*** PRODUCTION DECISIONS UNDER RISK
* Price Uncertainty: (Sandmo)
Under the EUH, consider the objective function of a competitive firm to be
EU(w + py - r'x) = EU(w + π),
where y denotes output sold at price p, x is a vector of inputs purchased at price r, f(x) is the production function, w denotes either initial wealth, the negative of fixed cost, or exogenous income, and π = (py - r'x) is firm profit. We will assume that the entrepreneur / decision maker has risk preferences represented by the utility function U(×) which satisfies U' > 0 and U" < 0 (implying risk aversion).
Assume that, because of production lags, output price p is not known at the time of the production decisions. Thus, p is a random variable with some given subjective probability distribution. Let m = E(p), and p = μ + se, where e is a random variable with mean zero. We will characterize the probability distribution of p by the mean μ and the mean preserving spread parameter σ.
Under the EUH, the production decisions can be represented by
Maxx,y {EU(w + py - r'x: y = f(x)}.
1- The firm minimizes cost:
To see that, note that the above maximization problem can be written as
Maxy {Maxx {EU(w + py - r'x): y = f(x)}}
= Maxy {EU(w + py + Maxx {- r'x: y = f(x)})}
= Maxy {EU(w + py - Minx {r'x: y = f(x)})}
= Maxy {EU(w + py - C(r,y))},
where C(r,y) = [Minx {r'x: y= f(x)}] is the cost function in a standard cost minimization problem under certainty.
The first order necessary condition associated with the choice of y is:
F(y,.) = EU'(p - C') = 0, (1)
or
m - C' + Cov(U',p)/EU' = 0,
where C' = ¶C/¶y denotes the marginal cost, and COV(U',p) = E(U'σe).
The associated second order sufficient condition is:
D = ¶F/¶y = EU'(-C") + EU"(p-C')2 < 0.
Note: The above objective function can also be written in terms of its certainty equivalent as:
Maxy {μy - C(r,y) - R(w,y,.)},
where R(w,y,.) is the Arrow-Pratt risk premium, with R > 0 under risk aversion. The associated first order condition is
μ - C' - R' = 0,
where R' = ¶R/¶y is the marginal risk premium. Comparing this result with the first order condition derived above, it follows that R' = -Cov(U',p)/EU'.
2- The Supply Function: The supply function if the function y*(w,μ,σ) that satisfies the first order condition F(y,.) = 0 in (1), or
μ = C' + R',
where R' = -Cov(U',p)/EU' is the marginal risk premium. This implies that, at the optimum supply y*, expected price μ is equal to marginal cost C' plus marginal risk premium R'.
But Cov(U',p) = sign(¶U'/¶p) = sign(U"y) < 0 under risk aversion. It follows that the marginal risk premium is positive under risk aversion: R' > 0. This in turn implies that μ > C' at the optimum.
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3- Comparative Static Analysis:
Let α = (w,μ,s) be a vector of parameters of the supply function y*(a). Differentiate the first order condition F(y,a) = 0 at the optimum y = y*(α) yields
¶F/¶a + ¶F/¶y ¶y*/¶a = 0,
or, with D = ¶F/¶y < 0,
¶y*/¶a = -D-1 ¶F/¶a
= -D-1 ¶[EU'(p - C')]/¶a
= sign {¶[EU'(p - C')]/¶a}.
a/ The effect of initial wealth w:
¶y*/¶w = -D-1 {¶[EU'(p - C')]/¶w} = -D-1 {EU"(p - C')}.
But E[U"(p - C')] >, =, or < 0 under DARA, CARA, or IARA. To see that, consider the Arrow-Pratt absolute risk aversion coefficient r = -U"/U'. Let π0 denote the value profit π when evaluated at p = C'. Under DARA,
r(π) < (>) r(π0) if p > (<) C'.
It follows that
-U"/U' < (>) r(π0) for p > (<) C',
or
U" > (<) -r(π0) U' for (p - C') > 0 (< 0),
or
U" (p - C') > -r(π0) U' (p - C'),
or, taking expectation,
E[U"(p - C')] > -r(π0) E[U' (p - C')] = 0,
from the first order condition (1). Following similar steps, it can be shown that E[U"(p - C')] < 0 (= 0) under IARA (CARA). This implies that
¶y*/¶w > 0 under DARA
= 0 under CARA
< 0 under IARA.
Thus, under DARA, changing initial wealth, fixed cost or exogenous income (w) influences supply.
Note: Consider an income transfer to farmers under DARA. An increase in w would tend to reduce the risk premium R, thus increasing the certainty equivalent and making farmers better off. But it would also stimulate production by shifting the supply schedule to the right. Given a downward sloping aggregate demand function, this would put downward pressure on food prices p. The associated decline in p would make consumers better off, but would reduce farmers' welfare. Note that such effects would not exist under certainty.
b/ The effect of expected price μ:
¶y*/¶m = -D-1{¶[EU'(p - C')]/¶μ} = -D-1{EU' + yEU"(p - C')}.
