# Chapter 7 PRODUCTION FUNCTIONS.

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Chapter 7 PRODUCTION FUNCTIONS

Objectives So far, we have studied consumer’s behaviour
In this lecture, we will study the supply side of the market (firms…) It will be easier because many concepts used when studying the firm have a clear counterpart in the previous chapters dedicated to the consumer

To have in mind… Utility function Production function Goods Inputs Indifference curve Isoquant Marginal rate of substitution Marginal rate of technical substitution Max utility Max profits Min expenditure Minimize costs

Production Function A firm produces a particular good (q) using combinations of inputs The most common used inputs are capital (k) and labor (l) One could think in introducing different inputs: skilled labour, unskilled labour, raw materials, intermediate products, technology…

Production Function The firm’s production function for a particular good (q) shows the maximum amount of the good that can be produced using alternative combinations of inputs (for example, capital (k) and labor (l)) q = f(k,l)

Marginal Productivy (equivalent to marginal utility). The marginal productivity (also called marginal physical product) is the additional output that can be produced by employing one more unit of that input while holding other inputs constant. Given by the first derivative of the production function with respect the input under consideration

Diminishing Marginal Productivity
The marginal physical product of an input depends on how much of the other inputs is being used In general, we assume diminishing marginal productivity Notice that we also assume that the second derivative of the Utility function with respect a good is negative

Diminishing Marginal Productivity
Notice, that the Marginal Productivity generally depends on all the inputs For instance, the marginal productivity of labor also depend on changes in other inputs such as capital we need to consider flk which is often > 0 An additional unit of labor will increase the production of the good more if there is more capital available

Average Physical Product
Another concept is average productivity of a certain input This is the ratio of the production and the amount of input used For instance, average labour productivity is: Note that APl also depends on the amount of capital employed

Isoquant Maps To illustrate the possible substitution of one input for another, we use an isoquant map An isoquant shows those combinations of k and l that can produce a given level of output (q0) (rings a bell… indifference curves) f(k,l) = q0

Isoquant Map Each isoquant represents a different level of output
output rises as we move northeast q = 30 q = 20 k per period l per period

Marginal Rate of Technical Substitution (RTS)
The slope of an isoquant shows the rate at which l can be substituted for k lA kA kB lB A B k per period - slope = marginal rate of technical substitution (RTS) RTS > 0 and is diminishing for increasing inputs of labor q = 20 l per period

Marginal Rate of Technical Substitution (RTS)
The marginal rate of technical substitution (RTS) shows the rate at which labor can be substituted for capital while holding output constant along an isoquant

RTS and Marginal Productivities
Take the total differential of the production function: Along an isoquant dq = 0, so

Decreasing RTS Because MPl and MPk will both be nonnegative, RTS will be positive (or zero) When will we have a diminishing RTS? If fkl is positive, then the RTS is diminishing (sufficient condition), which translates in isoquants being convex (math proof in the book) This mirrors utility theory…

Returns to Scale So far, we have studied how the production changes when we change the amount of one input, leaving the rest constant (marginal productivity) However…How does output respond to increases in all inputs together? suppose that all inputs are doubled, would output double? Returns to scale have been of interest to economists since the days of Adam Smith

Returns to Scale Smith identified two forces that come into operation as inputs are doubled greater division of labor and specialization of function loss in efficiency because management may become more difficult given the larger scale of the firm

Returns to Scale If the production function is given by q = f(k,l) and all inputs are multiplied by the same positive constant (t >1), then

Returns to Scale Whether there are constant, decreasing or increasing returns to scale depend on the production function… It is possible for a production function to exhibit constant returns to scale for some levels of input usage and increasing or decreasing returns for other levels

Production Function… Linear… Fixed proportions: q = f(k,l) = ak + bl
q = min (ak,bl) a,b > 0 Cobb Douglas: q = f(k,l) = Akalb A,a,b > 0 CES, see the book

The Linear Production Function
Capital and labor are perfect substitutes k per period q1 q2 q3 l per period

Fixed Proportions k/l is fixed at b/a  = 0
No substitution between labor and capital is possible. They must always be used in a fixed ratio k/l is fixed at b/a q3/b q3/a k per period q1 q2 q3  = 0 l per period

