Elasticity of Substitution
Recently, while studying production functions, I revisited a simple yet remarkably insightful concept. I would like to briefly summarize it here.
1. Marginal Rate of Technical Substitution (MRTS)
Before diving into the concept of elasticity of substitution, it would be helpful to first review the marginal rate of technical substitution.
The Marginal Rate of Technical Substitution (MRTS) measures the rate at which a firm can substitute capital for labor while keeping output constant. It is defined as the slope of an isoquant:
$$ MRTS_{LK} = -\frac{dK}{dL}\bigg|_{Q} = \frac{MP_L}{MP_K} $$
Under the cost-minimization condition, the MRTS equals the ratio of input prices: \( MRTS = \frac{w}{r} \), where \( w \) is the wage rate (price of labor) and \( r \) is the rental rate of capital.
2. Elasticity of Substitution
The elasticity of substitution measures how easily one input can be substituted for another while maintaining the same level of output. It is formally defined as the percentage change in the input ratio divided by the percentage change in the MRTS:
$$ \sigma = \frac{d \ln (K/L)}{d \ln (MRTS_{LK})}$$
or else,
$$ \sigma = \frac{d \frac{K}{L} / \frac{K}{L}}{d MRTS / MRTS}$$
Given that the cost-minimization condition holds,
$$ \sigma = \frac{d \frac{K}{L} / \frac{K}{L}}{d MRTS / MRTS} = \frac{d \frac{K}{L} / \frac{K}{L}}{d \frac{w}{r} / \frac{w}{r}} $$
In the Leontief production function, the elasticity of substitution is zero, meaning that inputs are perfect complements. For a Cobb–Douglas production function, \( Q = A L^{\alpha} K^{1-\alpha} \), the elasticity of substitution is constant and equals 1. For a CES production function \( Q = A [ \delta L^{-\rho} + (1-\delta) K^{-\rho} ]^{-1/\rho} \), the elasticity of substitution is given by \( \sigma = \frac{1}{1+\rho} \). For linear production function the elasticity of substitution is infinite, indicating that the inputs are perfect substitutes.
A quick tip: the closer an isoquant is to a straight line, the higher the elasticity of substitution. The more it curves toward the origin, the lower the elasticity.
3. Intertemporal Elasticity of Substitution (IES)
The intertemporal elasticity of substitution (IES) captures how a consumer’s current consumption responds to changes in the real interest rate — that is, how willing individuals are to substitute consumption between periods.
$$ IES = \frac{d \ln (C_{t+1}/C_t)}{d \ln (MRS)} $$
Here, MRS refers to the marginal rate of substitution, which is often represented as \( 1 + r_t \), in intertemporal choice models. Here, \( r_t \) denotes the interest rate.
A higher IES means that people are less willing to smooth their consumption over time.
In the context of the CRRA (Constant Relative Risk Aversion) utility function \( U(C) = \frac{C^{1-\theta}-1}{1-\theta} \), the intertemporal elasticity of substitution is \( \frac{1}{\theta} \), where \( \theta \) represents the coefficient of relative risk aversion.
4. Output Elasticity of an Input
The output elasticity of an input measures the percentage change in output resulting from a one-percent change in that input, holding other inputs constant.
$$ \varepsilon_L = \frac{\partial Q / Q}{\partial L / L} = \frac{MP_L}{AP_L} $$
Here, \( MP_L \) stands for marginal product of labor and \( AP_L \) stands for average product of labor.
Similarly, for capital,
$$ \varepsilon_K = \frac{\partial Q / Q}{\partial K / K} = \frac{MP_K}{AP_K} $$
Also, there is a specific relationship between output elasticity of labor and that of capital. Here, \( r \) is the rental rate of capital.
$$ \varepsilon_L + \varepsilon_K = r $$
To add, In a Cobb–Douglas production function, these elasticities correspond to the exponents on each input (i.e., \( \alpha \) for labor and \( 1-\alpha \) for capital).
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