An analytical framework for retailer price and advertising decisions for products with temperature-sensitive demand

ABSTRACT

The demand for weather-sensitive products, such as beverages, ice creams, or chocolate varies with changes in temperature. Yet, retailers lack a framework to adapt the marketing mix elements, such as price and advertising, in line with such changes. We provide a theoretical framework to fill this gap by developing an analytical model to derive the optimal marketing mix when product demand depends on temperature. The model prescribes how price and advertising for different demand characteristics should be set following a temperature change. Integrating the temperature element in the marketing mix offers an original profit-enhancing strategy.

KEYWORDS:

• Pricing
• advertising
• temperature
• weather-sensitivity
• marketing mix

JEL CLASSIFICATION:

• D21
• D22
• M21
• M37
• L81

Acknowledgments

The authors thank Bruno Karoubi and Wassim Rajhi for useful discussions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

¹ To avoid confusion with time noted $t$, we note temperature $w$, recalling that it approximates weather conditions.

² Because the control variables $p$ and $a$ are static in the sense that they do not affect a state variable, we must not solve the problem with the more complex Pontryagin principle.

³ Recall $p(t),a(t),w(t),D(p(t),a(t),w(t))$, the dot notation $\dot{x}=\frac{dx}{dt}$ and the subscript notations $f_x=\frac{\mathrm{\partial }f}{\mathrm{\partial }x}$ and $f_{xy}=\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }x\mathrm{\partial }y}$. Then, the chain rule dictates $\frac{dD}{dt}=D_p\dot{p}+D_a\dot{a}+D_w\dot{w}$, $\frac{d{D}_p}{dt}=D_{pp}\dot{p}+D_{pa}\dot{a}+D_{pw}\dot{w}$ and $\frac{d{D}_a}{dt}=D_{ap}\dot{p}+D_{aa}\dot{a}+D_{aw}\dot{w}$. Note that the demand $D=D(p(t),a(t),w(t))$ changes over time $t$ if and only if $p,a$, or $w$ change over time.

⁴ More precisely, the demand function is price- and temperature-linear. Because of the second-order condition (5d), the function cannot also be assumed linear in advertising. It has to be concave in advertising, which is verified with the natural logarithm function $ln$.