Substitution elasticity

In microeconomics , the substitution elasticity is a special elasticity and, as such, a measure that indicates the relative change of a dependent variable to a relative change of one of its independent variables. It takes into account the idea of ​​substitution, mostly in production and utility theory, which means that in consumption or production, some goods or production factors can to a certain extent be exchanged for one another (see substitutional production technology or substitute good ).

definition

Isoquants two inputs (X and Y) and three output levels ( , , ). The elasticity of substitution can be interpreted geometrically as the measure of curvature of the isoquant in the relevant area. ${\displaystyle Q_{1}}$ ${\displaystyle Q_{2}}$ ${\displaystyle Q_{3}}$

In the context of production theory, such a quotient describes the relative change in the factor input ratio to the relative change in the factor price ratio. Assume that a production function depends on two inputs (labor and capital), for example (this function is continuously differentiable). From a formal point of view: ${\displaystyle Q(x_{1},x_{2})}$

${\displaystyle \sigma =-{\frac {\mathrm {d} \,\left({\frac {x_{1}}{x_{2}}}\right)}{\mathrm {d} \,\left({\frac {dx_{1}}{dx_{2}}}\right)}}\cdot {\frac {\frac {\mathrm {d} \,x_{1}}{\mathrm {d} \,x_{2}}}{\frac {x_{1}}{x_{2}}}}}$,

where denotes the production factor and the negative sign only serves to set the absolute value greater than zero, since either the numerator or denominator is always negative. It should be noted that ${\displaystyle x_{i}}$ ${\displaystyle i}$

${\displaystyle {\frac {\mathrm {d} \,x_{1}}{\mathrm {d} \,x_{2}}}}$

corresponds to the marginal rate of substitution (MRS), which under the assumption of perfect competition (prices are assumed as given) is optimally equal to the price ratio of the input factors:

${\displaystyle {\frac {\mathrm {d} \,x_{1}}{\mathrm {d} \,x_{2}}}=-{\frac {p_{2}}{p_{1}}}}$.

From the latter it follows that the elasticity of substitution in the optimum indicates the percentage by which the quantitative ratio between two production factors (e.g. labor and capital input) changes if the price ratio between the corresponding production factors changes by one percent:

${\displaystyle \sigma =-{\frac {\mathrm {d} \,\left({\frac {x_{1}}{x_{2}}}\right)}{\mathrm {d} \,\left({\frac {p_{2}}{p_{1}}}\right)}}\cdot {\frac {\frac {p_{2}}{p_{1}}}{\frac {x_{1}}{x_{2}}}}}$.

The relationships apply analogously if one considers a household that optimizes its utility. The elasticity of substitution then indicates by how many percent the consumption ratio changes in the optimum with a constant level of use between two quantities of goods, if the price ratio between the corresponding goods changes by one percent. The larger the goods, the easier they can be substituted. ${\displaystyle \sigma }$

Constant elasticity of substitution

→ Main article : CES function

The so-called CES functions , whose elasticity of substitution is constant, are an important class of functions .

Intertemporal elasticity of substitution

In macroeconomics , intertemporal decision-making situations are to be analyzed that extend the concept of elasticity of substitution to several periods.

The intertemporal elasticity of substitution is the percentage change in the ratio of consumption at two points in time or in relation to the relative change in the slope of the intertemporal indifference curve (here corresponds to the marginal rate of substitution ). ${\displaystyle c(t_{1})/c(t_{2})}$ ${\displaystyle C_{1}/C_{2}}$

${\displaystyle \sigma _{1,2}=-{\frac {\frac {d(C_{1}/C_{2})}{C_{1}/C_{2}}}{\frac {d[U'(C_{1})/U'(C_{2})]}{U'(C_{1})/U'(C_{2})}}}={\frac {d\ln(C_{2}/C_{1})}{d\ln GRS}}}$

Properties of elasticity of substitution ${\displaystyle \sigma }$

The elasticity of substitution is symmetrical. For example

${\displaystyle \sigma _{LK}=\sigma _{KL}}$

The special cases:

• ${\displaystyle \sigma =1}$: Cobb-Douglas function
• ${\displaystyle \sigma =0}$: Leontief production function
• ${\displaystyle \sigma \rightarrow \infty }$: Linear production function.

