Producer Welfare

PRODUCER WELFARE

Preliminaries

Consider a production process involving the production of outputs y  R using inputs x  R under the technology represented by the feasible set F, z ≡ (-x, y)  F  Rn+m. Throughout, we assume that set F is closed. Given a reference netput bundle g  R with g ≠ 0, the associated shortage function is

S(z, g) = min {: (z-  g)  F}, if there is a  satisfying (z -  g)  F,(1)

= + otherwise.

We have seen that z  F implies that S(z, g)  0 (meaning that F  {z: S(z, g)  0}), that S(z, g) = 0 implies that z is on the upper bound of the feasible set F, and that F = {z: S(z, g)  0} under free disposal.

Consider that the production process is managed by a competitive firm facing market prices r  R for inputs x and p  R for outputs y. Let q ≡ (r, p)  R denote the prices for the netputs z ≡ (-x, y). Under profit maximization (and assuming that a maximum exists), the firm would choose the netputs z ≡ (-x, y) as follows

(q) = (r, p) = Maxz {q z: z  F},(2a)

= Maxx,y {p y - r x: (-x, y)  F},

= Maxy {p y - C(r, y)}, (2b)

= Maxx {R(p, x) - w x},(2c)

where (r, p) is the profit function, C(r, y) = Minx {r x: (-x, y)  F} is the cost function, and R(p, x) = Maxy {p y: (-x, y)  F} is the revenue function. Under differentiability, the first-order necessary condition for an interior solution for outputs y in (2b) is:

p = C(r, y)/y,(3)

implying "marginal cost pricing," where output prices p equals marginal cost C(r, y)/y. This identifies the marginal cost, C(r, y)/y, as the supply functions for outputs.

Similarly, under differentiability, the first-order necessary condition for an interior solution for inputs x in (2c) is:

r = R(p, x)/x, (4)

implying "marginal revenue pricing," where input prices r equals marginal revenue R(p, x)/x. This identifies the marginal revenue, R(p, x)/x, as the demand functions for inputs.

Let z*(q) = (-x*(r, p), y*(r, p)) denote the optimal solution in (2a), with (q) = q z*(q). Then, x*(r, p) are the profit maximizing input demand functions, and y*(r, p) are the profit maximizing supply functions. These functions are linear homogeneous in prices q ≡ (r, p), implying that only relative prices influence production behavior. In addition, under differentiability,

z*/q = is a symmetric, positive semi-definite matrix, (5)

implying that yj*/pj 0, j = 1, …, m, (supply functions being "upward-sloping" with respect to own output price), that xi*/ri 0, i = 1, …, n (input demand functions being "downward-sloping" with respect to own input price), and that zi*/qj = zj*/qi, for all i ≠ j (the symmetry or "integrability" conditions).

Duality

Result 1: The profit function  (q) given in (2a) can be alternatively written as

(q) = Maxz {q z - S(z, g) q g: z  Rn+m}. (6)

Proof: First, we show that

(q)  q z - S(z, g) q g,(7)

for any z.

There are three possibilities: a/ S(z, g) = -; b/ S(z, g) = +; and c/ - < S(z, g) < +.

Case a/: z satisfies S(z, g) = -. Then, (z - S(z, g) g)  F, (zj - S(z, g) gj) =  for the netputs where gj > 0, implying that (q) = . Thus, (7) holds.

Case b/: z satisfies S(z, g) = +. Then, given q g > 0, we have (q)  q z - S(z, g) q g = -, and (7) holds.

Case c/: z satisfies -  < S(z, g) < +. Then, (z - S(z, g) g)  F, implying that (q)  q [z - S(z, g) g]. Thus, (7) holds.

It follows from (7) that

(q)  Maxz{q z: z  F}

 Maxz{q z - S(z, g) q g: z  Rn+m}.

