Production Function

q = f ( L, K)

q= units of output

L, K = labor and capital inputs

Marginal product

the additional output gained from one extra unit of an input, holding the other inputs constant

The marginal product of labor

the additional output gained from one extra unit of labor, holding the other inputs constant

MPL = dQ/dL

The marginal product of capital

the additional output gained from one extra unit of capital, holding the other inputs constant

MPK = dQ/dK

Isoquants

slices of the production function that show combinations of K and L that produces the same level of output “q”

Isoquants are analogous to indifference curves

Their shape is determined by the substitutability between K and L

What is the slope of isoquants called?

the marginal rate of technical substitution (MRTS)

The isoquant exhibits diminishing margins - each additional unit of labor/capital increases q less than the previous unit and is worth less in terms of foregone capital/labor

MRTS equation

MRTS = -MPl / MPk = - (dq/dL)/(dq/dK)

How do you derive MRTS from a production function?

- q = f(L,K)
- Take the total derivative of our production function to see how total output is changing with respect to changes in inputs: dq = (dq/dL) * dL + (dq/dK) * dK
- Then, set dq = 0

So, 0 = (dq/dL) * dL + (dq/dK) * dK - Then rearrange the terms:

dL/dK = - (dq/dL) / (dq/dK) = MRTS

What has to be true in short-run production?

At least one input is fixed; for this course we assume that capital (K) is fixed & that labor is variable

What has to be true in long-run production?

All inputs are variable, firms can fully decide how much capital (K) and labor (L) to hire

Constant returns to scale equation

f ( 2L, 2K ) = 2 f(L,K)

Decreasing returns to scale equation

f (2L, 2K) < 2 f(L,K)

Increase returns to scale equation

f (2L, 2K) > 2 f(L,K)

Fixed costs

costs of inputs that can’t be varied in the short-run; capital

Variable costs

costs of inputs that can be varied in the short-run; labor

Total costs

C = F + VC; sum of fixed and variable costs

Marginal costs

the extra cost for another unit of output

MC = dC/dq where C is the total cost MC = w * 1/MPL => Marginal costs move inversely with marginal product of labor

What is the marginal cost in the short term?

SR MC = dVC/dq

The marginal cost is determined by the increase in the variable cost (since fixed costs do not vary with output)

Average cost

the average cost of production per unit produced

AC = C/q

The average variable cost equation

AVC = VC/q

The average fixed cost equation

AFC = FC/q

Long-run costs

In the very long run, all input costs are variable - so choose is over input mix to maximize production efficiency, or minimize costs

How do you (generally) determine how much a firm will want to produce?

- Derive the cost function (short or long run) that produces a given quantity of “q” most efficiently by combining K and L for a given set of factor prices “w” and “r’
- Choose a quantity that maximizes profits for a given price

How do you derive the cost function?

For a given unit of “q”, what is the cheapest way for me to combine “K” and “L”?

- We start by finding the isocost line: C = w
*L + r*K - Find the tangency between the isocost and isoquant (the lowest cost for a given isoquant)

MRTS = -MPl/MPk = - w/r - Use your production function & point of tangency to derive relationships between inputs and “q”
- Plug these relationships back into the total cost function C = w
*L + r*K

(remember SR K is fixed, so incorporate that as needed)

Isocost line

combinations of labor and capital that can produce an output at the same cost

C = w*L + r*K

Analogous to a budget constraint

Long-run average cost curves (LRAC) vs. short-run average cost curves (SRAC)

The LRAC is the lower envelope of the SRAC for different plant sizes. The LR cost of production is lower than the SR cost of production

In perfect competition, firms and consumers are price takers. Why?

Many small buyers and sellers Identical products Symmetric information No transactions costs Free exit and entry in the long run

Does the fact that a firm is a price taker (I..e the demand curve they face is perfectly elastic) mean that the market demand is also perfectly elastic?

No, the residual demand (or the demand curve a firm faces) is much more elastic than the overall demand curve of the market

Profit equation

pi = R(q) - C(q)

where R(q) is the total revenues the firm receives from selling output “q” and C(q) is the cost

In order to maximize profits, what should a firm do? (mathematically)

Set MR = MC or dR(q)/dq = dC(q)/dq & produce outputs at this point

What is a key part of understanding how a perfectly competitive firm will find the MR and MC to maximize profits?

Competitive firms face a perfectly elastic demand curve, MR = P, hence for a perfectly competitive firm P = MC

When will firms shut down in the SR?

