How do you compare the elasticity of substitution and the returns to scale from isoquants?

1 Elasticity of substitution

The elasticity of substitution measures how easily one input can be substituted for another, holding the output constant. It is defined as the percentage change in the ratio of inputs divided by the percentage change in the marginal rate of technical substitution (MRTS), which is the slope of the isoquant. A high elasticity of substitution means that inputs are easily substitutable, and a low elasticity means that inputs are complementary. The shape of the isoquant reflects the elasticity of substitution: the more curved the isoquant, the lower the elasticity, and the more linear the isoquant, the higher the elasticity.

2 Returns to scale

The returns to scale measure how the output changes when all inputs are increased proportionally. It is defined as the percentage change in output divided by the percentage change in inputs. There are three types of returns to scale: increasing, decreasing, and constant. Increasing returns to scale mean that output increases more than proportionally to inputs, decreasing returns to scale mean that output increases less than proportionally to inputs, and constant returns to scale mean that output increases proportionally to inputs. The shape of the isoquant map reflects the returns to scale: the closer the isoquants are to each other, the lower the returns to scale, and the farther the isoquants are from each other, the higher the returns to scale.

3 Comparing elasticity and returns

To compare the elasticity of substitution and the returns to scale from isoquants, we can use two methods: the ray method and the expansion path method. The ray method involves drawing a ray from the origin through a point on an isoquant, and observing how the MRTS changes along the ray. The expansion path method involves finding the optimal input mix for each level of output, and tracing a curve that connects these points. The ray method shows how the elasticity of substitution varies with the input ratio, while the expansion path method shows how the elasticity of substitution varies with the output level. The ray method also shows how the returns to scale vary with the input ratio, while the expansion path method shows how the returns to scale vary with the output level.

4 Optimal input mix

The optimal input mix is the combination of inputs that minimizes the cost of production for a given level of output. It is determined by the point where an isoquant is tangent to an isocost line, which is a line that shows all the combinations of inputs that have the same total cost. The slope of the isocost line is equal to the ratio of input prices, and the slope of the isoquant is equal to the MRTS. Therefore, the optimal input mix satisfies the condition that the MRTS is equal to the ratio of input prices. This condition ensures that the marginal product per dollar of each input is equalized.

5 Cost minimization

The cost minimization problem is to find the optimal input mix that minimizes the total cost of production for a given level of output. It can be solved by using either the graphical method or the mathematical method. The graphical method involves finding the point of tangency between an isoquant and an isocost line, as explained above. The mathematical method involves setting up a Lagrangian function that incorporates the production function, the total cost function, and a constraint that equates output to a given level. Then, taking the first-order conditions and solving for the optimal input levels and the Lagrange multiplier, which represents the marginal cost of production.

6 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?