Next, we prove the reverse inequality:

(q)  Maxz {q z - S(z, g) q g: z  Rn+m}. (8)

Start with (q)  Maxz{q z: z  F}. Note that z  F implies that S(z, g)  0. Given q g > 0, it follows that

(q)  Maxz{q z: z  F}

 Maxz{q z - S(z, g) q g: z  F}

 Maxz{q z - S(z, g) q g: z  Rn+m},

which gives (8).

Note 1: Note that the optimization problem in (6) is an unconstrained optimization problem. Under differentiability, the associated first-order necessary condition is

q = (q g) (∂S/∂z).

When the reference bundle g is chosen such that (q g) = 1 (i.e., when g is chosen such that one unit of g has a "unit price"), then, the first-order condition becomes

q = ∂S/∂z,

where profit maximization means that prices q must equal "marginal shortage" ∂S/∂z. This indicates that, when q g = 1, the marginal shortage ∂S/∂z provides a general measure of the "shadow price" (or marginal cost) of the netputs z. See below.

Note 2: The above results apply with or without free disposal. The reason why "free disposal" is not needed is that, under positive prices, there is no incentive for the firm to choose netputs z in the zone where free disposal may not hold. So, even if free disposal does not apply, the firm would choose not to locate there. If it did, it would fail to be allocatively efficient…

Measuring producer welfare

In general, changes in firm profit (q) give a simple measure of the effects of changes in the firm economic environment on firm welfare. It provides a monetary measure of the willingness to pay for the firm facing changes in its economic environment.

3.1.Welfare effects of price changes

Consider a change in market prices q  (r, p) from q0 (r0, p0) to q1 (r1, p1). The welfare impact of this price change on the firm can be measured by the change in firm profit: [(q1) - (q0)]. It measures the willingness to pay (or willingness to receive if negative) for the firm to face this price change.

Result 2: When the profit function (q) is differentiable, the change in profit due to a change in prices q from q0 to q1 can be written as

(q1) - (q0) = zi*(q) dqi.(9)

Proof: When (q) is differentiable, applying the fundamental theorem of calculus gives

(q1) - (q0) = (∂/∂q) dq,

= (∂/∂qi) dqi.

Under differentiability, applying the envelope theorem to (2a) or (6) gives ∂/∂q = z*(q). Substituting this result into the above expression yields (9).

Equation (9) provides a convenient measure of firm welfare effects due to price changes. It shows that profit change can be measured by the "areas to the left of the profit-maximizing netput decision rules and between prices before and after the change." To illustrate, for a change in output prices p from p0 to p1, this gives

(r, p1) - (r, p0) = yi*(r, p) dpi,

or in the context of a single output price change for the i-th output from pi0 to pi1,

(r, p1) - (r, p0) = yi*(r, p) dpi ,

which is the area to the left of the profit-maximizing supply curve yi*(r, p) and between the two prices pi0 and pi1. This means that any output price increase (p1 > p0) tends to make the firm better off, with (r, p1) - (r, p0) ≥ 0.

For a change in input prices from r0 to r1, this gives

(r1, p) - (r0, p) = -xi*(r, p) dri,

or in the context of a single input price change for the i-th input from ri0 to ri1,

(r1, p) - (r0, p) = -xi*(r, p) dri.

This shoes that the change in profit can be measured by the negative of the area to the left of the profit-maximizing input demand curve xi*(r, p) and between the two prices ri0 and ri1. This means that any increase in input price (r1 > r0) tends to make the firm worse off, with (r1, p) - (r0, p) ≤ 0.