Cobb-Douglas Production Function
Suppose that the production function is q = f(k,l) = Akalb A,a,b > 0 This production function can exhibit any returns to scale f(tk,tl) = A(tk)a(tl)b = Ata+b kalb = ta+bf(k,l) if a + b = 1  constant returns to scale if a + b > 1  increasing returns to scale if a + b < 1  decreasing returns to scale

Cobb-Douglas Production Function
The Cobb-Douglas production function is linear in logarithms ln q = ln A + a ln k + b ln l a is the elasticity of output with respect to k b is the elasticity of output with respect to l

Technical Progress Methods of production change over time
Following the development of superior production techniques, the same level of output can be produced with fewer inputs the isoquant shifts in

Technical Progress Suppose that the production function is
q = A(t)f(k,l) where A(t) represents technology that changes with time changes in A over time represent technical progress A is shown as a function of time (t) dA/dt > 0 We will not look more into this… just that you know what it is…

Chapter 8 COST FUNCTIONS

Definitions of Costs It is important to differentiate between accounting cost and economic cost the accountant’s view of cost stresses out-of-pocket expenses, historical costs, depreciation, and other bookkeeping entries economists focus more on opportunity cost Opportunity costs are what could be obtained by using the input in its best alternative use

Definitions of Costs Labour Costs
To both economist and accountants, labour costs are very much the same thing: labour costs of production (hourly wage)

Definitions of Costs Capital Costs (accountants and economists differ…) accountants use the historical price of the capital and apply some depreciation rule to determine current costs the cost of the capital is what someone else would be willing to pay for its use (and this is what the firm is forgoing by using the machine) we will use v to denote the rental rate for capital

Definitions of Costs Costs of Entrepreneurial Services
accountants believe that the owner of a firm is entitled to all profits revenues or losses left over after paying all input costs economists consider the opportunity costs of time and funds that owners devote to the operation of their firms

Example… IT programmer, she does a new software on her free time and sells it by £5000. Accounting profits=£5000. This seems like a good project Economist profits=£5000 minus what she could have earned working for a firm in her time. Might not seem such a good project any more… part of accounting profits would be considered as entrepreneurial costs by economists

Economic Cost The economic cost of any input is the payment required to keep that input in its present employment the remuneration the input would receive in its best alternative employment

Another example A shop owner
If her accounting profits are smaller than the rental price of the physical shop, it means that she is having losses as she could obtain more money by no running the shop but renting it

Two Simplifying Assumptions
There are only two inputs homogeneous labor (l), measured in labor-hours homogeneous capital (k), measured in machine-hours entrepreneurial costs are included in capital costs Firms cannot influence the input prices, they are given… (they do not depend on firms decisions on the inputs to be used…)

Economic Profits Total costs for the firm are given by
total costs = C = wl + vk Total revenue for the firm is given by total revenue = pq = pf(k,l) Economic profits () are equal to  = total revenue - total cost  = pq - wl - vk  = pf(k,l) - wl - vk

Economic Profits Economic profits are a function of the amount of capital and labor employed we could examine how a firm would choose k and l to maximize profit We will do later on…

Minimizing costs… For now…
we will assume that the firm has already chosen its output level (q0) and wants to minimize its costs We will examine the inputs that the firm will choose in order to minimize costs but produce q0 (kind of a compensated demand…)

Cost-Minimizing Input Choices
q0 We fix the isoquant of output q0. k per period Costs are represented by parallel lines with a slope of -w/v C1 C3 C2 C1 < C2 < C3 l per period

Cost-Minimizing Input Choices
The minimum cost of producing q0 is C2 This occurs at the tangency between the isoquant and the total cost curve k per period C1 C3 C2 k* l* The optimal choice is l*, k* q0 l per period

Cost-Minimizing Input Choices
Mathematically, we seek to minimize total costs given q = f(k,l) = q0 Setting up the Lagrangian: L = wl + vk + [q0 - f(k,l)] First order conditions are L/l = w - (f/l) = 0 L/k = v - (f/k) = 0 L/ = q0 - f(k,l) = 0

Cost-Minimizing Input Choices
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 This is subject to the same reservations as with utility (if RTS is strictly decreasing, no corner solutions…)

Cost-Minimizing Input Choices
Cross-multiplying, we get For costs to be minimized, the marginal productivity per dollar spent should be the same for all inputs The solution to this minimization problem for an arbitrary level of output q0 give us the contingent demand functions for inputs: lc=lc(w,v,q0); kc=kc(w,v,q0)

Contingent Demand for Inputs
The contingent demand for input would be analogous to the compensated demand function in consumer theory Notice that the contingent demand for input is based on the level of firm’s output. So, it is a derived demand.

