Cost Minimization.

Cost Minimization

Accountants vs. Economists
Economists think of costs differently from accountants. Economics Cost = Explicit Costs + Implicit Costs Accounting Cost = Explicit Costs Opportunity Cost is synonymous with Cost Measuring implicit opportunity costs is difficult generally, these are resources “owned” by the firm Includes the value of the entrepreneur’s time Includes accounting profit at next best alternative use of firm’s resources – or, accounting profit from selling all owned resources and investing the proceeds in the best possible alternative investment.

Fixed Cost and Sunk Cost
Fixed vs. Sunk cost “fixed” means its cost does not vary with output “sunk” means its cost cannot be avoided The security guard outside the parking lot is a fixed cost, but if they can be fired at a moment’s notice, the cost is not sunk A long term lease for renting a building is fixed and sunk Varian calls a fixed cost that is not sunk a “quasi-fixed cost” Sunk, but not fixed? No, because if it varies, it can be avoided.

Short Run – Long Run Key distinction in terms of optimizing behavior if everything can be controlled vs. not everything can be controlled. Short run, the quantities of some inputs used in production are fixed and some are variable. Short run, firms can shut down (q=0), but cannot exit the industry Long run, the quantity of all inputs used are variable. Long run, firms can enter or exit an industry.

To Distinguish Short Run from Long Run
We need two inputs, one always variable, and one that can only be adjusted periodically, but is then re-fixed. To accomplish that, we assume two inputs Homogeneous labor (L), measured in labor per time Homogeneous capital (K), measured in machine per time Entrepreneurial costs assumed to be zero (or included in fixed costs) Inputs are hired in perfectly competitive markets, so price of L, w, and price of K, v, are not a function of L and K

To be clear L is assumed always variable
K is always fixed when the firm is in production, but can periodically be adjusted to a new fixed amount (so in the long run it can be varied to a new fixed amount). That is, production always takes place in a short run situation.

(Typical Intermediate Micro Abbreviations in Parentheses)
Notation K, capital; v, rental rate of capital L, Labor; w, wage rate 5340 Nomenclature (Typical Intermediate Micro Abbreviations in Parentheses) Short Run Long Run Total Cost SC (TC) C (LRTC) Average Total Cost SAC (ATC) AC (LRAC) Variable Cost VC - Average Variable Cost AVC Fixed Cost FC Average Fixed Cost AFC Marginal Cost SMC (MC) MC (LRMC)

Cost Clearly, C can be stated as: C = vK + wL
But this is meaningless, we need cost as a function of output, not K and L. To get that, back to the production function

Graphically q q3 q2 K q1 q = f(K, L) q0
An entire range of input combinations can be used to produce every level of output. q3 q0 q1 q2 L

Short Run, K constant q = f(K=K3, L) q q = f(K=K2, L) q = f(K=K1, L)
Possible to produce the same level of output at different combinations of K and L. q1 Three different slices through the production function at different levels of K. L

Short Run, K constant Let’s look at just one slice. q q2
At some fixed level of K, for every q, we know how much L it will take. L2 L

Production to Cost Flip the axis q L L q

Production to Cost L L = L(q,K) q

Production to Cost SC = w·L(q)+vK Multiply L by w to get labor cost
$/time VC = w· L(q) Add in FC, SC=SVC+FC FC q

Production to Cost SC = w· L(q)+rK And we can figure a few things
$/time VC = w· L(q) FC q

Production to Cost SAC $ SMC AVC AFC q

What does that mean for cost curves?
Short Run, K constant q = f(K=K3, L) q q = f(K=K2, L) Inflection point, q3 q = f(K=K1, L) Inflection point, q2 With more K, you can produce more with a given L and the inflection point moves towards more L and q. Inflection point, q1 L What does that mean for cost curves?

Production to Cost $ SC = w· L2(q)+rK2 SC = w· L3(q)+rK3
Inflection Points SC = w· L1(q)+rK1 FC q Fixed cost is higher with more K, but the inflection point is further to the right, with a slower build-up of crowding.

C C $ SC = w· L2(q)+rK2 SC = w· L3(q)+rK3 SC = w· L1(q)+rK1 q
C is the lowest point on any of the SC curves for any q

Min value at higher q with more K.
SAC $ Min value at higher q with more K. SAC = C2/q SAC = C1/q SAC = C3/q LAC q AC is the lowest point on any of the SR ATC curves for any q

MC C $ MC is the slope of the C curve at any q q

Short Run – Long Run Firms ALWAYS produce in a short run situation with at least one fixed and one variable input. Being in the long run simply means the firm has adjusted to an optimal level of capital to minimize the cost of producing any chosen (profit maximizing) level of output.

