+b= 1 [note we invert the bracket when we bring it to the other side]

This is a different factor demand function to the ‘normal’, ‘Marshallian-like’ demand function we had earlier. What does it correspond to?

L K We are adjusting Costs to get to best point on target isoquant given w/r If have different w/r, have differently sloped line, tangent to w 0 /r 0 So picking out points of tangency along isoquant

This is a different factor demand function to the ‘normal’, ‘Marshallian-like’ demand function we had earlier. This is essentially a Hicksian Factor demand curve It is a Quantity-Constrained CONDITIONAL factor-demand curve

The constraint (3) was and Similarly we can solve for K. The easiest way to do this is to go back to the FOC Inverting (4) above

[note we again invert the bracket when we bring it to the other side] And with CRS a + b=1 as before,

So Minimising Cost subject to an output constraint yields a solution for the use of the two factors: Conditional on the level of output. Similar to Hicksian demand in consumer theory

Note, since So conditional factor-demand functions always slope down Constant – so no ‘income’ type effect < 0

Cost Function But the objective of this exercise is not just to derive conditional factor demands, but rather to derive a cost function To do this we return to the original C= wL + rK

Note can now formalise the cost function for the item C = wL +rK

L K Notice C-D production function takes the form: w r While its cost function for a given quantity Q takes the form:

Now we assumed earlier that there were CRS and that a+b=1 No reason why it should be of course, and more generally the Cobb-Douglas version of the cost function is: Where k is a constant that depends on a and b

TC AC Q Q Long Run total and average cost functions

TC AC If a + b =1, CRS and constant LR average costs Q Q

TC AC If a + b <1, DRS and increasing LR average costs Q Q

TC AC If a + b >1, IRS and decreasing LR average costs Q Q

What about the Short-Run? Derivation of short-run costs from an isoquant map –Recall in SR Capital stock is fixed Derivation of short-run costs from an isoquant map –Recall in SR Capital stock is fixed

Units of capital (K) O Units of labour (L) TC 1 TC 4 TC Deriving a SRAC curve from an isoquant map L0L0 K0K0

TC AC SR TC and AC curve lies above LR costs Q Q

The Short Run Cost Function Note So the actual L employed at any point is : So short run costs are:

The Short Run Cost Function for a=b=1/2 Since

The Short Run Marginal Cost Function for a=b=1/2 Since We take the derivative to find MC

Under Perfect Competition MC=MR=P Which gives us the short run supply curve of the firm i: Q P MC=Q s

Equilibrium : To find an equilibrium then we have to set D =S D P Q The remaining aspects of any production problem ask you to apply first year ideas to a specific problem

Gereralised version of Problem 3 Minimise cost : wL + w 2 K s.t. f(x 1, x 2 ) =

3 EQNS – 3 unknowns x 1, x 2, So solve for ‘Quantity Constrained’ Conditional factor demands X 1 = x 1 (w 1, w 2, ) X 2 = x 2 (w 1, w 2, )