Total Cost Function The total cost function gives the minimum cost incurred by the firm to produce any output level with given input prices. C = C(v,w,q) We can compute it as: C = C(v,w,q)=w*lc(v,w,q)+ v*kc(v,w,q)

Average Cost Function The average cost function (AC) is found by computing total costs per unit of output

Marginal Cost Function
The marginal cost function (MC) is found by computing the change in total costs for a change in output produced

Graphical Analysis of Total Costs
Suppose that k1 units of capital and l1 units of labor input are required to produce one unit of output C(q=1) = vk1 + wl1 To produce m units of output (assuming constant returns to scale) C(q=m) = vmk1 + wml1 = m(vk1 + wl1) C(q=m) = m  C(q=1)

Graphical Analysis of Total Costs
With constant returns to scale, total costs are proportional to output Total costs AC = MC Both AC and MC will be constant Output

Graphical Analysis of Total Costs
Suppose instead that total costs start out as concave and then becomes convex as output increases one possible explanation for this is that there is a third factor of production that is fixed as capital and labor usage expands total costs begin rising rapidly after diminishing returns set in

Graphical Analysis of Total Costs
Total costs rise dramatically as output increases after diminishing returns set in Output

Graphical Analysis of Total Costs
Average and marginal costs MC MC is the slope of the C curve AC If AC > MC, AC must be falling If AC < MC, AC must be rising min AC Output

Shifts in Cost Curves The cost curves are drawn under the assumption that input prices and the level of technology are held constant any change in these factors will cause the cost curves to shift

Some Illustrative Cost Functions
See Example 8.2 in the book

Properties of Cost Functions
If the input prices are multiplied by an amount t, the total cost is multiplied by the same amount cost minimization requires that the ratio of input prices be set equal to RTS, a doubling of all input prices will not change the levels of inputs purchased Nondecreasing in q, v, and w

Concavity of Cost Function
The cost function C(v,w1,q1) is concave in input prices. Why? Next slide… C(v,w1,q1) w1 Costs C(v,w,q1) w

Properties of Cost Functions
Concave in input prices The cost function increases less than proportionally when one input price increases because the firm can substitute it by other inputs As the expenditure function in consumer theory

Reaction to input prices increases
If the price of an input increases, the cost will increase The increase in costs will be largely influenced by the relative significance of the input in the production process If firms can easily substitute another input for the one that has risen in price, there may be little increase in costs It is important to measure the substitution of inputs in order to predict how much costs will be affected by an increase in the price of an input (possibly due to a tax increase)

Input Substitution A change in the price of an input will cause the firm to alter its input mix We wish to see how k/l changes in response to a change in w/v, while holding q constant

Input Substitution Rather than the derivative, we will use the elasticity: gives an 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. Hence, costs will not change so much It can me estimated using econometrics

Shephard’s Lemma Shephard’s lemma (a trick to obtain the contingent demand functions from the cost function) the contingent demand function for any input is given by the partial derivative of the total-cost function with respect to that input’s price See example 8.4 in the book As we obtained the compensated demand from the expenditure function in consumer theory

Short-Run, Long-Run Distinction
Economic actors might not be completely free to change the amount of inputs Economist usually assume that it takes time to change capital levels, while labor can be changed quickly Economist say that the “short run” is the period of time in which some of the inputs cannot be changed “Long-run” is when the period of time when all the inputs can be changed

Short-Run, Long-Run Distinction
Assume that the capital input is held constant at k1 and the firm is free to vary only its labor input The production function in the short run becomes q = f(k1,l)

Short-Run Total Costs Short-run total cost for the firm is
SC = vk1 + wl There are two types of short-run costs: short-run fixed costs are costs associated with fixed inputs (vk1) short-run variable costs are costs associated with variable inputs (wl)