Different Shapes for the long run C Curve
$ There are four different potential shapes q

C -- CRS C $ CRS, as K and L are scaled upwards, C rises at the same rate. That is, MC is constant. q

Min value at higher q with more K.
C -- CRS $ Min value at higher q with more K. ATC = TC2/q ATC = TC1/q ATC = TC3/q AC=MC q CRS, as K and L are scaled upwards, LAC rises at the same rate. That is, LAC is constant.

C -- IRS C $ q IRS, as K and L are scaled upwards, C rises more slowly
That is, MC is decreasing. q

C -- IRS $ AC MC q CRS, as K and L are scaled upwards, AC rises at a slower rate. AC is decreasing.

C -- DRS $ C q DRS, as K and L are scaled upwards, TC rises faster
That is, MC is increasing. C q

LTC -- DRS Note, it is not the low point of each ATC curve, but the simply the lowest point on any ATC. $ MC AC q CRS, as K and L are scaled upwards, AC rises at a faster rate. AC is increasing.

C – IRS, CRS, DRS C $ IRS, then CRS, then DRS q

C – IRS, CRS, DRS So we have a U-shaped AC curve for a completely different reason than the SAC curve is U-shaped. $ MC AC q

Firm Decisions So which short run curve should we be on? Two choices
That is, how much K (and then L) do we want to hire (with K fixed in the SR)? Two choices Profit Maximization: Firms maximize profit by choosing q*, K and L to minimize cost all at once. Cost minimization: Firms choose a q*, then choose K and L to minimize the cost of producing q*.

Profit Maximization Economic profits () are equal to
 = total revenue - total cost Total costs for the firm are given by C = wL + vK Total revenue for the firm is given by total revenue = R = p·q = p·f(K,L)  = p·f(K,L) - wL - vK

Profit Maximization Solving to maximize profit means jointly choosing q = q* along with K* and L*. Yields profit maximizing factor (input) demand functions which provide L* and K* that minimizes the cost of producing q*. When r, w, and p change, L*, K*, and q* all change. L*=L(w, v, p); K*=K(w, v, p); q*=q(w, v, p) These functions allow for a change in q* (the isoquant) when prices change.

Cost Minimization That firms minimize cost is a weaker hypothesis of firm behavior than profit maximization. Yields quantity constant (quantity contingent) factor (input) demand functions which provide L* and K* that minimize the cost of producing q0. When r and w change, L* and K* do change, but q does not, stay on one isoquant. Why bother? It is how we get the cost functions and curves (i.e. what we use in principles and intermediate micro)

New Direction in Graphing
In all the graphs above, we have illustrated the long run as a series of short run curves and traced out the envelope. Good for intuition, but not terribly tied to the math of the optimization Let’s switch to the isoquant graph.

Intuitively Isoquant, all combinations of factors that yield the same output. Slope is -dK/dL K Isocost (total cost), all combinations of factors that yield the same total cost of production. Slope is -w/v L

Cost-Minimizing Input Choices
When K and L change a small amount Along an Isoquant,

TRS is the change in K needed to replace one L while maintaining output. Minimum cost occurs where the TRS is equal to w/v the rate at which K can be traded for L in the production process = the rate at which they can be traded in the marketplace

Intuitively w=20, v=10 w/v=2 K TRS=.25 L
Firing one L has MB of $20 and MC of $2.5 Lowers cost by $17.50

Intuitively w=20, v=10 w/v=2 K TRS=1 L
Firing one L has MB of $20 and MC of $10 Lowers cost by $10

Intuitively w=20, v=10 w/v=2 K TRS=1.5 L
Firing one L has MB of $20 and MC of $15 Lowers cost by $5

Intuitively w=20, v=10 w/v=2 K TRS=2 L
Firing one L has MB of $20 and MC of $20 Lowers cost by $0 Further reductions in L require an increase in cost as TRS >w/r.

Intuitively w=20, v=10 w/v=2 K TRS=2.5 L
Firing one L has MB of $20 and MC of $25 Raises cost by $5

Plan Figure out the quantity of K and L that minimize total cost, holding q constant. Use the resulting factor demand curves to derive the cost functions.

We seek to minimize total costs given q =q0 q = f(K,L) = q0 Setting up the Lagrangian: FOCs are

Dividing the first two conditions we get The cost-minimizing firm should equate the RTS for the two inputs to the ratio of their prices But also Which tells us that for the last unit of all inputs hired should provide the same bang-for-the-buck.

