Incentive and Strategic Contracting: Implications for the Franchise Decision

Abstract

We examine theoretical predictions and econometric evidence concerning franchise contracting and sales-force compensation and discuss a number of factors that ought to influence the contracts that are written between principals and agents. For each factor, we construct the simplest theoretical model that captures what we view as its essence. The comparative statics from the theoretical exercise are then used to organize our discussion of the empirical evidence, where the evidence is taken from published studies that have attempted to assess each factor’s effect on the power of agent incentives. We also discuss theoretical issues and empirical results pertaining to a few topics that have been addressed in the literature but that do not fit easily into our simple modeling framework. Finally, we discuss a few recent models with endogenous prices. Unfortunately, the evidence that relates to strategic or game-theoretic agency models is scanty. Nevertheless, we discuss the findings from the few relevant studies that we have been able to uncover.

*This chapter is a revised and updated version of Lafontaine and Slade (2001), which was entitled “Incentive Contracting and the Franchise Decision” published in the earlier edition of this volume. This revision builds on some of our recent work, most notably Lafontaine and Slade (2007 and 2012).

Algebraic Derivations

In each case below, the agent (A) maximizes his certainty-equivalent income, $E(y)-(r/2)Var(y)$, whereas the principal (P) maximizes the expected total surplus—expected output minus the agent’s cost of effort minus the agent’s risk premium—$E(q)-{a^2}/2-(r/2)Var(y)$. With one exception, noted below, the agent’s compensation is given by $S(q)=\alpha q+W$. The cases differ according to the specification of the function that maps effort into output, $q=f(a,\varepsilon, \Theta )$.

1.1 Risk

$$ q=a+\varepsilon . $$

$$ A:\quad \mathop{\max}\limits_a \left[ {\alpha a+W-\frac{{{a^2}}}{2}-\frac{r}{2}{\alpha^2}{\sigma^2}} \right]. $$

The resulting first-order condition (foc) is: a = α. Substituting the agent’s effort choice into the principal’s problem yields:

$$ P:\quad \mathop{\max}\limits_{\alpha } \left[ {\alpha -\frac{{{\alpha^2}}}{2}-\frac{r}{2}{\alpha^2}{\sigma^2}} \right], foc: 1-\alpha -r\alpha {\sigma^2}=0, $$

$$ {\alpha^{*}}=\frac{1}{{1+r{\sigma^2}}}, $$

$$ \frac{{d{\alpha^{*}}}}{dr }=-\frac{{{\sigma^2}}}{{{\Gamma^2}}}<0, \frac{{d{\alpha^{*}}}}{{d{\sigma^2}}}=-\frac{r}{{{\Gamma^2}}}<0,\;\;\;\mathrm{where}\;\;\;\Gamma =1+r{\sigma^2}. $$

1.2 Agent Effort

$$ q=\eta a+\varepsilon . $$

$$ A:\quad \mathop{\max}\limits_a \left[ {\alpha \eta a+W-\frac{{{a^2}}}{2}-\frac{r}{2}{\alpha^2}{\sigma^2}} \right], foc: a=\alpha \eta . $$

$$ P:\quad \mathop{\max}\limits_{\alpha } \left[ {{\eta^2}\alpha -\frac{{{\alpha^2}{\eta^2}}}{2}-\frac{r}{2}{\alpha^2}{\sigma^2}} \right], foc: {\eta^2}-\alpha {\eta^2}-r\alpha {\sigma^2}=0, $$

$$ {\alpha^{*}}=\frac{{{\eta^2}}}{{{\eta^2}+r{\sigma^2}}}, $$

$$ \frac{{d{\alpha^{*}}}}{{d\eta }}=\frac{{2\eta r{\sigma^2}}}{{{\Gamma^2}}}>0,\;\;\;\mathrm{where}\;\;\;\Gamma =\left( {{\eta^2}+r{\sigma^2}} \right). $$

1.3 Outlet Size

$$ q=\eta a+(\gamma +\varepsilon )k. $$

$$ A:\quad \mathop{\max}\limits_a \left[ {\alpha (\eta a+\gamma k)+W-\frac{{{a^2}}}{2}-\frac{r}{2}{\alpha^2}{\sigma^2}{k^2}} \right], foc: a=\alpha \eta . $$

$$ P:\quad \mathop{\max}\limits_{\alpha } \left[ {{\eta^2}\alpha -\frac{{{\alpha^2}{\eta^2}}}{2}-\frac{r}{2}{\alpha^2}{\sigma^2}{k^2}} \right], foc: {\eta^2}-\alpha {\eta^2}-r\alpha {\sigma^2}{k^2}=0, $$

$$ {\alpha^{*}}=\frac{{{\eta^2}}}{{{\eta^2}+r{\sigma^2}{k^2}}}, $$

$$ \frac{{d{\alpha^{*}}}}{dk }=-\frac{{2{\eta^2}r{\sigma^2}k}}{{{\Gamma^2}}}<0,\;\;\;\mathrm{where}\;\;\;\Gamma ={\eta^2}+r{\sigma^2}{k^2}. $$

1.4 Costly Monitoring

This result is derived in Lafontaine and Slade (1996).