Short-Run Total Costs In the Short-run:
Cannot decide the amount of fixed inputs. The firm does not have the flexibility of input choice to vary its output in the short run, the firm must use nonoptimal input combinations the RTS will not be equal to the ratio of input prices Consequently, short-run costs will be equal to or larger than long-run costs

Short-Run Total Costs Because capital is fixed at k1,
the firm cannot equate RTS with the ratio of input prices. Notice that short run costs will be Equal or larger than the long run cost k per period q2 q1 q0 l per period

Short-Run and Long-Run costs
The short run cost depend on the amount of fixed capital available There is no a unique short run cost curve. There will be as many as possible levels of capitals are The short run cost of producing q1 will be equal to the long run cost when the available capital in the short run is the same as the optimal level of capital for the long run (see q1 and k1 in the previous graph).

Short-Run Marginal and Average Costs
Remember that short run costs are given by: SC = vk1 + wl The short-run average total cost (SAC) function is SAC = total costs/total output = SC/q The short-run marginal cost (SMC) function is SMC = change in SC/change in output = SC/q

Relationship between Short-Run and Long-Run Costs
SC (k2) Total costs SC (k1) q0 q1 q2 C SC (k0) The long-run C curve is the minimum of Short-run ones Output

Chapter 9 PROFIT MAXIMIZATION

The Nature of Firms A firm is an association of individuals who have organized themselves for the purpose of turning inputs into outputs Each individual will have different objectives... Modeling the relation among all types of workers, managers, shareholders can be very complicated

Modeling Firms’ Behavior
The simplest approach in economics is the following: the decisions are made by a single dictatorial manager who rationally pursues some goal usually profit-maximization The dictatorial manager can monitor perfectly that everyone is working according to her guidelines (this assumption will be relaxed later in the course)

Profit Maximization A profit-maximizing firm chooses both its inputs and its outputs with the sole goal of achieving maximum economic profits seeks to maximize the difference between total revenue and total economic costs

Market Demand Curve In the previous lectures, we studied the demand curve for the individual consumer If we add up the demand curve of all the consumers of the market, then we will obtain the Market demand curve Its derivative with respect to price will be negative (less price, more quantity demanded) We will denote the market demand curve by D(p) or q(p)

(q) = R(q) – C(q) = p(q)q –C(q)
Output Choice Total revenue for a firm is given by R(q) = p(q)q In the production of q, certain economic costs are incurred [C(q)]->cost function Economic profits () are the difference between total revenue and total costs (q) = R(q) – C(q) = p(q)q –C(q)

(q) = R(q) – C(q) = p(q)q –C(q)
Output Choice The necessary condition for choosing the level of q that maximizes profits can be found by setting the derivative of the  function with respect to q equal to zero (q) = R(q) – C(q) = p(q)q –C(q)

Output Choice To maximize economic profits, the firm should choose the output for which marginal revenue is equal to marginal cost

Second-Order Conditions
MR = MC is only a necessary condition for profit maximization For sufficiency, it is also required that This is standard maths… at the max the second derivative must be negative

Profit Maximization Profits are maximized when the slope of
q* Profits are maximized when the slope of the revenue function is equal to the slope of the cost function revenues & costs C R The second-order condition prevents us from mistaking q0 as a maximum q0 output

Marginal Revenue R(q) = p(q)q
The second term (dp/dq) is negative for firms that face a downward-sloping curve. Typically these are large firms so that prices decreases when they produce more. The second term (dp/dq) is zero for competitive or price taking firms. Typically these are small firms so that the price does not depend on the quantity that the firm produces. In this case we have that the marginal revenue will be equal to the price.

Marginal Revenue Suppose that the demand curve for a sub sandwich is
q = 100 – 10p Solving for price, we get p = -q/ This means that total revenue is R = pq = -q2/ q Marginal revenue will be given by MR = dR/dq = -q/5 + 10

Profit Maximization To determine the profit-maximizing output, we must know the firm’s costs If subs can be produced at a constant a marginal cost of \$4, then MR = MC -q/ = 4 q = 30 We should also check that the second derivative of the profit function in this point is negative

Marginal Revenue Curve
The marginal revenue curve shows the extra revenue provided by the last unit sold For a firm that faces a downward-sloping demand curve, the marginal revenue curve will lie below the demand curve (see the formula before)