3.2.Welfare effects of quantity changes

Consider the case where z = (za, zb)  F, where the netputs za (-xa, ya) are market goods with price qa (ra, pa) > 0. The other netputs zb (-xb, yb) can have two interpretations. First, the netputs zb can be non-market goods (e.g., public goods without a market price). Second, under rationing, the netputs zb can be rationed. In this case, the rationed netputs zb are chosen outside the firm. In either case, profit maximization typically would not apply with respect to zb. However, it can still apply with respect to za. The profit maximization problem (2a) then becomes choosing za conditional on zb as follows

a(qa, zb) = qa za*(qa, zb) = Maxza {qa za: (za, zb)  F}. (2a')

Let ga denote a reference bundle of netputs za satisfying ga ≥ 0, ga 0. Define the shortage function

Sa(z, ga) = min {: (za -  ga, zb)  F}, if there is a  satisfying (za -  ga, zb)  F,(1')

= + otherwise.

Using equation (6) (result 1), the profit maximization problem (2a') can be alternatively written as

a(qa, zb) = qa za*(qa, zb) = Maxza {qa za - Sa(z, ga) qa ga},(6')

where z ≡ (za, zb)  Rn+m.

Result 3: When the profit function a(qa, zb) is differentiable in zb, the change in profit due to a change in rationed quantities zb from zb0 to zb1 can be written as

a(qa, zb1) - a(qa, zb0) = -(qa ga) (za*(qa, zb), zb, ga) dzbi.(10)

Proof: When a(qa, zb) is differentiable in zb, applying the fundamental theorem of calculus gives

a(qa, zb1) - a(qa, zb0) = (∂a/∂zb) dzb,

= (∂a/∂zbi) dzbi.

Under differentiability, applying the envelope theorem to (6') gives ∂a/∂zb = -(za*(qa, zb), zb, ga) (qa ga). Substituting this result into the above expression yields (10).

Equation (10) provides a convenient measure of the firm welfare effects of changes in the rationed quantities zb. It becomes even simpler in situations where the reference bundle ga satisfies qa ga = 1, i.e., when ga is chosen such that ga has "unit price." Then, with qa ga = 1, (10) becomes

a(qa, zb1) - a(qa, zb0) = -(za*(qa, zb), zb, ga) dzbi.(10')

Given qa ga = 1, equation (10') shows that a change in profit due a change in the rationed quantities zb from zb0 to zb1 is measured by the negative of the area below the marginal shortage function (or shadow price function) (za*(qa, zb), zb, ga) and between the rationed quantities before and after the change. To illustrate, in the context of a single quantity change of the i-th netput from zbi0 to zbi1, (10') becomes

a(qa, zb1) - a(qa, zb0) = -(za*(qa, zb), zb, ga) dzbi,

which is the negative of the area below the marginal shortage function (measuring the “shadow price” or marginal cost function of zb) (za*(qa, zb), zb, ga) and between the rationed quantities zbi0 and zbi1. This means that any increase in rationed quantity (zbi1 > zbi0) tends to make the firm better off (worse off) whenever (za*(qa, zb), zb, ga) < 0 (> 0). This illustrates the usefulness of the shortage function in welfare analysis.

Note 3: Note that (10) requires at least one market good. However, it allows for an arbitrary number of "rationed goods". This provides some flexibility for analyzing either rationing situations or the production value of non-market goods (e.g., environmental services).

Note 4: Note that equation (10) applies with or without free disposal. First, consider the case where free disposal does not apply to the market goods za. Then profit maximization under positive prices means that the firm has no incentive to choose netputs za in the zone where free disposal may not hold. Indeed, if the firm decided to locate in this zone, it would fail to be allocatively efficient. Second, consider the case where free disposal does not apply to the non-market goods zb. Then, in the zone where free disposal does not apply, S(za, zb, g) would be decreasing in zb (while S(za, zb, g) would be non-decreasing in zb in the regions where free disposal does apply). Thus, under differentiability, the sign of S/zb can be used to investigate whether "free disposal in zb" applies (when S/zb 0) or does not apply (when S/zb < 0).