Let ¶yc/¶μ = -D-1[EU'] > 0 denote the compensated expected price effect. We have just shown that ¶y*/¶w = -D-1E[U"(p - C')]. This generates the following "Slutsky type"relationship
¶y*/¶μ = ¶yc/¶μ + (¶y*/¶w) y*,
where the expected (uncompensated) price slope ¶y*/¶μ, is equal to the compensated price slope ¶yc/¶μ > 0, plus a wealth (or income) effect, (¶y*/¶w) y*.
We have also shown that ¶y*/¶w >, =, < 0 under DARA, CARA or IARA. Thus, DARA preferences generate a positive wealth effect, which is a sufficient condition for a positive uncompensated price slope, ¶y*/¶μ > 0.
c/ The effect of price risk s: (evaluated at s = 1)
¶y*/¶s = -D-1{¶[EU'(p - C')]/¶s} = -D-1{E(U'e) + yE[U"(p - μ)(p - C')]}
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= -D-1{E(U'e) + yE[U"(p - C' + C' - μ)(p - C')]}
= -D-1{E(U'e) + yE[U"(p - C')2] + y(C' - μ)E[U"(p - C')]}.
but E(U'e) = Cov(U', p) = sign(U"y) < 0 under risk aversion. Also, E[U"(p - C')2 < 0 under risk aversion. Finally, we have shown that (C' - μ) < 0 under risk aversion, and that E[U"(p - C')] > 0 under DARA. It follows that
¶y*/¶s < 0 under DARA,
i.e. an increase in risk (as measured by s) has a negative effect on supply under DARA.
d/ The effect of a profit tax t: EU[(1-t)π], where t is the tax rate on profit π.
¶y*/¶t = -D-1{¶[EU'(p - C')]/¶t} = -D-1{E[-U"(p - C')π]}
But E[U"(p - C')π] >, =, or < 0 under DRRA, CRRA, or IRRA. To see that, consider the relative risk aversion coefficient `r = -p U"/U'. Let π0 denote the value profit π when evaluated at p = C'. Under DRRA,
`r(π) < (>) `r(π0) if p > (<) C'.
It follows that
-πU"/U' < (>) `r(π0) for p > (<) C',
or
U"π > (<) - `r(π0) U' for (p - C') > 0 (< 0),
or
U"(p - C')π > - `r(π0) U' (p - C'),
or, taking expectation,
E[U"(p - C')π] > - `r(π0) E[U' (p - C')] = 0,
from the first order condition (1). Following similar steps, it can be shown that E[U"(p - C')π] < 0 (= 0) under IRRA (CRRA). This implies that
¶y*/¶t < 0 under DRRA
= 0 under CRRA
> 0 under IRRA.
Thus, a change in profit tax t influences supply under non-constant relative risk aversion.
4- Long run analysis:
Consider an industry made of identical firms facing free entry and exit. Then the industry equilibrium must satisfy
EU(w + py - C(r,y)] = U(w). (2)
Indeed, if EU(w + py - C(r,y)] > U(w), then there is an incentive for potential entrants to enter the industry. Under free entry, they would do so, implying a disequilibrium situation. And if EU(w + py - C(r,y)] < U(w), then there is an incentive for current firms to exit the industry. Under free exit, they would do so, implying a disequilibrium situation. Thus, equation (2) must be satisfied in a long run situation.
Using the certainty equivalent, note that (2) can be written as
w + μy - C(r,y) - R(y,.) = w,
or
μy = C(r,y) + R(y,.),
or
μ = C(r,y)/y + R(y, ×)/y.
where R is the Arrow-Pratt risk premium, with R > 0 under risk aversion. It follows that, in a long run equilibrium, the expected price μ is equal to the average cost, C/y, plus the average risk premium, R/y. Under risk aversion, the risk premium is positive,R > 0, implying that
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μ > C(y, ×)/y,
or
E(π) = μy - C(y, ×) > 0.
Thus, risk aversion in the long run implies that expected price exceeds the average cost of production, and that expected profit is positive.
Note: Consider the minimization of [C(y, ×)/y + R(y, ×)/y] with respect to y. The associated necessary first order condition is
C'/y - C/y2 + R'/y - R/y2 = 0,
or
C' + R' = C/y + R/y.
Recall the short run equilibrium condition: μ = C' + R'; and the long run equilibrium condition: μ = C/y + R/y. It follows that both short run and long run equilibrium conditions are satisfied at the minimum of [C/y + R/y]. If the [C/y + R/y] function has a U-shape, it follows that the short run as well as long run equilibrium conditions satisfy
μ = Miny [C(y, ×)/y - R(y, ×)/y] > Miny [C(y, ×)/y]
under risk aversion (R > 0). Thus, under free entry, equilibrium prices are higher under risk and risk aversion (compared to the riskless case). This suggests that consumers would benefit from reducing price uncertainty...
