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Consider a production function of the following form: Q(KL)=5K/12 a) What would you call this particular type of functional form? b) Is there any assumption which you would usually make regarding K and L, and why is this economically sensible? c) Setequal to 10 and rearrange the function so that it is in the form K = S(L). What does this function represent? d) Using your answer from e) find an expression for the marginal rate of substitution (MRS) between K and L, and explicitly state, given the assumptions you made in

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Final Answer:

a) The given production function,
\(Q(K,L) = (5K)/(12)\), represents a Cobb-Douglas functional form.

b) The Cobb-Douglas functional form assumes constant returns to scale, implying that the exponents on capital
(\(K\)) and labor (\(L\)) sum to 1. This assumption is economically sensible as it allows for flexibility in adjusting inputs while maintaining a consistent output scale.

c) Setting
\(Q = 10\) and rearranging the function, we get \(K = (24)/(5)L\).

Explanation:

a) The Cobb-Douglas functional form is characterized by a production function of the form
\(Q(K, L) = A K^(\alpha) L^(1-\alpha)\), where \(A\) is a positive constant and
\(\alpha\) is the output elasticity of capital. In the given production function
\(Q(K, L) = (5K)/(12)\), the exponents sum to 1 (\(\alpha = (1)/(2), 1-\alpha = (1)/(2)\)), confirming it as a Cobb-Douglas form.

b) The assumption of constant returns to scale is integral to the Cobb-Douglas model. It means that if both inputs, capital
(\(K\)) and labor (\(L\)), are increased by a certain factor, output
(\(Q\)) will increase by the same factor. This assumption aligns with economic intuition, as it allows for easy interpretation of the impact of input changes on output.

c) Setting
\(Q = 10\) and rearranging the given function \(Q(K, L) = (5K)/(12)\) results in \(K = (24)/(5)L\). This expression represents a linear relationship between capital
(\(K\)) and labor (\(L\)) with a slope of \((24)/(5)\).

In conclusion, the Cobb-Douglas functional form provides a useful framework for analyzing production relationships, and the constant returns to scale assumption simplifies the interpretation of input-output dynamics in economic production models. The rearranged function
\(K = (24)/(5)L\) expresses the proportional relationship between capital and labor in the given production context.

User Jeremiah Winsley
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