1.5 Franchisor Effort

$$ q=\eta a+\theta b+\varepsilon . $$

This problem has two incentive constraints:

$$ A:\quad \mathop{\max}\limits_a \left[ {\alpha (\eta a+\theta b)+W-\frac{{{a^2}}}{2}-\frac{r}{2}{\alpha^2}{\sigma^2}} \right], foc: a=\alpha \eta . $$

$$ P:\quad \mathop{\max}\limits_b \left[ {(1-\alpha )(\eta a+\theta b)-\frac{{{b^2}}}{2}} \right], foc: b=(1-\alpha )\theta . $$

The franchisor chooses α to maximize total surplus, equal to $\eta a+\theta b-\frac{{{a^2}}}{2}-\frac{{{b^2}}}{2}-\frac{r}{2}{\alpha^2}{\sigma^2},$ subject to the two incentive constraints. After substituting, we have:

$$ \mathop{\max}\limits_{\alpha } \left[ {\alpha {\eta^2}+(1-\alpha ){\theta^2}-\frac{{{\alpha^2}{\eta^2}}}{2}-\frac{{{{{(1-\alpha )}}^2}{\theta^2}}}{2}-\frac{r}{2}{\alpha^2}{\sigma^2}} \right], $$

$$ foc:{\eta^2}-{\theta^2}-\alpha {\eta^2}+(1-\alpha ){\theta^2}-r\alpha {\sigma^2}=0, $$

$$ {\alpha^{*}}=\frac{{{\eta^2}}}{{{\eta^2}+{\theta^2}+r{\sigma^2}}}, $$

$$ \frac{{d{\alpha^{*}}}}{{d\theta }}=-\frac{{2{\eta^2}\theta }}{{{\Gamma^2}}}<0, \frac{{d{\alpha^{*}}}}{{d\eta }}=\frac{{2\eta ({\theta^2}+r{\sigma^2})}}{{{\Gamma^2}}}>0,\;\;\;\mathrm{where}\;\;\;\Gamma =\left( {{\eta^2}+{\theta^2}+r{\sigma^2}} \right). $$

1.6 Multiple Tasks

$$ q=a+\varepsilon,\quad \varepsilon \sim N(0,\varSigma ),\quad C(a)=\frac{{{a^T}a}}{2}, $$

where q, a, and ε are vectors, as is α. However, W remains a scalar.

$$ A:\quad \mathop{\max}\limits_a \left[ {{\alpha^T}a+W-\frac{{{a^T}a}}{2}-\frac{{r{\alpha^T}\varSigma \alpha }}{2}} \right],\quad foc: a=\alpha . $$

After substituting, we have:

$$ P:\quad \mathop{\max}\limits_{\alpha } \left[ {{\alpha^T}j-\frac{{{\alpha^T}\alpha }}{2}-\frac{{r{\alpha^T}\varSigma \alpha }}{2}} \right], foc: j-\alpha -r\varSigma \alpha =0. $$

Hence, ${\alpha^{*}}={{(I+r\varSigma )}^{-1 }}j,$ where j is a vector of ones. When n = 2, this becomes

$$ \alpha_i^{*}=\frac{{1+r{\sigma_{jj }}-r{\sigma_{12 }}}}{{(1+r{\sigma_{11 }})(1+r{\sigma_{22 }})-{r^2}\sigma_{12}^2}}. $$

Setting ${\sigma_{11 }}={\sigma_{22 }}={\sigma^2}$ yields:

$$ \alpha_i^{*}=\frac{{1+r{\sigma^2}-r{\sigma_{12 }}}}{{{{{(1+r{\sigma^2})}}^2}-{r^2}\sigma_{12}^2}}=\frac{1}{{1+r({\sigma^2}+{\sigma_{12 }})}} $$

so that

$$ \frac{{d\alpha_i^{*}}}{{d{\sigma_{12 }}}}=-\frac{r}{{{{{(1+r({\sigma^2}+{\sigma_{12 }}))}}^2}}}<0. $$

1.7 Spillovers Within Chains

$$ q=1-p-\mu \bar{p}+a+\varepsilon . $$

where $\bar{p}$ is the price at another outlet in the same chain.

$$ A:\quad \mathop{\max}\limits_a \left[ {\alpha (1-p-\mu \bar{p}+a)+W-\frac{{{a^2}}}{2}-\frac{r}{2}{\alpha^2}{\sigma^2}} \right],\quad foc: a=\alpha . $$