In the short run, firms shut down if P < min AVC

To find this, we can derive individual firm supply curve using P = MC and Q = 0 (shut down) for P < min AVC

SR market supply curve

the horizontal sum of individual firm SR supply curves

Are industry profits positive or negative in SR?

Can be either positive or negative

How do firms maximize profits in the long run? (i.e. what will happen to profits in a perfectly competitive market)

In LR, free entry and exit drives economic profits to 0, i.e. P = MC = AC. Hence, LR industry supply curve is perfectly elastic at P = min AC and each firm produces at q = min AC

Which supply curves are more elastic - short run or long run?

SR supply less elastic than LR supply with barriers to entry

LR supply w/ barriers to entry is less elastic than LR supply curves with free entry

Potential implication on the LR supply curve when input prices increase?

Could lead to an upward sloping LR supply curve, even with free entry

MRPL equation

In the SR and LR, demand for labor will be its marginal revenue product

MRPL = MR * MPL

MR = marginal revenue from an additional unit of output (MR = p if competitive output market); however MPL in LR will take into account optimal capital adjustmnets

Which is more elastic - LR or SR labor demand?

LR labor demand is more elastic than SR labor demand

Explain the difference between the short run and the long run

Capital is fixed in the SR, and variable in the LR

Explain why average costs are at a minimum when they cross the marginal cost curve

When average costs are falling, they must be below the marginal costs

When they rising, they must be above the marginal cost

If the average cost > marginal cost, you can always drive average cost down by producing more (left of the intersection)

If the average cost < marginal cost, the more you produce, the more the average costs go up (right of intersection)

Explain when a firm will shut down in the short run

In the short run, a firm should shut down when total variable cost exceeds total revenue, which is also when average variable cost exceeds price. Fixed costs cannot be avoided in the short run, so they are irrelevant to the shutdown decision.

You only shut down if:

“pq” < VC

“p” < AVC

Explain when a firm will shut down in the long run

Key difference is the shutdown rule

For SR competition: Firm will shutdown when price is less than AVC (fixed costs can’t be changed)

For LR competition: Firms will shutdown when price is less than Average Costs (fixed and variable together)

Explain when firms will enter/exit in the long run

We’re going to argue that this decision will be a function of the profits being made in that market

Enter if profits being made; exit if money is being lost

In a perfectly competitive market, firms will enter & exit in the long run until profits are zero

Entry & exit drives profit to zero (in the LR)

Explain why, in theory, long-run supply in a perfectly competitive market will be flat at min ATC when there are identical firms

All the firms will be producing at marginal cost equals minimum average cost

Long run supply, with identical firms, are free entry and exit, will be perfectly elastic at minimum average cost.

Competition leads to cost minimization.

Why does ATC = MC = p in the long run for a firm in a perfectly competitive market?

The market supply curve becomes increasingly elastic with firm entry. It does not ever become perfectly inelastic.

Firms enter until average cost equals price (=MC), which is the point at which profits are driven to zero. This is also the point at which average cost is minimized, which is by definition the point at which the firm is operating at greatest efficiency.

Explain three cases in which the long-run supply may be upward sloping. Do firms earn profits in each of these cases? Why?

Barriers to entry and exit, such as large sunk costs, mean that there is no longer free entry and exit. This makes entry difficult even for firms for which it would be profitable to produce in the long run.

If firms are not identical — if some firms can produce at a lower minimum average cost — then they can stay in the market and make profits. However they might not out-compete all other firms because they cannot supply the market quantity demanded; that is, they are capacity-constrained.

Input costs that rise with output, such as wages, increase the minimum average cost of higher quantities of output. This creates an upward-sloping output supply curve.

Input prices are not driven to zero by competition. Firms do minimize the cost of their inputs, but this does not explain why input prices are not fixed.

With non-identical firms, with costly entry and exit, or with rising input costs, not all of these results are necessarily true. But with identical firms, free entry and exit, and fixed input costs, all these results are true.

Market price will be equal to the long-run minimum average cost of each firm in the market, driving their profits to zero, and squeezing out every firm that charges above or below that price.

Sunk costs

They’re forgone no matter what you produce at any level of production

Profit equation

price – average cost * number of units (level)

If ATC > MC then, ATC is…

decreasing

If ATC < MC, then ATC is

increasing

If ATC = MC, then ATC

is at its minimum

Derive LR cost curve, 7 steps

- Write down the production function.
- Solve for the optimum by setting MRTS = w/r; solve for either K or L
- Rewrite the production function, substituting solved K or L
- Solve for the other L/K
- Plug L/K back into the production function to solve for K/L
- Write cost function = C = rK +w L
- Then put those numbers back into the cost function, as a function of the quantity