A N T H O L O G Y

Optimal Pricing Under Uncertainty

YURI ARENBERG Brooklyn College of CUNY

In a seminal paper, Sandmo [AER, 61, 1971] considered a competitive finn which must choose its level of output before observing the product price. In contrast, this study is concerned with a firm which specifies the price and faces uncer- tainty about the quantity demanded. It is shown that two price-setting rules are possible.

Consider a firm which produces a (perishable) commodity at a cost of c per each additional unit. The firm faces a linear demand function:

y = a - b p + u (1)

where y is the quantity of its product demanded, p is the price per unit of output, u is the random variable whose expected value, E(u), is equal to 0, and a > 0, b > 0, and a / b > c. Assume that u can take on either a value ofu ' (u' > 0) with probability o fq (0 < q < 1) or u" with probability of 1 - q. Since E(u) = 0, u" = - (q/(1 - q))u' .

Suppose that the finn must specify p and de- cide how many units to produce before u is observed. Had it known with certainty that u = 0, a profit maximizing finn would have set a price of (1/2b) (a + bc) and produced (and sold) (1/2) (a - bc) units. Under certainty, depending on the pa- rameters of the model, it may be in the interest of the firm maximizing expected profit, E(n), to set thatp at which (I)y > 0 irrespective whether u' or u" occurs, (II) y > 0 only if u' occurs. If case (I) is warranted, E (rt) takes this form:

(qp - c) (a - bp + u') + (1 - q )p

[a - b p - (q/(1 - q))u'] - F (2)

where F is the fixed cost. Maximizing equation (2) with respect top, it can be shown that, as under certainty, the price is (1/2b) (a + bc); however, more units of output, (1/2) (a - b c + 2u'), are produced, all or (1/2) [a - bc - (2q/(1 - q))u'] of which are sold depending upon whether u' or u" occurs. The latter result sharply contrasts with Sandmo' s finding that a risk averse firm produces fewer units of output under uncertainty.

Under case (II) , in v iew of the sales nonnegativity constraint, if u" were to occur, zero units of output would be sold, and E(n) becomes:

(qp - c) (a - bp + u') - F . (3)

Then, the firm specifies a price of (1/2b) [a + (bc /q) + u'] (which is higher than under certainty) and produces (1/2) [a - (bc /q ) + u'] units.

Now, the higher price is warranted if E(n) under case (II), ( 1 /4qb ) [q(a + u') - bc] 2 - F ,

exceeds that under case (I), (1/4b) [(a - bc) 2 -

4bcu '] - F . This happens when demand fluctua- tions are large (i.e., u' > (I/q) [(1 - q0.5) (q0.5 a _ bc)]). The intuition behind this result is clear. In the presence of large fluctuations, the benefit of a low price is small even if a bad state occurs, but the opportunity loss is substantial if its good counterpart results.

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