The principal chooses p =  $\bar{p}$ and α to

$$ P:\quad \mathop{\max}\limits_{{p,\alpha }} \left[ {\left( {1-(1+\mu )p+\alpha } \right)p-\frac{{{\alpha^2}}}{2}-\frac{r}{2}{\alpha^2}{\sigma^2}} \right], $$

$$ \mathop{foc}\limits_p: 1-2(1+\mu )p+\alpha =0, {p^{*}}=\frac{{1+\alpha }}{{2(1+\mu )}}, $$

$$ \mathop{foc}\limits_{\alpha }: p-\alpha -r\alpha {\sigma^2}=0. $$

Substituting for p yields:

$$ {\alpha^{*}}=\frac{1}{{2(1+r{\sigma^2})(1+\mu )-1}}, $$

$$ \frac{{d{\alpha^{*}}}}{{d\mu }}=-\frac{{2(1+r{\sigma^2})}}{{{\Gamma^2}}}<0\;\;\;\mathrm{where}\;\;\;\Gamma =2\left( {1+r{\sigma^2}} \right)(1+\mu )-1. $$

1.8 Product Substitution

$$ q=1-p+\delta \bar{p}+a+\varepsilon $$

where $\bar{p}$ is now the price at a rival chain.

$$ A:\quad \mathop{\max}\limits_a \left[ {\alpha (1-p+\delta \bar{p}+a)+W-\frac{{{a^2}}}{2}-\frac{r}{2}{\alpha^2}{\sigma^2}} \right],\quad foc: a=\alpha . $$

$$ P:\quad \mathop{\max}\limits_{{p,\alpha }} \left[ {\left( {1-p+\delta \bar{p}+\alpha } \right)p-\frac{{{\alpha^2}}}{2}-\frac{r}{2}{\alpha^2}{\sigma^2}} \right], $$

$$ \mathop{foc}\limits_p: 1-2p+\delta \bar{p}+\alpha =0. $$

Using symmetry to set p =  $\bar{p}$ yields:

$$ {p^{*}}=\frac{{1+\alpha }}{{2-\delta }}. $$

$$ \mathop{foc}\limits_{\alpha }: p-\alpha -r\alpha {\sigma^2}=0. $$

Substituting for p yields:

$$ {\alpha^{*}}=\frac{1}{{(1+r{\sigma^2})(2-\delta )-1}}, $$

$$ \frac{{d{\alpha^{*}}}}{{d\delta }}=-\frac{{(1+r{\sigma^2})}}{{{\Gamma^2}}}>0,\;\;\;\mathrm{where}\;\;\;\Gamma =\left( {1+r{\sigma^2}} \right)(2-\delta )-1. $$

1.9 Strategic Delegation of the Pricing Decision

$$ q=1-p+\delta \bar{p}+a+\varepsilon, $$

where $\bar{p}$ is again the price at a rival chain. In this case, the agent is compensated by residual claims after he pays a royalty ρ per unit to the franchisor, as well as a franchise fee F. Thus we have:

$$ A:\quad \mathop{\max}\limits_{a,p } \left[ {(p-\rho )(1-p+\delta \bar{p}+a)-F-\frac{{{a^2}}}{2}-\frac{r}{2}{{{(p-\rho )}}^2}{\sigma^2}} \right] $$

$$ \mathop{foc}\limits_a: a=p-\rho, $$

$$ \mathop{foc}\limits_p: 1-2p+\delta \bar{p}+a+\rho -r(p-\rho ){\sigma^2}=0. $$

Substituting for a yields:

$$ p=\frac{{1+\delta \bar{p}+r\rho {\sigma^2}}}{{1+r{\sigma^2}}}, $$

$$ \frac{dp }{{d\bar{p}}}=\frac{\delta }{\Gamma}>0, \frac{dp }{{d\delta }}=\frac{\bar{p}}{\Gamma}>0,\ \frac{dp }{dr }=\frac{{{\sigma^2}(\rho -1-\delta \bar{p})}}{{{\Gamma^2}}}<0, $$

$$ \frac{dp }{{d{\sigma^2}}}=\frac{{r(\rho -1-\delta \bar{p})}}{{{\Gamma^2}}}<0,\ \frac{dp }{{d\rho }}=\frac{{r{\sigma^2}}}{\Gamma}>0, $$

where $\Gamma =1+r{\sigma^2}.$ Using symmetry to set p =  $\bar{p}$ yields:

$$ p_D^{*} =\frac{{1+r\rho {\sigma^2}}}{{1-\delta+r{\sigma^2}}} \\ =\frac{1}{{1-\delta}}\;\;\;\;\mathrm{when}\;\;\;r{\sigma^2}=0.\\$$

By contrast, under vertical integration, assuming that a = 0 and ρ = 0, we have

$$ \begin{array}{llll} p_D^{*} & =\frac{{1+r\rho {\sigma^2}}}{{1-\delta +r{\sigma^2}}} \hfill \\ & =\frac{1}{{1-\delta }}\;\;\;\;\mathrm{when}\;\;\;r{\sigma^2}=0. \hfill \\ \end{array} $$

Setting p =  $\bar{p}$ yields $p_I^{*}=\frac{1}{{2-\delta }} <1< \frac{1}{{1-\delta }}=p_D^{*}.$ Thus $p_I^{*}<p_D^{*}$ when $r{\sigma^2}$ is small.