Marginal Revenue Curve
As output increases from 0 to q1, total revenue increases so MR > 0 price As output increases beyond q1, total revenue decreases so MR < 0 p1 D (p) output q1 MR

Marginal Revenue Curve
When the demand curve shifts, its associated marginal revenue curve shifts as well a marginal revenue curve cannot be calculated without referring to a specific demand curve

Supply Curve for a price taking firm
It is the curve that tell us how much a price taking firm will produce for each possible price In the next few slides, we will study how to determine it

Analysis of the supply of a price taking firm
Notice that if a firm is price taker, the second derivative of the profit function is equal to minus the derivative of the marginal cost So, the condition that the second derivative of the profit function is negative is equivalent to increasing the marginal cost must be increasing) A price taker firm will only maximize profits when marginal costs are increasing

Supply by a Price-Taking Firm
SMC SAC p* = MR q* Maximum profit occurs where p = SMC SAVC output

Supply by a Price-Taking Firm
SMC SAC p* = MR SAVC Since p > SAC, profit > 0 output q* Show in the whiteboard a situation with losses !!!!

Supply by a Price-Taking Firm
SMC If the price rises to p**, the firm will produce q** and  > 0 q** p** SAC p* = MR SAVC output q*

Supply by a Price-Taking Firm
We can see from previous graphs that the positively-sloped portion marginal cost function is very useful to determine how much the firm will produce because the firm maximizes profits using the rule p=MC So there will be a strong connection between the positively-sloped portion of the Marginal Cost curve and the Supply curve of the firm

Supply by a Price-Taking Firm
However, it could happen that the firm will have very large losses even if p=MC (maximizing profits is not equivalent to have large positive profits) and hence the firm might decide not to produce at that price Consequently, not all the positively-sloped portion of the marginal cost curve is part of the firm supply curve To be part of the firm supply curve we must ensure that the firm is better of producing at that level than not producing at all!!! This will be different depending on whether we are in the short run or in the long run !!! So far, we will study the short-run (the long run one will be studied in the next set of slides)

Short-Run Supply by a Price-Taking Firm
In the short run, we have “fixed costs” that the firm cannot eliminate them even if it decides not to produce. That is, in the short-run, the firm will be ready to produce and have losses, as long as the losses are not larger than the fixed costs. Consequently, there will be production even if price is below the minimum of the average cost in the short run Firms will only produce in the short run as long as the price is larger or equal to the minimum short run variable average cost.

Short-Run Supply by a Price-Taking Firm
If p > SAVC, the firm operates because the difference between p and SAVC contributes towards covering fixed costs If p< SACV, the firm has the fixed costs PLUS a cost of (SAVC-p) for each unit produced… It is better not to produce and only have the fixed costs Thus, the price-taking firm’s short-run supply curve is the positively-sloped portion of the firm’s short-run marginal cost curve above the point of minimum average variable cost for prices below this level, the firm’s profit-maximizing decision is to shut down and produce no output.

Short-Run Supply by a Price-Taking Firm
SMC The firm’s short-run supply curve is the SMC curve that is above SAVC SAC SAVC output

Short-Run Supply Study example 9.3 in the book

Profit Functions for price taking firms
So far, we have studied the optimal level of output of the firm. For this, we have formulated the firms problem of maximizing profits in terms of input prices, output prices, and level of output. However, there are other ways of studying the behaviour of the firm

Profit Functions for price taking firms
A firm’s economic profit can be expressed as a function of inputs  = pq - C(q) = pf(k,l) - vk - wl Only the variables k and l are under the firm’s control the firm chooses levels of these inputs in order to maximize profits treats p, v, and w as fixed parameters in its decisions

The first-order conditions for a maximum are
The firm solves the following problem: The first-order conditions for a maximum are /k = p[f/k] – v = 0 /l = p[f/l] – w = 0 A profit-maximizing firm should hire any input up to the point at which its marginal contribution to revenues is equal to the marginal cost of hiring the input

Profit Maximization and Input Demand
These first-order conditions for profit maximization also imply cost minimization they imply that RTS = w/v

Profit Maximization and Input Demand
To ensure a true maximum, second-order conditions require that kk = fkk < 0 ll = fll < 0 kk ll - kl2 = fkkfll – fkl2 > 0 capital and labor must exhibit sufficiently diminishing marginal productivities so that marginal costs rise as output expands

