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Homework answers / question archive / Consider the production function for tomatoes: y = AFaSb, where y is the quantity of tomatoes, F is the quantity of fertilizer and S is the number of sprayings of pesticide during the growing season

Consider the production function for tomatoes: y = AFaSb, where y is the quantity of tomatoes, F is the quantity of fertilizer and S is the number of sprayings of pesticide during the growing season

Economics

Consider the production function for tomatoes: y = AFaSb, where y is the quantity of tomatoes, F is the quantity of fertilizer and S is the number of sprayings of pesticide during the growing season. All other inputs are fixed and represented by A > 0. The constants a and b are each positive, but each is less than 1.

1. Determine the returns to scale. (2 points)
2. Find the marginal product of fertilizer, and then determine how it responds to the quantity of sprayings by finding

and interpreting the relevant derivative. (3 points)
3. Derive the elasticity of output with respect to sprayings? (2 points)

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Solution:

Given that,

\rightarrow Consider the production function for tomatoes: y = AFaSb, where y is the quantity of tomatoes, F is the quantity of fertilizer and S is the number of sprayings of pesticide during the growing season.

\rightarrowAll other inputs are fixed and represented by A > 0. The constants a and b are each positive, but each is less than 1.

i.e,

Production function, y = A*F^a * S^b

(a).

Suppose both quantity of fertilizers and frequency of spraying are increased by proportion, t, then , new output,

y' = A*(tF)^a (tS)^b = t^(a+b) A*F^a * S^b = t^(a+b)*y

If a + b > 1, then the output increases by more than proportion, t, hence, production function exhibits increasing returns to scale.

If a + b < 1, then the output increases by less than proportion, t, hence, production function exhibits decreasing returns to scale.

If a + b = 1, then the output increases by equal to proportion, t, hence, production function exhibits constant returns to scale.

(b).

Marginal product of fertilizer, MPF = dy/dF = a*A*F^(a-1) * S^b

To determine how MPF responds to increase in frequency of spraying,

dMPF/dS = a*b*A F^(a-1) S^(b-1)

As both a and b are positive constants, so, the marginal product of fertilizers increases with increase in frequency of spraying.

(c).

Elasticity of output with respect to spraying = (dy/dS)*S/y = (b*A*F^a S^(b-1))*S/(A*F^a S^b) = b

Thus, elasticity of output with respect to spraying is b.