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Cost Constraint/Isocost Line

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.

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Cost Constraint/Isocost Line

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  1. Cost Constraint/Isocost Line

  2. COST CONSTRAINT C= wL + rK (m = p1x1 +p2x2) w: wage rate (including fringe benefits, holidays, PRSI, etc) r: rental rate of capital Rearranging: K=C/r-(w/r)L

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

  4. EQUILIBRIUM K e L

  5. EQUILIBRIUM We can either Minimise cost subject to equilibrium or Maximise output subject to equilibrium

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

  7. Lagrange 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 Labour by maximising output subject to a cost constraint. Let L* be Lagrange, L be labour and K be capital. Set up the problem Multiply out the part in brackets

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

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

  11. 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.)

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

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

  14. Cobb-Douglas: Equilibrium

  15. Cobb-Douglas: Equilibrium Simplify again. Demand function for Labour

  16. Cobb-Douglas: Equilibrium Rearrange so that wL is on the left hand side Now go back to the 3rd FOC and replace wl and follow the same procedure as before to solve for the demand for Capital Demand for capital

  17. EQUILIBRIUM Slope of isoquant = Slope of isocost MRTS = (-) w/r or MPL/MPK = (-) w/r

  18. Aside: General Equilibrium MRTSA = – w/r K Firm A L

  19. Aside: General Equilibrium MRTSB = – w/r K Firm B L

  20. Aside: General Equilibrium MRTSA = – w/r MRTSB = – w/r MRTSA = MRTSB 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 ³ 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 isy = f(K,L) • Take the output level y ³ 0 as given. • Given the input prices r and w, the cost of an input bundle (K,L) is rK + wL.

  23. The Cost-Minimization Problem • 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.

  24. The Cost-Minimization Problem • 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. • The (smallest possible) total cost for producing y output units is therefore

  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 • Input prices are r and w. • What are the firm’s conditional input demand functions? K*(r,w,y), L*(r,w,y)

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

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

  29. A Cobb-Douglas Example of Cost Minimization (a) (b) From (b), Now substitute into (a) to get So is the firm’s conditionaldemand for Capital.

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

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

  32. Conditional Input Demand Curves Cond. demandfor Labour Fixed r and w y1 y2 y3 y3 y2 outputexpansionpath y1 L* L*(y3) L*(y1) L*(y2) L*(y3) Cond.demandfor Capital y3 L*(y2) y2 L*(y1) y1 K*(y3) K* K*(y1) K*(y1) K*(y3) NB K*(y2) K*(y2)

  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 L

  35. Total Price Effect K Total effect is a to c, more labour is used L1 to L3 c Y2 a Y1 L1 L3 L

  36. Total Price Effect K L1 to L2 is the substitution effect. More labour is being used. c a Y2 b Y1 L1 L2 L

  37. Total Price Effect K L2 to L3 is the output (scale) effect. c a Y2 b Y1 L2 L3 L

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

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