Cost Constraint/Isocost Line

COST CONSTRAINT C= wL + 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)L

COST CONSTRAINT L K Represents society’s willingness to trade the factors of production C=wL+rK C/w C/r Slope of the isocost =  K/  L=  K/  L = (-) w/r (-) w/r

EQUILIBRIUM L K e

We can either Minimise cost subject to or Maximise output subject to equilibrium

EQUILIBRIUM Minimise cost subject to output or Maximise output subject to costs Lagrangian method

Lagrange Method u Set up the problem (same as with utility maximisation subject to budget constraint). u Find the first order conditions. u 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.

Example: Cobb-Douglas Equilibrium Set up the problem Multiply out the part in brackets Derive a demand function for Capital and Labour by maximising output subject to a cost constraint. Let L* be Lagrange, L be labour and K be capital.

Cobb-Douglas: Equilibrium Find the First Order Conditions by differentiation with respect to K, L and 1 st FOC 2 nd FOC 3 rd FOC

Cobb-Douglas: Equilibrium Rearrange the 1 st FOC and the 2 nd FOC so that is on the left hand side of both equations 1 st FOC 2 nd FOC

Cobb-Douglas: Equilibrium We now have two equations both equal to - so we can get rid of (Aside: Notice that we can find Y nested within these equations.)

Cobb-Douglas: Equilibrium We have multiplied by L/L=1

Cobb-Douglas: Equilibrium Note: K -1 =1/K Return to the 3 rd FOC and replace rk

Cobb-Douglas: Equilibrium

Simplify again. Demand function for Labour

Cobb-Douglas: Equilibrium Rearrange so that wL is on the left hand side 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 Demand for capital

EQUILIBRIUM Slope of isoquant = Slope of isocost MRTS = (-) w/r or MP L /MP K = (-) w/r

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

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

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

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

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

The Cost-Minimization Problem u For given r, w and y, the firm’s cost- minimization problem is to solve subject to Here we are minimising costs subject to a output constraint. Usually you will not be asked to work this out in detail, as the working out is tedious.

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

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

A Cobb-Douglas Example of Cost Minimization u A firm’s Cobb-Douglas production function is u Input prices are r and w. u What are the firm’s conditional input demand functions? K*(r,w,y), L*(r,w,y)

A Cobb-Douglas Example of Cost Minimization At the input bundle (K*,L*) which minimizes the cost of producing y output units: (a) (b) and

A Cobb-Douglas Example of Cost Minimization (a)(b)

A Cobb-Douglas Example of Cost Minimization (a)(b) From (b), Now substitute into (a) to get Sois the firm’s conditional demand for Capital.

A Cobb-Douglas Example of Cost Minimization is the firm’s conditional demand for input 2. Sinceand

A Cobb-Douglas Example of Cost Minimization So the cheapest input bundle yielding y output units is

Fixed r and w Conditional Input Demand Curves output expansion path Cond. demand for Labour Cond. demand for Capital L*(y 1 ) L*(y 2 ) L*(y 3 ) K*(y 2 ) K*(y 1 ) K*(y 3 ) y2y2 y3y3 y3y3 y1y1 y2y2 y3y3 y1y1 y2y2 y1y1 L*(y 1 ) L*(y 2 ) L*(y 3 ) L* K* K*(y 1 ) K*(y 2 ) K*(y 3 ) NB

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

Total Price Effect K L a Q1

Total Price Effect K L L1L1 c a L3L3 Total effect is a to c, more labour is used L 1 to L 3 Y2Y2 Y1Y1

Total Price Effect K L L1L1 b c a L2L2 L 1 to L 2 is the substitution effect. More labour is being used. Y2Y2 Y1Y1

Total Price Effect K L b c a L2L2 L3L3 L 2 to L 3 is the output (scale) effect. Y2Y2 Y1Y1

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