Cost Functions Dr. Andrew McGeeSimon Fraser University
Where we stand…
What we know…• Production function: (K,L) → Q• Suppose we know – w = wage rate of L – R = rental rate of K• Then we can determine how much it costs for a firm to produce an amount Q=f(K,L) assuming that the firm endeavors to minimize its costs• That is, we can figure out TC(Q), so all that remains to solve the firm’s profit maximization problem is to determine, P(Q), which depends on market structure
Types of costs• Accounting costs – Things for which you pay money – E.g., bills, labor costs, rent – Other things like depreciation (i.e., lost value of assets that lose value over time)• Economic costs – Opportunity costs of doing business • Values of highest foregone options of inputs – Workers (how much they could earn elsewhere) – Machines (how much they could earn if put to some other use) – Entrepreneur (how much he/she could earn as a salaried employee at some other firm)
Overlap between Accounting & Economic Costs• There is significant overlap between accounting & economic costs: – E.g., labor costs=wages=opportunity cost of labor in a competitive labor market• Economic costs > accounting costs – Entrepreneur’s salary < opportunity costs because entrepreneur receives a large share of firm’s accounting profits as owner of firm – Investors’ opportunity costs are not reflected in firm’s costs (compensated through accounting profits by way of dividends, capital gains)
Economic & Accounting Profit
Economic Costs (forget about accounting costs for now) Price-takers in input marketsw r L K
Deriving Total Cost Function
Cost Minimization Problem
Depicting Cost Minimization KTC3=rK3 All input combinations on the iso-expenditureTC2=rK2 (iso-cost) lines are input combinations that result in the same total cost of production w/r=MRTSTC1=rK1 Total cost is declining in the southwesterly directionTC0=rK0 A Isoquant q0 -w/r This firm can produce q0 for less than L TC3, but it cannot produce q0 for TC1 or TC2. Notice that for this and every output level and given input prices (w & r) Isocost or isoexpenditure lines there is a unique minimum cost of production. Our goal is to derive TC(w,r,q)
Duality: Output maximization• For every constrained maximization problem there exists a constrained minimization problem yielding the same solution for appropriate parameter values (“duality”)• Dual problem of cost minimization: maximize output subject to expenditure constraint
Output Maximization K w/r=MRTS Solving the Lagrangian will yield theE=rK same optimal condition: w/r=MRTS A q3 q2 q1 q0 L E=wL
Demand for inputs?K A q3 q2 q1 q0 L Demand for L at different wage rates? NO!
Demand for inputs?• Can we derive firm’s demand for inputs (K,L) using solutions to cost minimization problem (much like we derived demand for goods using the solution to the consumer choice problem)?• No!• Cost minimization holds output constant, but firm’s demand for K & L obviously depends on how much output it chooses to produce.
Deriving Total Cost Function K q ETC3=rK3 Cost expansion path: from this expansion path we q0 TC0TC2=rK2 can obtain the TC q1 TC1 associated with each TC(w,r,q)TC1=rK1 q2 output level q TC2TC0=rK0 q3 TC3 q3 q2 q0 q1 L
Example: Cobb-Douglass production function
Example: Cobb-Douglass production function• →wL=rK• Suppose w=r=$4. Then L=K. If L=K & q=40, we have• This implies that TC=$4*4+$4*4=$32 (lowest possible cost of producing 40 units)• &• (extra output for last $1 spent on inputs)
Example: Cobb-Douglass production function
Example: Deriving the Cost Function for a Cobb-Douglass production function
Other important cost functions
Graphing the Cost Functions $ TC(q) Observations: 1. Ceteris paribus invoked (w & r held constant) 2. MC initially declining through q0 (may not always be true) -Results from IRS (benefits to specialization) -Shape of the cost functions q depends on the production q0 q1 function$/unit MC(q)=dTC/dq 3. MC intersects AC at minimum value of AC -MC<AC for q<q1 -MC>AC for q>q1 4. For q>q0, MC is increasing. This is AC(q)=TC(q)/q the region of DRS in the production function. q q0 q1
Example: Deriving MC and AC functions
CRS production functions & cost functions CRS production functions have$ linear TC functions because MC TC(q) is constant: $ TC(q) IRS CRS DRS q q
CRS production functions & cost functions • What is AC when production function exhibits CRS everywhere? In previous example and in all such examples, AC(q)=TC(q)/q=c, a constant. When f(K,L) exhibits CRS everywhere, AC(q)=MC(q)=constant.$/unit When you see constant marginal costs, you immediately know something about the AC=MC productions function. Likewise, when you see CRS, you immediately know something about the MC> q
Effects of input price increase• When the price of one of the two inputs goes up, what happens if producers wish to maintain the same level of output? – Producers will substitute the now relatively cheaper input for some of the input whose price increased – Total costs definitely do not decrease following an input price increase. If they do, the producer could not have been minimizing costs to start with
Partial elasticity of substitution between inputs
Partial elasticity of substitution between inputs• Partial elasticity of substitution (s) measures how firms change input mix in response to price changes• High s → firms change input mix (K/L ratio) substantially in response to small changes in relative input prices• Low s → firms change input mix (K/L ratio) little in response to changes in relative input prices
Long vs. Short Run Costs• Short run = period of time during which at least one input cannot be changed• Long run = any time horizon during which all input can be changed• Variable input = input that can be changed in the short run• Fixed input = input that cannot be changed in the short run – Some inputs take time to be delivered or made; this time-to-delivery defines the short run• All inputs are variable in the long run
Short run cost functions
Short run cost functions
Enveloping the Short-run cost curves• Envelope of the STC: The set of lowest costs of production on any STC (i.e., for any fixed level of K) for every possible output level• The envelope of the STC curves defines the TC curve• Similarly, the envelope of the SATC curves defines the AC curve
Enveloping the Short-run cost curvesTC TC(w,r,q) Imagine that each STC curve corresponds to the firm’s cost schedule with a different number of plants. How many plants should the firm build to produce q1, q2, q3, and q4 units of output? q q1 q2 q3 q4 This TC curve is linear. What does that tell us?
Enveloping the Short-run cost curves TC TC The envelope of the STC curves need not be linear. q q1 q2 q3 q4
Enveloping the Short-run cost curvesCost perunit AC=MC q This is a CRS production function How are we deriving the MC curve from the SMC curves?
Enveloping the Short-run cost curves Not CRSCost perunit MC AC At MES, AC reaches its minimum values and AC=MC=SATC=SMC q MES=minimum efficient scale For each output level, find the SMC curve corresponding to the SATC just tangent to the AC curve. From this SMC curve we derive the MC (in the LR) of producing that output level.
Example: Short run Cobb Douglass Costs Fixed input STC =1 =4 =9
Example: Short run Cobb Douglass Costs• If in the short run you happen to be using the capital level that minimizes the costs of producing output level q in the long run (K1*), then you must also be minimizing costs in the short run, meaning that the derivative of the STC at this capital level is zero. Use this fact to solve for K1*:• Plug this back into the STC. You are now effectively using the K level that minimizes LR costs while using the L-level that minimizes costs for any given K-level. Thus you must be minimizing costs in the LR, so this is the TC function:• You can check that this is indeed the same as the TC we derived earlier for this Cobb Douglass production function