Economics of the Firm: Production and Cost

Firms and Production

What are Firms' Problems?

• Firms employ inputs to produce and sell outputs.

• Three key jobs of a typical firm:

  • Master how to produce (production technology).

  • Determine what inputs are required.

  • Control total cost of production at the lowest level (cost-minimization).

  • Attain the highest profit (profit-maximization).

Production

• Production function represents the relationship between inputs and output.

• General form: $q = F(x<em>1, x</em>2, …, x_n)$

• $q$: quantity of output

• $x_i$: quantity of input $i$

• Producing output at a quantity of $q$ requires $n$ kinds of inputs at respective quantities.

• Two-input model: $q = F(K, L)$

  • Capital (K): machine or factory building.

  • Labor (L): workers.

  • To achieve the same output target, we can use more of one input but fewer of another.

  • Some inputs (L) are more flexible and its quantity employed can be changed even in the short run but some (K) are fixed.

  • Economists often work with a two-input model for simplicity.

Short Run vs. Long Run

Short Run

• Time period where levels of some inputs are fixed (capacity constraint).

• In a two-input model, capital (K) is fixed, while labor (L) can change.

  • Example: Firm already signed a contract for renting a machine/flat for a year.

Long Run

• Time period where employment levels of all inputs can change.

• Example: Selling noodles.

  • In the short run, shop space is a fixed input, while flour and materials are variable inputs.

  • In the long run, shop space becomes a variable input (can rent more or shut down).

Fixed vs. Variable

• Inputs are either fixed or variable in the short run.

• Fixed Cost (FC):

  • Costs that do not change with output.

  • Cost of employing the fixed inputs is exactly the fixed cost

• Variable Cost (VC):

  • Costs that change with output.

  • Producing more requires more variable inputs, not fixed inputs.

  • The cost of employing the variable inputs is exactly the variable costs.

• In the long run, all inputs are variable, so all costs are variable costs.

• The shop rent is a fixed cost in the short run, while the cost of noodles, food materials, and staff (including overtime) are variable costs.

Theory of the Firm and Production

• Economists can't tell you how to make noodles, but the general pattern of fixed vs. variable inputs/costs is applicable to all businesses.

• This general pattern provides useful insights to decision-makers.

Application: Fixed vs. Variable Costs

• Running a noodle shop: Fixed (rent) and variable (materials and staff) costs are often equally important.

• Running an app: Almost all costs are fixed (expenditure on inventing the app).

• Businessmen need to know the share of fixed vs. variable costs for different business strategies.

• Different patterns matter very much to business and competition.

Law of Diminishing Marginal Product (MP) or Marginal Return

• Consider a production function $q = F(K, L)$, where $K$ is fixed in the short run and $L$ is variable.

  • Total Product (TP) = $q$

  • Average Product of Labor (AP or $AP_L$) = $q/L$

  • Marginal Product of Labor (MP or $MP_L$) = $\frac{\Delta q}{\Delta L}$ (when $K$ is unchanged)

• The law states that $MP$ is decreasing with $L$ when $L$ is large enough, other things being equal.

• "Other things" include other inputs and the production technology.

• Generally speaking, $MP$ of an input will be diminishing when its quantity is large enough, other things being equal.

Why is this a Law?

• Suppose you need two inputs (K and L) to produce a good.

• For example, craftsmen (L) produce wooden toys using a knife (K).

• If you increase only one input but not the others, the sharing of the fixed inputs will eventually take effect.

Application: Law of Diminishing MP

• If a producer runs a firm, it may need more K to avoid the falling MP. A single firm's perspective.

• If all firms have already done that, what happens? Can the diminishing MP be avoided? A whole-economy perspective.

• The resources on earth are limited. So, there are always some inputs that are fixed. When all firms expand, the law will apply sooner or later.

• Thomas Malthus (1766-1834) used the law to predict massive hunger due to population growth and fixed land (Economics is called a dismal science due to this analysis).

• The law applied in growing economies: without balanced growth, output per person cannot grow forever.

Total Product Curve

• The TP curve generally looks concave.

• Up to point B, output increases at an increasing rate. From point B, output increases at a decreasing rate.

• The segment beyond D need not exist (the downward sloping segment is possible).

Marginal Product Curve

• There must be a downward sloping segment of the MP curve.

• The upward sloping segment of MP may or may not exist.

