© 2005 Pearson Education Canada Inc. 7.1 Chapter 7 Production and Cost: Many Variable Inputs

© 2005 Pearson Education Canada Inc. 7.2 Isoquants and Input Substitution  An isoquant is a curve composed of all bundles that produce some fixed quantity of output.  An example: Y=(1200Z 1 Z 2 ) 1/2  Setting y =120 and simplifying gives 12=Z 1 Z 2 (see Figure 7.1).

© 2005 Pearson Education Canada Inc. 7.3 Figure 7.1 Isoquants for courier services

© 2005 Pearson Education Canada Inc. 7.4 Marginal Rate of Technical Substitution (MRTS)  The MRTS measures the rate at which one input can be substituted for the other, with output remaining constant.  The MRTS is the absolute value of the slope of the isoquant.

© 2005 Pearson Education Canada Inc. 7.5 Perfect Substitutes and Perfect Compliments  Inputs are perfect substitutes when one output can always be substituted for the other on fixed terms and the MRTS is constant.  With perfect compliments, substitution is impossible and the MRTS cannot be defined for the bundle at the kink in the isoquant.

© 2005 Pearson Education Canada Inc. 7.6 Figure 7.2 Some illustrative isoquants

© 2005 Pearson Education Canada Inc. 7.7 Diminishing Rate of Technical Substitution  Most cases fall between perfect substitutes and perfect compliments. In these cases, one input can be substituted for the other but the MRTS is not constant.  In such cases, it becomes increasingly difficult to substitute one input for the other.  This means the MRTS diminishes moving fro left to right along the isoquant.

© 2005 Pearson Education Canada Inc. 7.8 Figure 7.3 The marginal rate of technical substitution, MRTS

© 2005 Pearson Education Canada Inc. 7.9 MRTS as a Ratio of Marginal Products  When the quantity of input 1 is decreased by ΔZ 1, the change in y is (approx) the marginal product of the input times the change in the quantity of input 1.  Therefore: Δy =MP 1 Δyz 1  Similarly: Δy =MP 2 Δyz 2

© 2005 Pearson Education Canada Inc MRTS as a Ratio of Marginal Products  When Z 1 is very small, MRTS can approximated by Δz 2 /Δz 1  Solving for Z 1 & Z 2 and substituting from above yields MRTS = (Δy/MP 2 )(Δy/MP 1 )  Reducing gives MRTS = MP 1 /MP 2  Therefore MRTS is equal to the marginal product of input 1 divided by the marginal product of input 2.

© 2005 Pearson Education Canada Inc Returns to Scale  Increasing returns to scale occurs when increasing all inputs by X% increases output by more than X%.  Constant returns to scale occurs when an increase in all inputs of X% increases output by X%.  Decreasing returns to scale occurs when an increasing all inputs by X% increases output by less than X%.

© 2005 Pearson Education Canada Inc Figure 7.4 Constant returns to scale

© 2005 Pearson Education Canada Inc The Cost Minimization Problem: A Perspective  The cost function shows the minimum cost of producing any level of output in the long-run.  The long-run cost minimizing problem is: minimize w 1 z 1 +w 2 +z 2 choosing z 1 and z 2 subject to constraint y=F(z 1, z 2 )

© 2005 Pearson Education Canada Inc Conditional Input Demand Functions  The solution to the cost minimization problem gives the values of the endogenous variables (z 1 * & z 2 * ) as a function of the exogenous variables (y, w 1 and w 2 ).  Since z 1 * & z 2 * are dependent upon the level of y chosen, the input demand functions are described as conditional demand functions.

© 2005 Pearson Education Canada Inc The Long-run Cost Function  Once we know the input demand functions, the long-run cost function is the sum of the input quantities and their respective prices. TC(y,w 1,w 2 ) = w 1 z 1 * +w 2 z 2 *

© 2005 Pearson Education Canada Inc Solving Cost Minimization Problems  The isocost line shows all bundles of inputs that cost the same. It can be expressed as: c=w 1 z 1 +w 2 z 2.  The absolute value of the slope of the isocost line is w 1 /w 2.  This slope says that w 1 /w 2 of input 2 must be given up to get an additional unit of input 1.  The slope is the opportunity cost of input 1 in terms of input 2.

© 2005 Pearson Education Canada Inc Figure 7.5 The cost-minimizing bundle

© 2005 Pearson Education Canada Inc Principles of Cost Minimization 1. The cost minimizing input bundle is on the isoquant: y Ξ F(z 1 * +z 2 * ). 2. The MRTS is equal to w 1 /w 2 at the cost minimizing bundle: MRTS(z 1 * z 2 * ) Ξ w 1 /w 2 The second principle can be generalized by stating the marginal product per dollar must be identical for all inputs.

© 2005 Pearson Education Canada Inc Comparative Statics for Input Prices  If all input prices change by the same factor of proportionality (a): 1. The cost of minimizing the input bundle for y units of output does not change. 2. The minimum cost pf producing y units of output changes by (a).

© 2005 Pearson Education Canada Inc Figure 7.7 Costs and input prices

© 2005 Pearson Education Canada Inc From Figure 7.7  If the cost-minimizing quantity of both inputs (i and j) is positive and there is diminishing MRTS, if p i increases and p j does not, the cost minimizing quantity of i increases and j decreases.  If the price of an input increases and the quantity demanded of that input is positive, the minimum cost of producing any level of output rises.

© 2005 Pearson Education Canada Inc Comparative Statics: Level of Output  The expansion path connects the cost minimizing bundles that are generated as output increases.  A normal input is one where the quantity demanded increases when output rises.  An inferior input is one where the quantity demanded decreases when output rises.

© 2005 Pearson Education Canada Inc Figure 7.8 The output expansion path

© 2005 Pearson Education Canada Inc Homothetic Production Functions  A homothetic production function is a type of function where the expansion path is a ray through the origin.  For these types of functions the MRTS is constant along any ray from the origin.

© 2005 Pearson Education Canada Inc Long-run Costs and Output  Long-run average costs (LAC) is equal to the total cost of output (TC) divided by the quantity of output (y): LAC(y)=TC(y)/y  As output rises, LAC is constant, decreasing, or increasing as there are constant, increasing, or decreasing returns to scale.

© 2005 Pearson Education Canada Inc Figure 7.9 Costs and returns to scale

© 2005 Pearson Education Canada Inc Figure 7.10 More on costs and returns to scale

© 2005 Pearson Education Canada Inc Long-run Marginal Cost  Long-run marginal cost (LMC) is the rate at which costs increase as output increases (the slope of TC).  When LMC lies below LAC, LAC is decreasing, when LMC exceeds LAC, LAC is rising, LMC intersects LAC at the LAC minimum.

© 2005 Pearson Education Canada Inc Figure 7.11 Deriving LAC and LMC from TC

© 2005 Pearson Education Canada Inc Figure 7.12 Comparing TC and STC

© 2005 Pearson Education Canada Inc Figure 7.13 Relationships between long-run and short-run cost functions

© 2005 Pearson Education Canada Inc Figure 7.14 A cost-based theory of market structure

© 2005 Pearson Education Canada Inc From Figure 7.14  U-shaped cost curves reflect initial increasing and subsequent decreasing returns to scale.  If LAC attains its minimum at a relatively large level of output, we expect to see a monopoly or oligopoly.  If LAC attains its minimum at a relatively small level of output, we expect to see a competitive market.