Final answer:
The elasticities of supply and demand can be calculated using the derivative of the supply and demand functions multiplied by the price over quantity.
Supply becomes more elastic as price increases, while demand becomes less elastic as price increases.
Step-by-step explanation:
To find the elasticity of supply and demand, we can use the formula for elasticity that involves the natural logarithm. The elasticity formula is defined as the percentage change in quantity over the percentage change in price. Mathematically, the price elasticity of supply (Es) and demand (Ed) can be calculated using the following formulae:
Price Elasticity of Supply (Es):
Es = (dq/dp) × (p/q)
Price Elasticity of Demand (Ed):
Ed = (dq/dp) × (p/q)
Given the supply function q=10p⁴+2p², we can find the derivative with respect to price (dq/dp), which would be dq/dp = 40p³+4p. To calculate elasticity at a given price point, simply plug in the value for p in both the derived supply function and the original supply function and solve for Es.
Similarly, we have the demand function represented by q= 100−p²/p². The derivative of this demand function is slightly tricky due to the term p² in the denominator, but once you work through this, you would plug in the values for p to find Ed.
For our case, when p=1, supply elasticity (Es) is calculated as (40×1³+4×1) × (1/(10×1⁴+2×1²)), and when p=5, Es is calculated similarly by substituting p with 5 in the formula.
As for the demand elasticity, when p=1, demand elasticity (Ed) is calculated by substituting p with 1 in the derived demand function and original demand function, and for p=5, similarly substituting p with 5.
To conclude, supply gets more elastic as price increases because the derivative of the supply function increases at an increasing rate. Conversely, you will find that demand gets less elastic as price increases, since higher prices usually result in a less than proportional change in quantity demanded.