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Stochastic Demand

Stochastic Demand. Uncertainty in price setting Preferences Competing technologies Income Prices of other goods Weather Sometimes don’t produce enough Reliability issues. Demand. Demand is a function of Price Some random variable, u , that account for uncertainty.

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Stochastic Demand

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  1. Stochastic Demand • Uncertainty in price setting • Preferences • Competing technologies • Income • Prices of other goods • Weather • Sometimes don’t produce enough • Reliability issues

  2. Demand • Demand is a function of • Price • Some random variable, u, that account for uncertainty. • Distribution of this random variable, f(u), is known by firm • Examine the multiplicative form of Uncertainty • Q=X(p)u • Where X is the mean demand function

  3. Multiplicative Demand • 0<u<∞ • The value of u pivots the demand curve around the vertical axis • E(u)=1 • X(p) is the expected demand curve • Prices is a parameter in the distribution of demand

  4. Expected Value of Welfare • E[W]=E[CS-L]+TR-TC] • L>0 implies excess demand • L=0 otherwise • E[CS]=area under the demand, X(p)u, for all i periods weighted by the probability of, u, occurring. • E[L]=area L weighted by the probability of, u, occurring. • Assumes customers are ranked by WTP, customers with highest WTP are served first, and ranking is costless

  5. Expected Value of Welfare • E[R]=all possible revenue weighted by the, u, possibilities • Broken into parts by excess supply and excess demand • E[C]=all possible costs weighted by the, u, possibilities • Broken into operating costs and capacity costs • Assumes a fixed coefficient technology, with parameters, b, β

  6. Maximize Welfare • Max E[W] • Yields pi=b • Problems with these pricing • When u is large leads to excess demand • Not guaranteed R>C

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