Empirical research

Information about elasticities is important in order to be able to assess the effects of possible political interventions.

Estimating the elasticity of substitution [σ] is particularly important when analyzing growth and other economic problems.

Empirical estimates for the United States :

• Chirinko (2008) discusses different σ values ​​(see Table 1 'Estimates of the elasticity of substitution' in Chirinko 2008) and points to particular evidence of σ values ​​between 0.40-0.60.
• Young (2013) gives σ values ​​below 0.620 and generally σ <1 for a large majority of SIC in the United States.
• Chirinko and Mallick (2017) say that the substitution values ​​between labor and capital are a crucial prerequisite for understanding the substantial decrease in the labor share of income. Their benchmark estimate of σ is 0.406.
• Knoblach, Roessler and Zwerschke (2019) estimate σ as part of a meta-regression analysis and arrive at σ values ​​= 0.45-0.87.

The determinants of have been insufficiently researched so far. The empirical estimates for other countries, especially for the developing countries, show that there is a lot of catching up to do. ${\displaystyle \sigma }$

Individual evidence

1. ↑ Anton Frantzke: Fundamentals of Economics. Microeconomic theory and tasks of the state in the market economy, Schäffer-Poeschel, Stuttgart, 1999, S80
2. ↑ Schweitzer, M., & Küpper, HU (2013). Production and Cost Theory: Fundamentals — Applications. Springer publishing house. P. 97.
3. ^ Harms, P. (2008). International Macroeconomics. Mohr Siebeck. P. 59.
4. ↑ Christiaans, T. (2004). Neoclassical Growth Theory: Representation, Criticism, and Extension. P. 66.
5. ↑ León-Ledesma, Miguel A., Peter McAdam, and Alpo Willman. 2010. "Identifying the Elasticity of Substitution with Biased Technical Change." American Economic Review, 100 (4): 1330-57. DOI: 10.1257 / aer.100.4.1330
6. ↑ Fahl, Ulrich, ed. Energy models for climate protection in Germany: structural and macroeconomic effects from a national perspective. Vol. 33. Physica-Verlag, 1999. p. 198.
7. ↑ Chirinko, Robert S., 2008. "[sigma]: The long and short of it," Journal of Macroeconomics, Elsevier, vol. 30 (2), pages 671-686, June. < https://ideas.repec.org/a/eee/jmacro/v30y2008i2p671-686.html >
8. ↑ Chirinko, Robert S., 2008. "[sigma]: The long and short of it," Journal of Macroeconomics, Elsevier, vol. 30 (2), pages 671-686, June. < https://ideas.repec.org/a/eee/jmacro/v30y2008i2p671-686.html >
9. ↑ Young, Andrew T., 2013. "US Elasticities Of Substitution And Factor Augmentation At The Industry Level," Macroeconomic Dynamics, Cambridge University Press, vol. 17 (04), pages 861-897, June. < https://ideas.repec.org/a/cup/macdyn/v17y2013i04p861-897_00.html >
10. ↑ Robert S. Chirinko & Debdulal Mallick, 2017. “The Substitution Elasticity, Factor Shares, and the Low-Frequency Panel Model,” American Economic Journal: Macroeconomics, American Economic Association, vol. 9 (4), pages 225-253, October. < https://ideas.repec.org/a/aea/aejmac/v9y2017i4p225-53.html >
11. ^ Knoblach, Michael & Roessler, Martin & Zwerschke, Patrick. (2016). The Elasticity of Factor Substitution Between Capital and Labor in the US Economy: A Meta-Regression Analysis. 10.13140 / RG.2.2.13719.78244. https://www.researchgate.net/publication/316544066_The_Elasticity_of_Factor_Substitution_Between_Capital_and_Labor_in_the_US_Economy_A_Meta-Regression_Analysis
12. ↑ Knoblach, Michael & Stöckl, Fabian. (2019). What determines the elasticity of substitution between capital and labor? A literature review. https://www.researchgate.net/publication/330563679_What_determines_the_elasticity_of_substitution_between_capital_and_labor_A_literature_review

Web links

• Substitution elasticity - Article in the Gabler Wirtschaftslexikon