To illustrate, consider the single output case (m = 1) where z = (-xb, ya), output ya is the only market good with price pa > 0, and inputs xb are all non-market goods. Let g = (0, …, 0, 1). Then, S(-xb, ya) = ya - f(xb), where f(xb) is the standard production function. First, if output ya does not satisfy free disposal, then the revenue maximizing firm would always choose ya to be on the production function (i.e., away from any feasible point where free output disposal may not hold). Second, if inputs xb do not satisfy free disposal, then the production function would exhibit a maximum. This maximum can be used to identify the zones where free input disposal applies and where it does not. In particular, if the maximum point is unique and in the absence of "other local maxima," free input disposal would apply for input quantities lower than this point (where marginal productivity is non-negative: f/xb 0), and free input disposal would fail to hold for input quantities higher than this point (where marginal productivity is negative: f/xb < 0).

3.3.Welfare effects of technical change

Consider a change in technology represented by a change in the feasible set from F(t0) to F(t1), where t is a technology index. In the context of technological progress from t0 to t1, we have F(t0)  F(t1). Then, the shortage function (1) becomes

S(z, t, g) = min {: (z-  g)  F(t)}, if there is a  satisfying (z -  g)  F(t),(1")

= + otherwise.

Note that technological progress from t0 to t1 means that S(z, t0, g) ≥ S(z, t1, g). The profit maximization problem (2a) becomes

(q, t) = q z*(q, t) = Maxz{q z: z  F(t)}. (2")

Using equation (6), this can be alternatively written as

(q, t) = q z*(q, t) = Maxz {q z - S(z, t, g) q g: z  Rn+m}.(6")

Result 4: When the profit function (q, t) is differentiable in the technology index t, the change in profit due to a change in t from t0 to t1 can be written as

(q, t1) - (q, t0) = -(q g) (z*(q, t), t, g) dt.(11)

Proof: When (q, t) is differentiable in t, applying the fundamental theorem of calculus gives

(q, t1) - (q, t0) = (∂/∂t) dt.

Under differentiability, applying the envelope theorem to (6") gives ∂/∂t = -(z*(q, t), t, g) (q g). Substituting this result into the above expression yields (11).

Equation (11) provides a convenient measure of the welfare effects of technical change on the firm. It becomes even simpler in situations where the reference bundle g satisfies q g = 1, i.e., when g is chosen such that g has "unit price." Then, with q g = 1, (11) becomes

(q, t1) - (q, t0) = -(z*(q, t), t, g) dt.(11')

Given q g = 1, equation (11') shows that a change in firm profit due technical change t from t0 to t1 is measured by the negative of the area below the marginal shortage function (z*(q, t), t, g) and between t0 and t1. When technical progress is associated with an increase in t from t0 to t1 (with t1 > t0), then we expect ∂S/∂t ≤ 0 (since technical progress implies that S(z, t0, g) ≥ S(z, t1, g)). It follows from (11') any technical progress tends to make the firm better off, with [(q, t1) - (q, t0)] ≥ 0. Again, this illustrates the usefulness of the shortage function in welfare analysis.

Aggregation across firms

All results presented above were presented for a single firm. However, note that they can be easily aggregated across firms in an industry. The aggregation is particularly simple if all firms face the same prices q. Then, profit functions can be simply added across firms to obtain the industry profit function. Similarly, supply/demand functions can be added across firms to obtain aggregate supply/demand functions. And as long as the reference bundle g contains private goods and remains constant for all firms, the firm shortage functions can also be added across firms to generate an aggregate shortage function. At the industry level, this aggregate shortage function simply measures the aggregate number of units of the reference bundle g that the industry can generate by moving to the upper-bound of the feasible set. Importantly, note that the analysis would still apply at the aggregate even in the presence of heterogeneous technology across firms. It would allow different firms in the industry to have access to different technologies (e.g., different farms producing under different agro-ecological conditions; or different firms being at different stages of the adoption process for new technologies).

This means that all results presented above would apply as well at the industry level, working with an aggregate profit function, aggregate supply/demand functions, and an aggregate shortage function. This "nice aggregation property" makes the shortage function particularly convenient for economic and warfare analysis of production activities.