* Production uncertainty:
1- The general case:
Let the production function be denoted by y(x,e), where y is output, x is a vector of inputs, and e is a random variable (with a given subjective probability distribution) reflecting production uncertainty (e.g. the weather). Under the EUH, assume that the objective function of the decision maker is
Maxx {EU[w + py(x,e) - r'x]}
where π = py(x,e) - r'x denotes profit. We assume that, because of production lags, the decision maker does not know e and p at the time of the input decisions, e and p being treated as random variables. Thus, the firm faces both price and production uncertainty.
The necessary first order conditions for x are:
E[U' (p ¶y(x,e)/¶x - r] = 0,
or
E[p ¶y(x,e)/¶x] = r - Cov[U', p ¶y(x,e)/¶x]/EU',
or
E(p) E[¶y(x,e)/¶x] + Cov[p, ¶y(x,e)/¶x] = r - Cov[U', p ¶y(x,e)/¶x]/EU'.
Note: Consider the maximization of the certainty equivalent
Maxx {w + E[p y(x,e)] - r'x - R(x, ×)}
where R(x, ×) is the Arrow-Pratt risk premium. The associated necessary first order conditions are
¶E[p y(x,e)]/¶x - r - ¶R(x, ×)/¶x = 0,
or
¶E[p y(x,e)]/¶x = r + ¶R(x, ×)/¶x,
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where ¶R(x, ×)/¶x is the marginal risk premium. Comparing this result with the first order condition derived above indicates that the marginal risk premium takes the form: ¶R(x, ×)/¶x = - Cov[U', p ¶y(x,e)/¶x]/EU'. This result also shows that, at the optimal input use, the expected marginal value product, ¶E[p y(x,e)]/¶x, is equal to the input cost r, plus the marginal risk premium, ¶R(x,.)/¶x. In general, the marginal risk premium can be either negative or positive depending on the nature of the stochastic production function y(x,e).
2- Some special cases:
a/ Multiplicative production uncertainty: y(x,e) = e f(x), where E(e) = 1.
Let q = pe denote the revenue per unit of expected output. Then the above maximization problem becomes
Maxx {EU[w + q f(x) - r'x]},
which is the same as Sandmo's model (after replacing p by q). Thus, the Sandmo's results apply. However, note that this specification implies that Var(y) = Var(e) f(x)2, and that ¶Var(y)/¶x = 2 Var(e) f(x) ¶f(x)/¶x. Given f(x) > 0 and ¶f(x)/¶x > 0, it follows that ¶Var(y)/¶x > 0. Thus, this stochastic production function specification restricts inputs to be always variance increasing.
b/ Additive production uncertainty: y(x,e) = f(x) + e, where E(e) = 0.
This simple specification implies that Var(y) = Var(e), and ¶Var(y)/¶x = 0. Thus, this stochastic production function specification restricts input use to have no impact on the variance of output.
c/ The Just-Pope specification: y(x,e) = f(x) + e h1/2(x), where E(e) = 0.
This specification implies
. E(y) = f(x) and ¶E(y)/¶x = ¶f(x)/¶x,
. Var(y) = Var(e) h(x) and ¶Var(y)/¶x = Var(e) ¶h(x)/¶x >, =, < 0 as ¶h(x)/¶x >, =, < 0.
Note that the production function y = f(x) + e h1/2(x) corresponds to a regression model with heteroscedasticity.
Just and Pope have presented evidence that fertilizer use tends to increase expected yield (¶f(x)/¶x > 0), as well as the variance of yield (¶h(x)/¶x > 0).
d/ The moment-based approach: (Antle and Goodger)
Let μ(x) = E[y(x,e)],
and
Mi(x) = E{[y(x,e) - μ(x)]i} = the i-th central moment of the distribution of output y, i = 2, 3, ...
Consider:
1. y = μ(x) + u, where E(u) = 0 and Var(u) = M2(x),
2. [y - μ(x)]i = ui = Mi(x) + vi, i = 2, 3, ...
where E(vi) = 0, and
Var(vi) = E[ui - Mi]2 = E(u2i) + Mi2 - 2 E(ui) Mi = M2i - Mi2.
Interpreting models 1. and 2. as regression models, they can be estimated by weighted least squares (to correct for heteroscedasticity).
Antle and Goodger have found some evidence that input use can influence μ, M2 as well as M3...
* The multiproduct firm:
1- Price uncertainty:
Consider the n product case where y = (y1, ..., yn)' = (nx1) output vector with corresponding prices p = (p1, ..., pn)'. Let pi = μi + σi ei, where E(ei) = 0, i = 1, ..., n. Under the EUH, the firm makes decisions by maximizing
Maxy [EU(w + p'y - C(r,y)],
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where C(r,y) denotes the cost function. Let y* be the optimal supply decisions associated with the above maximization problem. Some properties of y* generalize from the Sandmo single product model:
. ¶y*/¶w = 0 under CARA,
. ¶y*/¶t = 0 under CRRA, where t = tax rate,
. The Slutsky decomposition: ¶y*/¶μ = ¶yc/¶μ + (¶y*/¶w) y*, where ¶yc/¶μ is a symmetric, positive semi-definite matrix of compensated price effects, and (¶y*/¶w) y* denotes the income effect.