Input Demand Functions
The first-order conditions can be solved to yield input demand functions Capital Demand = k(p,v,w) Labor Demand = l(p,v,w) These demand functions are unconditional they implicitly allow the firm to adjust its output to changing prices

Substitution and Output effect
Now we are equipped to study how input choices changes with input prices… When w falls, two effects occur substitution effect if output is held constant, there will be a tendency for the firm to want to substitute l for k in the production process output effect the firm’s cost curves will shift and a different output level will be chosen

Substitution Effect If output is held constant at q0 and w falls, the firm will substitute l for k in the production process k per period Because of diminishing RTS along an isoquant, the substitution effect will always be negative q0 l per period

Output Effect A decline in w will lower the firm’s MC
Price Consequently, the firm will choose a new level of output that is higher P q0 q1 Output

Output Effect Output will rise to q1
k per period Thus, the output effect also implies a negative relationship between l and w q0 l per period

Cross-Price Effects No definite statement can be made about how capital usage responds to a wage change a fall in the wage will lead the firm to substitute away from capital the output effect will cause more capital to be demanded as the firm expands production

Substitution and Output Effects
We have two concepts of demand for any input the conditional demand for labor, lc(v,w,q) the unconditional demand for labor, l(p,v,w) At the profit-maximizing level of output lc(v,w,q) = l(p,v,w)

Substitution and Output Effects
Differentiation with respect to w yields substitution effect output effect total effect

Profit Functions for price taking firms
A firm’s profit function shows its maximal profits as a function of the prices that the firm faces We use the unconditional input demands to obtain the profit function

Properties of the Profit Function
If output and input prices change in the same percentage, the profits change in the same percentage with pure inflation, a firm will not change its production plans and its level of profits will keep up with that inflation

Properties of the Profit Function
Nondecreasing in output price Nonincreasing in input prices

Properties of the Profit Function
Convex in output prices the profits obtainable by averaging those from two different output prices will be at least as large as those obtainable from the average of the two prices

Graphical analysis of profits
Because the profit function is non-decreasing in output prices, we know that if p2 > p1 (p2,…)  (p1,…) The welfare gain to the firm of this price increase can be measured by welfare gain = (p2,…) - (p1,…)

Graphical analysis of profits
price SMC p1 q1 If the market price is p1, the firm will produce q1 A If the market price rises to p2, the firm will produce q2 p2 q2 B output Increase in Total Revenue= p2Aq2q1Bp1 !!!!!!!!!!!!!!!

The increase in costs is the area: Aq2q1B
Remember that:

Graphical analysis of profits
price SMC The firm’s change in profits rise by the shaded area p2 p1 output q1 q2

Producer Surplus in the Short Run
Let’s measure how much the firm values producing at the prevailing price relative to a situation where it would produce no output This is given by: This difference in profits is known as the producer surplus

Producer Surplus in the Short Run
price SMC Producer surplus at a market price of p1 is the shaded area p1 p0 output q1 Shut down price

Producer Surplus in the Short Run
Producer surplus is the extra return that producers make by making transactions at the market price over and above what they would earn if nothing was produced the area below the market price and above the supply curve

Producer Surplus in the Short Run
Because the firm produces no output at the shutdown price, (p0,…) = -vk1 profits at the shutdown price are equal to the firm’s fixed costs This implies that producer surplus = (p1,…) - (p0,…) = (p1,…) – (-vk1) = (p1,…) + vk1 producer surplus is equal to current profits plus short-run fixed costs

Important Points to Note:
The rule: MR=MC The firm will never produce in the downward sloping part of the marginal cost curve The supply curve of the firm has to take into account that the firm might be better of not producing if the losses at the level of output that maximizes profits are larger than the fixed costs.

Important Points to Note:
Short-run changes in market price result in changes in the firm’s short-run profitability these can be measured graphically by changes in the size of producer surplus the profit function can also be used to calculate changes in producer surplus

Important Points to Note:
Profit maximization provides a theory of the firm’s derived demand for inputs the firm will hire any input up to the point at which the value of its marginal product is just equal to its per-unit market price increases in the price of an input will induce substitution and output effects that cause the firm to reduce hiring of that input