Cost Constraint/Isocost Line

Transcription

1 Cost Constraint/Isocost ine

2 COST CONSTRAINT C= w + rk (m = p 1 x 1 +p 2 x 2 ) w: wage rate (including fringe benefits, holidays, PRSI, etc) r: rental rate of capital Rearranging: K=C/r-(w/r)

3 COST CONSTRAINT K C/r (-) w/r C=w+rK Represents society ss willingness to trade the factors of production Slope of the isocost = C/w K/=K/ = (-) w/r

4 EQUIIBRIUM K Y e C w rk

5 EQUIIBRIUM U We can either Minimise cost subject to Y Y C and e equilibrium or Maximise output subject to C C Y and e equilibrium

6 agrangian method EQUIIBRIUM Minimise cost subject to output w rk Y f K, or Maximise output subject to costs f (, K ) C w r

7 agrange Method Set up the problem (same as with utility maximisation subject to budget constraint). Find the first order conditions. It is easier to maximise output subject to a cost constraint than to minimise costs subject to an output constraint. The answer will be the same (in essence) either way.

8 Example: Cobb-Douglas Equilibrium Derive a demand function for Capital and abour by maximising output subject to a cost constraint. et be agrange, be labour and K be capital. Set up the problem AK a 1a C w rk Multiply out the part in brackets AK a 1a C w rk

9 Cobb-Douglas: Equilibrium Find the First Order Conditions by differentiation with respect to K, and a1 1a K AaK r 0 1 st FOC A a a 1 a K w 0 2 nd FOC C w rk 0 3 rd FOC

10 Cobb-Douglas: Equilibrium Rearrange the 1 st FOC and the 2 nd FOC so that is on the left hand side of both equations AK AaK 1 1 a a 1a r a a 1 st FOC A 1 a K 2 nd FOC w

11 Cobb-Douglas: oug Equilibrium u We now have two equations both equal to - so we can get rid of a a 1 1 A 1 a K r w AaK a a (Aside: Notice that we can find Y nested within these equations.)

12 Cobb-Douglas: oug Equilibrium u AaK a a 1 1 A1 ak r w AaK a a a 1 1aa a aa 1 K r 1 A1 ak w ak 1 r We have multiplied by /=1 AK AaK 1 a 1 w 1 a a K A 1 a K r w a 1 a

13 Cobb-Douglas: oug Equilibrium u a 1 a rk w Note: K -1 =1/K rk aw ( 1 a ) Return to the 3 rd FOC and replace rk

14 Cobb-Douglas: oug Equilibrium u rk aw ( 1 a ) C w rk 0 w aw ( 1 a ) C w a 1 ( 1 a ) C

15 Cobb-Douglas: oug Equilibrium u w a Simplify 1 C again. ( 1 a ) C w ( 1 a ) ( 1 a ) C w Demand function for abour

16 Cobb-Douglas: oug Equilibrium u a 1 a rk w Rearrange so that w is on the left hand side Demand for capital w K ( 1 a a C r a ) rk Now go back to the 3 rd FOC and replace wl and follow the same procedure as before to solve for the demand for Capital

17 EQUIIBRIUM U Slope of isoquant = Slope of isocost or MRTS = (-) w/r MP /MP K = (-) w/r

18 Aside: General Equilibrium K MRTS A = w/r Firm A

19 Aside: General Equilibrium K MRTS B = w/r Firm B

20 Aside: General e Equilibrium u MRTS A = w/r MRTS B = w/r MRTS A = MRTS B We will return to this later when doing general equilibrium.

21 The Cost-Minimization Problem A firm is a cost-minimizer if it produces any given output level y 0 at smallest possible total cost. c(y) denotes the firm s smallest possible total cost for producing y units of output. c(y) is the firm s total cost function.

22 The Cost-Minimization Problem Consider a firm using two inputs to make one output. The production function is y = f(k,) Take the output level y 0 as given. Given the input prices r and w, the cost of an input bundle (K,) is rk + w.

23 The Cost-Minimization Problem For given r, w and y, the firm s cost- minimization problem is to solve subject to min K,0 ( rk w) f ( K, ) y Here we are minimising costs subject to a output t constraint. t Usually you will not be asked to work this out in detail, as the working out tis tedious.

24 The Cost-Minimization Problem The levels K(r,w,y) and (r,w,y) in the least-costly input bundle are referred to as the firm s conditional demands d for inputs 1 and 2. The (smallest possible) total cost for producing y output units is therefore c ( r, w, y ) rk ( r, w, y ) w ( r, w, y )

25 Conditional Input Demands Given r, w and y, how is the least costly input bundle located? And how is the total cost function computed?

26 A Cobb-Douglas Example of Cost Minimization A firm s Cobb-Douglas production function is 1 / 3 2 / 3 f ( K, ) K Input prices are r and w. y What are the firm s conditional input demand functions? K(r,w,y), (r,w,y)

27 A Cobb-Douglas Example of Cost Minimization At the input bundle (K,) which minimizes the cost of producing y output units: (a) 1 / 3 2 / 3 and y ( K ) ( ) (b) w r y / y / K (2 / 3)( K (1/ 3)( K ) ) 1/3 2/3 ( ) ( ) 1/3 2/3 2K

28 A Cobb-Douglas Example of Cost Minimization (a) 1 / 3 2 / 3 (b) w r 2K K y ( K ) ( )

29 A Cobb-Douglas Example of Cost Minimization 1 / 3 2 / 3 (a) (b) From (b), 2K K y ( K ) ( ) 2r K w Now substitute t into (a) to get w r y 2/3 2/3 1/3 2 r 2 r ( K ) K K w w So K w 2r 2 / 3 y is the firm s conditional demand d for Capital.

30 A Cobb-Douglas Example of Cost Minimization r w w 2r 2 / 3 Since 2 K and K y 2r w w 2r 2 / 3 is the firm s conditional demand for input 2. y 2 r w 1 / 3 y

31 A Cobb-Douglas Example of Cost Minimization So the cheapest input bundle yielding y output units is output units is y w r y w r K ),, ( ),,, ( r w y y 3 1 / 3 2 / 2 ),, ( ),,, ( y w r y r w 2, 2

32 Conditional Input Demand Curves y 1 Fixed r and w y y 2 y 3 y 3 y 2 Cond. demand for abour (y 3 ) (y 2 ) (y (y) 1 NB output expansion path y 1 y y 3 y 2 y 1 K(y 1 ) K(y 3 ) (y ) K(y 2 ) (y 1 ) (y 3 ) (y 2 ) K(y 1 ) K(y 3 ) K(y 2 ) Cond. demand for Capital K

33 Total Price Effect Recall the income and substitution effect of a price change. We can also apply this technique to find out what happens when the price of a factor of production changes, e.g. wage falls Substitution and output (or scale ) effects

34 Total Price Effect K a Q1

35 Total Price Effect K Total effect is a to c, more labour is used 1 to 3 c a Y 2 Y 1 1 3

36 Total Price Effect K a b c Y 1 1 to 2 is the substitution effect. More labour is being used. Y 2 1 2

37 Total Price Effect K 2 to 3 is the output t (scale) effect. a b c Y 1 Y 2 2 3

38 Total Price Effect What about perfect substitutes and perfect complements? (Homework)