The inverse of this equation is also of interest The value of the Lagrange multiplier is the extra costs that would be incurred by increasing the output constraint slightly by hiring enough L or K to increase output by 1. That is λ = MC of increasing production by one unit.

SOC to ensure costs are RISING, along the isoquant, away from the tangency: Bordered Hessian The part in brackets is the same condition required for strict quasi concavity of the production function

Intuitively Cost minimized for q0 when L=L* and K=K * w=20, r=10
SOC satisfied as moving along the isoquant means increasing TC Total Cost RTS=2 K* L* L Cost minimized for q0 when L=L* and K=K *

SOC fail, Intuitively Cost maximized for q0 when L=L* and K=K *
w=20, r=10 w/r = 2 K SOC not satisfied as moving along the isoquant means decreasing TC Total Cost RTS=2 K* L* L Cost maximized for q0 when L=L* and K=K *

Conditional Factor Demand (aka Constant Output)
Solve the FOC to derive K * =Kq (w, v, q0) L * =Lq (w, v, q0) Earlier, we considered an individual’s expenditure-minimization problem to develop the compensated demand for a good In the present case, cost minimization leads to a demand for capital and labor to produce a constant quantity of output.

The Firm’s Expansion Path
The expansion path is the locus of cost-minimizing tangencies K The curve shows how inputs increase as output increases K=KE(w, v, L) K* K* K* L L* L* L*

The cost-minimizing combinations of K and L for every level of output (according to the quantity constant factor demand functions) If input prices remain constant for all amounts of K and L, locus of cost-minimizing choices is the expansion path Solve the minimization condition for K to get the expansion path

The expansion path does not have to be a straight line the use of some inputs may increase faster than others as output expands depends on the shape of the isoquants The expansion path does not have to be upward sloping if the use of an input falls as output expands, that input is an inferior input

Cost Functions Long Run Cost Short Run Cost
The factor demand curves we have just derived are not appropriate as they each assume L and K are variable.

Short Run Cost Functions
With only two inputs, once K is fixed, the production function dictates how much L is needed for each q. I am just using the “s” superscript here to denote Short Run

Short Run & Long Run Pick a level of output, q1 and hold w and v constant C q1 q

Short Long Relationship
$ MC SMC ATC AC AVC q

SMC vs. LMC $ When q rises in SR, cost rises by more than in the LR
When q falls in SR, cost falls by less than in the LR; Means from q1 to q2 the SMC is lower as C starts higher at q1. q2 q3 q1 q

Short Long Relationship (Easier to see if assume CRS)
SMC actually lower from q1 to q2 in the short run as total cost is higher to start $ SMC (K=K1) ATC (K=K1) MC=AC AVC (K=K1) q1 q2 q3 q

Fixed Proportions Factor Demand and Cost Functions
Suppose we have a fixed proportions technology such that q = f(K,L) = min(aK,bL) To minimize cost, production will occur at the vertex of the L-shaped isoquants where q = aK = bL (any extra K or L only drives up cost) Expansion path: Factor Demand Curves Kq=q/a and Lq=q/b Cost function: C(w,v,q) = vK + wL = v(q/a) + w(q/b)

Cobb-Douglas Cost Minimization
Suppose that the production function is Cobb-Douglas: q = K L The Lagrangian expression for cost minimization of producing q0 is ℒ = vK + wL + (q0 - K  L ) The FOCs for a minimum are ℒK = v - K -1 L = 0 ℒL = w - K  L -1 = 0 ℒ = q0 - K L = 0

Dividing the first equation by the second gives us This production function is homothetic the TRS depends only on the ratio of the two inputs

Expansion path equation: the expansion path is a straight line The K/L ratio is a function of w and v.

Cobb-Douglas Input Demand
Using the remaining FOC (production function), solve for the input demand equations K* = Kq (v,w,q) L* = Lq (v,w,q)

Cobb-Douglas Cost Now we can derive total costs as Where MC AC

CES Cost Minimization Suppose that the production function is CES:
q = (K +L)/ The Lagrangian expression for cost minimization of producing q0 is ℒ = vK + wL + [q0 - (K  +L )/] The FOCs for a minimum are ℒ L = w - (/)(K +L)(-)/()L-1 = 0 ℒ K = v - (/)(K +L)(-)/()K-1 = 0 ℒ  = q0 - (K + L)/ = 0

CES Cost Minimization Dividing the first FOC equation by the second gives us This production function is also homothetic

CES Expansion Path Expansion path Capital Labor ratio

CES Input Demand Using the remaining FOC (production function), solve for the input demand equations K* = Kq (v,w,q) L* = Lq (v,w,q)

CES Cost Functions To derive the total cost, we would use the inputs demand functions and get