Average Product and Marginal Product

• The relation between AP and MP is useful in economics.

• Imagine the average height of students in a classroom (AP).

• A newcomer's height is MP.

• If MP > AP, AP is increasing.

• If MP < AP, AP is decreasing.

Costs in the Short Run

• Consider $q = F(K, L)$, where L is variable and K is fixed.

  • Total Variable Cost (TVC) = $wL$, where $w$ is the wage rate.

  • Total Fixed Cost (TFC) = $rK$, where $r$ is the rental rate of capital.

  • Total Cost (TC) = $wL + rK$

  • Average Total Cost (ATC) = $\frac{TC}{q}$

  • Average Variable Cost (AVC) = $\frac{TVC}{q}$

  • Average Fixed Cost (AFC) = $\frac{TFC}{q}$

  • Marginal Cost (MC) = $\frac{\Delta TC}{\Delta q}$

• If $w$ and $r$ are unchanged, the change in cost reflects the change in $L$.

Total Cost Curves

• TP curve must have a concave segment.

• Correspondingly, TC (and TVC) curve must have a convex segment.

• TFC is a horizontal line.

• TC is above TVC by the same vertical distance, which is TFC.

Average and Marginal Cost Curves

• If MP > AP, AP is increasing. If MP < AP, AP is decreasing.

• If MC > AC, AC is increasing. If MC < AC, AC is decreasing.

• The vertical gap between ATC and AVC is AFC. AFC is decreasing with q.

Long Run: No Specific Knowledge

• In a two-input model q = F(K, L), both K and L are variable in the long run.

• It is still useful to understand why a pattern may appear (or not).

Returns to Scale

Increasing Returns to Scale

• If all inputs increase proportionally, output increases more than proportionally.

• In a two-input model, if K and L double, q is more than doubled.

• Reasons:

  • Larger scale enables specialization.

  • Better coordination between inputs.

Constant Returns to Scale

• If all inputs increase proportionally, output increases proportionally.

• In a two-input model, if K and L double, q is doubled.

• This is true only if there are no any inputs that can be shared between the two factories.

Decreasing Returns to Scale

• If all inputs increase proportionally, output increases less than proportionally.

• In a two-input model, if K and L double, q is less than doubled.

• Reasons:

  • Communication becomes difficult.

  • Expansion reduces productivity.

Applying Returns to Scale

• A producer should be mindful of the output-input ratio when running a firm at a larger scale.

• Whenever more inputs do not pay off much, we say it is decreasing returns.

• Whenever more inputs pay off greatly, we say it is increasing returns.

Returns to Scale vs. Scale Effects of Cost

• If input prices are constant, then returns to scale and scale effects of cost are closely related.

• Economies of scale. Total cost increases less than proportionally.

• Diseconomies of scale. Total cost increases more than proportionally.

• Neither one of above. Total cost increases proportionally.

• When a firm is small relative to the whole market of inputs, it is realistic to assume that input prices will not be affected by a firm's expansion.

Long-Run Average Cost (LAC) Curve

• LAC exhibits all three patterns.

• Downward sloping segment reflects scale economy.

• Upward sloping segment reflects scale diseconomy.

• Horizontal segment reflects neither.

• It is U-shaped.

Avoiding Fallacious Applications

• Focus on the conditions for scale economy/diseconomy instead of merely stating that a firm is big and therefore has scale economy.

• Just scale economy does not imply low prices.

Economy of Scope

• So far, assumed a firm produces only one type of good.

• Sometimes a firm produces more than one product because economy of scope exists.

• Degree of scope economy: $\frac{[C(x, 0) + C(0, y) - C(x, y)]}{C(x, y)}$

• Unless otherwise specified, we don't assume economy of scope for our discussion.

Formal Definition:

• C(x, y) < C(x, 0) + C(0, y)

• For x, y > 0

• Why scope economy exists? Some inputs can be shared for more than one type of production.

Firm's Rational Choice

• How will a firm choose inputs to produce output?

• A firm will choose the plan that costs least (cost minimization).

• Best Producers also try to make a profit (profit maximization).

• If a firm generates a higher revenue but also a higher total cost, it will likely produce and sell more.

Isoquants

• Isoquants: different input plans that can produce the same output.

• Plan A: more K and fewer L.

• Plan B: fewer K and more L.

• Plan D: same K and fewer L than Plan B; produces less than $q_2$.

