© 2009 Pearson Education Canada 6/1 Chapter 6 Production and Cost: One Variable Input

© 2009 Pearson Education Canada 6/2 Production Function  The production function identifies the maximum quantity of good y that can be produced from any input bundle (z 1, z 2 ).  A production function is stated as: y=F(z 1, z 2 ).

© 2009 Pearson Education Canada 6/3 Production Functions  In a fixed proportions production function, the ratio in which the inputs are used never varies.  In a variable proportion production function, the ratio of inputs can vary.

© 2009 Pearson Education Canada 6/4 Figure 6.1 Finding a production function

© 2009 Pearson Education Canada 6/5 From Figure 6.1  The production function is: F(z 1 z 2 )=(1200z 1 z 2 ) 1/2 F(z 1 z 2 )=(1200z 1 z 2 ) 1/2  This is a Cobb-Douglas production function. The general form is given below where A, u and v are positive constants.

© 2009 Pearson Education Canada 6/6 Costs  Opportunity cost is the value of the highest forsaken alternative.  Sunk costs are costs that, once incurred, cannot be recovered.  Avoidable costs are costs that need not be incurred (can be avoided).  Fixed costs do not vary with output.  Variable costs change with output.

© 2009 Pearson Education Canada 6/7 Long-Run Cost Minimization  The goal is to choose quantities of inputs z 1 and z 2 that minimize total costs, subject to being able to produce y units of output.  That is: 1. Minimize w 1 z 1 +w 2 z 2 (w 1,w 2 are input prices). 2. Choosing z 1 and z 2 subject to the constraint y=F(z 1, z 2 ).

© 2009 Pearson Education Canada 6/8 Production: One Variable Input  Total production function TP (z 1 ) (z 2 fixed at 105) defined as: TP (z 1 )=F(z 1, 105)  Marginal product MP(z 1 ) is the rate of output change when the variable input changes (given fixed amounts of all other inputs).  MP (z 1 )=slope of TP (z 1 )

© 2009 Pearson Education Canada 6/9 Figure 6.2 A total product function

© 2009 Pearson Education Canada 6/10 Figure 6.3 From total product to marginal product

© 2009 Pearson Education Canada 6/11 The Free-disposal Assumption  Because a production function gives the maximum output from any input combination, we assume that increased amounts of inputs will not be used if they negatively impact output.  This is sometimes called the free- disposal assumption and combined with our definition of the production function, implies that marginal product cannot be negative.

© 2009 Pearson Education Canada 6/12 The Free-disposal Assumption  Given the free-disposal assumption, the marginal product of any input is always greater than or equal to zero.  Furthermore, for any input bundle, the marginal product of at least one input is positive.

© 2009 Pearson Education Canada 6/13 Diminishing Marginal Productivity  Reflects the assumption that at some point the rate of increase in total output (marginal product) will begin to decline.  Suppose the quantities of all inputs except one - say, input1- are fixed. There is a quantity of input 1- say, z” 1 - such that whenever z 1 exceeds z” 1, the marginal product of input 1 decreases as z 1 increases.

© 2009 Pearson Education Canada 6/14 Figure 6.4 From total product to marginal product: Another illustration

© 2009 Pearson Education Canada 6/15 Average Product  Average product (AP) of the variable input equals total output divided by the quantity of the variable input. AP(z 1 )=TP(z 1 )/z 1

© 2009 Pearson Education Canada 6/16 Figure 6.5 From total product to average product

© 2009 Pearson Education Canada 6/17 Figure 6.6 Comparing the average and marginal product functions

© 2009 Pearson Education Canada 6/18 Marginal and Average Product 1. When MP exceeds AP, AP is increasing. 2. When MP is less than AP, AP declines. 3. When MP=AP, AP is constant.

© 2009 Pearson Education Canada 6/19 Costs of Production: One Variable Input  The cost-minimization problem is: Minimize w 1 z 1 by choice of z 1. Subject to constraint y=TP(z 1 ).  The variable cost function, VC(y) is: VC(y)=the minimum variable cost of producing y units of output.

© 2009 Pearson Education Canada 6/20 Figure 6.7 Deriving the variable cost function

© 2009 Pearson Education Canada 6/21 More Costs  Average variable cost is variable cost per unit of output. AV(y)=VC(y)/y  Short-run marginal cost is the rate at which costs increase in the short- run. SMC(y)=slope of VC(y)

© 2009 Pearson Education Canada 6/22 Figure 6.8 Deriving average variable cost and short-run marginal cost

© 2009 Pearson Education Canada 6/23 Short-run Marginal Costs and Average Variable Costs 1. When SMC is below AVC, AVC decreases as y increases. 2. When SMC is equal to AVC, AVC is constant (its slope is zero). 3. When SMC is above AVC, AVC increases as y increases.

© 2009 Pearson Education Canada 6/24 Average Product and Average Cost AVC (y’)=w 1 /AP(z 1 ’)  The average variable cost function is the inverted image of the average product function.

© 2009 Pearson Education Canada 6/25 Marginal Product and Marginal Cost SMC (y’)=(w 1 Δz 1 )/(MP(z’))  The short-run marginal cost function is the inverted image of the marginal product function.

© 2009 Pearson Education Canada 6/26 Figure 6.9 Comparing cost and product functions

© 2009 Pearson Education Canada 6/27 Figure 6.10 Seven cost functions

© 2009 Pearson Education Canada 6/28 Figure 6.11 The costs of commuting

© 2009 Pearson Education Canada 6/29 Figure 6.12 Total commuting costs

© 2009 Pearson Education Canada 6/30 Figure 6.13 The allocation of commuters to routes

© 2009 Pearson Education Canada 6/31 Application: The Allocation of Output Among Different Plants  Let us reinterpret the lessons from traffic congestion to solve a problem related to multi-plant firms.  The firm’s problem is to allocate output to its two plants so as to minimize its total variable cost.

© 2009 Pearson Education Canada 6/32 Application: The Allocation of Output Among Different Plants  Think of Figure 6.13 as the variable and marginal cost functions associated with two plants (TCC 1, TCC 2 and MC 1, MC 2 ) respectively.  The curve TCC in Figure 6.13a tells us that the firm can minimize the variable costs of producing 5000 units by allocating 2000 units to the first plant and 3000 units to the second plant.

© 2009 Pearson Education Canada 6/33 Application: The Allocation of Output Among Different Plants  Using the logic from Figure 6.13b: –Total costs can be reduced by allocating output from the high-marginal cost plant to the low-cost marginal plant. –To minimize the total variable cost of producing a given output in two or more plants, a firm allocates output to the plants so that short-run marginal cost is the same in all plants.