Law of Demand? With Cobb-Douglass, CES etc. we can take the partial derivative w.r.t. price of the inputs and see how quantity demanded responds. But if we don’t know the functional form, how do we know how demand for L and K will respond to changes in w and v? Comparative Statics

Comparative Statics Setting up the Lagrangian:
ℒ = wL + vK + [q0 - f(K,L)] FOCs are ℒL = w - ·fL = 0 ℒK = v -  ·fK = 0 ℒλ = q0 - f(K,L) = 0 Solve for L* = Lq(w,v,q) K* = Kq (w,v,q) λ* = λq (w,v,q)

Comparative Statics Plug solutions into FOC
w - λ(w,v,q)·fL(Lq(w,v,q), Kq (w,v,q)) ≡ 0 v - λ(w,v,q)·fK(Lq (w,v,q), Kq (w,v,q)) ≡ 0 q0 - f(Kq (w,v,q), Lq (w,v,q)) ≡ 0 These are identities because the solutions (FOC) are substituted into the equations from which they were solved. Whatever prices and output may be, the firm will instantly adjust K and L to minimize cost of that level of output.

Comparative Statics Differentiate these w.r.t. w To get:
Notationally, replace, Lq(w,v,q) with L*, etc.

Matrix Notation and Cramer’s Rule

With a higher wage, the amount of L used will definitely fall. It is possible that K becomes more productive with less L being used (if fKL< 0). If this effect is large enough, the higher productivity of capital can more than compensate for the higher cost of labor and MC can fall. Unlikely in the real world.

Comparative Statics Differentiate these w.r.t. q To get:

If labor is an inferior input, we COULD minimize cost with more K and less L. I.e., a “backward bending” expansion path.

If there is IRS, although total cost is still rising, MC may well be falling along the expansion path.

Shephard’s Lemma (Again)
Remember how we used Shephard’s Lemma to derive the compensated demand curves from the expenditure function? We can do it again from the cost function

Marginal Cost Function
And also: That is, λ*= λ(v,w,q) tells us the MC of increasing production along the expansion path

Fixed Proportions Shephard’s Lemma Results
Suppose we have a fixed proportions technology The cost function is For this cost function, output constant demand functions are quite simple:

Cobb-Douglas Shephard’s Lemma Result
Suppose we have a Cobb-Douglas technology The cost function is where then the output constant factor demand functions are:

CES Shephard’s Lemma Result
Suppose we have a CES technology The cost function is The quantity constant demand functions for capital and labor are

Properties of Cost Functions
Homogeneity cost functions are all homogeneous of degree one in the input prices a doubling of all input prices will double cost. As derivatives of HD1 functions are HD0, the contingent demand functions must be HD0, that is, a doubling of w and v will not affect the cost minimizing input mix. inflation will shift the cost curves up and will not change Kc, Lc

Properties of AC and MC functions
Both AC and MC are HD1 meaning a doubling of input prices means a doubling of MC and AC True enough, C=C(v, w, q) is HD1 and However, C is HD1 in input prices, and MC is the derivative w.r.t. q. So the derivative of an HD1 function being HD0 does not apply here.

Properties of Cost Functions
The total cost function is non-decreasing in q, v, and w. Fixed factor production has a linear cost function w.r.t. w and v To the extent that one factor can be substituted for another, the function will be concave to input prices.

Concavity of Cost Function
At w1, the firm’s costs are C(v,w1,q1) If the firm continues to buy the same input mix as w changes, its cost function would be Cpseudo C=wL1+v1K1 C C*=C(v,w,q1) C(v,w1,q1) Since the firm’s input mix will likely change, actual costs will be less than C such as C*=C(v,w,q1) w w1

Measuring Input Substitution
A change in the price of an input will cause the firm to alter its input mix The change in K/L in response to a change in w/v, while holding q constant is

Input Substitution In the Production lecture:
But more usefully, because RTS = w/v at cost minimum

Input Substitution This alternative definition of the elasticity of substitution in the two-input case, s must be nonnegative large values of s indicate that firms change their input mix significantly if input prices change s = 0 for fixed factor production

Size of Shifts in Costs Curves
The increase in cost caused by a change in the price of an input will be largely influenced by the relative significance of the input in the production process the ability of firms to substitute another input for the one that has risen in price – easy substitution means little change in costs.

Appendix, Envelope Derivation
The marginal cost function (MC) is found by computing the change in total costs for a change in output produced

Marginal Cost Function and 
Back to the cost function

Marginal Cost Function
So as long as cost are being minimized, this is true

Marginal Cost Function and 
And combine:

Cost Minimization.

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