Properties of Isoquants

• Along the same isoquant, output is the same.

• An isoquant is downward sloping.

• A higher isoquant represents a higher output.

• The shape of an isoquant is normally convex.

Marginal Products and Marginal Rate of Technical Substitution (MRTS)

• Marginal Product of Labor (MPL): The extra output brought about by employing one more unit of input L (other inputs unchanged).

• Marginal Product of Capital (MPK): $\frac{\Delta q}{\Delta K}$. The extra output brought about by employing one more unit of input K (other inputs unchanged).

• Marginal Rate of Technical Substitution: $MRTS_{LK} = -\frac{\Delta K}{\Delta L}$. To produce the same output, how many units of input K can be saved when an extra unit of input L is used.

Properties of MRTS and Isoquants

• Along the same isoquant, $0 = \frac{\Delta q}{\Delta L} \Delta L + \frac{\Delta q}{\Delta K} \Delta K$

• $\frac{\Delta K}{\Delta L} = - \frac{MP<em>L}{MP</em>K}$

• An isoquant is downward sloping: [-\frac{MPL}{MPK}] < 0

• The shape of an isoquant is normally convex (MRTSLK is normally diminishing with L and normally diminishing with K).

Total Cost and Isocost Line

• Isocost line: $C_0 = wL + rK$

• Rearranging, $K = \frac{C_0}{r} - \frac{w}{r}L$

• If K = 0, then L = $\frac{C_0}{w}$

• If L = 0, then K = $\frac{C_0}{r}$

• The slope of the isoquant is -w/r

Cost Minimization

• When the output target is $q<em>1$, Plan A minimizes cost at $C</em>1$.

• If output target is different, another plan should be chosen and another cost level will be incurred.

Conditions for Cost Minimization

• The slope of the isoquant equals the slope of the isocost line: $\frac{MP<em>L}{MP</em>K} = \frac{w}{r}$

• The target output can be produced: $(L, K)$ must satisfy $q_1 = F(K, L)$

• The first condition above is applicable only for the following scenarios:

  • Interior solutions (both K > 0 and L > 0) exist at the minimization point.

  • The isoquant and isocost line are continuous (smooth curve or line).

Short Run vs. Long Run Cost Minimization

• If any K and L can be employed, Plan A is cost-minimizing.

• If the firm cannot increase K more than $K<em>1$, it can only adopt Plan B to produce $q</em>2$.

• The cost of producing $q_2$ in the short run is represented by the blue line, but in the long run by the green line.

• When the output target is $q_1$, however, Plan D is cost-minimizing. Both the short run and long run cost is the same.

Long-Run vs. Short-Run Cost

• When the output target is $q_2$, short-run average cost (SAC) is higher than long-run average cost (LAC).

• When the output target is $q_1$, both the short-run and long-run cost are the same.

Impact of Input Price Changes

• The demand curve for labor is downward sloping: a lower wage is accompanied by a larger quantity demanded for labor.

• The cost of producing $q_1$ at a higher wage is represented by the green line; lower wage is represented by the blue line.

Short Run vs. Long Run with Fixed Capital

• A lower wage does not change the quantity demanded for labor in the short run.

• The costs (green vs. blue) of producing $q_1$ in the short run are different.

Cost Minimization vs. Output Maximization

• We have so far discussed the choices for cost-minimization, which explores the lowest total cost at a given output quantity.

• Why don't we discuss output maximization at a given total cost level?

• If a firm maximizes output at a given cost, how does it choose when input prices change? Is its demand curve for an input downward sloping (in the long run)?

Special Functions of Isoquants

• Perfect Substitutes

  • It is hard to imagine an example of perfect substitutes in production. The example is Bottles sold by a vending machine vs Bottles sold by a sales person

• Fixed Proportions

  • Mobile Phones is an example of fixed-proportion function (also called perfect complements)

Examples and Class Discussion

• Some questions will be handled in lecture class. You may try it before the lecture class.

• The answer will be given in lecture class.

Cost Curve Derivation

• Given the TP curve, students can try to do the same for TC, AC, and MC with each other.

• Combining the AP curve and MP curve found from above, their relation can be seen.

• These relations are derived from a TP curve. If the TP curve is of a different shape, the AP and MP curves may also be different.

Appendix

• The information contained in pages 51-62 are extra examples and practice problems. The user is encouraged to further explore the concepts within.