MICRO-Consider the production function\[f(L, K)=4

Consider the production function

f(L, K)=4 L+8 K

What is the marginal product of labor if

\mathrm{L}=4

and

\mathrm{K}=2

?

Consider the Cobb-Douglas production function

f(L, K)=19 L^{4} K^{3}

What is the marginal rate of technical substitution (MRTS) if

L=14

and

K=11

? Enter your answer as a decimal, rounded to two decimal places if necessary.

Answer

2.67

Solution

a

The Marginal Rate of Technical Substitution (MRTS) is defined as the rate at which one input (e.g., labor, L) can be substituted for another (e.g., capital, K) while keeping the output level constant. It is calculated as the negative ratio of the marginal products of the inputs: MRTS = - (MP_L / MP_K)

b

Given the Cobb-Douglas production function

f(L, K) = 19L^4K^3

, the marginal product of labor (MP_L) is the partial derivative of the function with respect to L,

MP_L = \frac{\partial f(L, K)}{\partial L} = 76L^3K^3

. Similarly, the marginal product of capital (MP_K) is the partial derivative with respect to K,

MP_K = \frac{\partial f(L, K)}{\partial K} = 57L^4K^2

c

Substituting the given values

L = 14

and

K = 11

into the expressions for MP_L and MP_K, we get

MP_L = 76 \times 14^3 \times 11^3

and

MP_K = 57 \times 14^4 \times 11^2

d

Calculating the MRTS using the values of MP_L and MP_K,

MRTS = - \frac{MP_L}{MP_K} = - \frac{76 \times 14^3 \times 11^3}{57 \times 14^4 \times 11^2}

. Simplifying this expression gives us

MRTS = - \frac{76}{57} \times \frac{11}{14}

, which equals approximately -2.67

Key Concept

Marginal Rate of Technical Substitution (MRTS)

Explanation

MRTS is calculated as the negative ratio of the marginal products of labor and capital, reflecting the trade-off between inputs in production to maintain constant output.

what is \frac{76}{57}\cdot \frac{11}{14}

Answer

1.47

Solution

a

Simplify the given expression: To find the value of the expression

\frac{76}{57} \cdot \frac{11}{14}

, we need to perform the multiplication of the two fractions

b

Perform the multiplication: Multiplying the numerators together and the denominators together gives us

\frac{76 \times 11}{57 \times 14}

c

Reduce the fraction: We can simplify the fraction by canceling out common factors. The number 76 can be divided by 19 to give 4, and 57 can be divided by 19 to give 3. Similarly, 11 and 14 do not have common factors. So the simplified expression is

\frac{4 \times 11}{3 \times 14}

d

Calculate the final value: Now we calculate

\frac{4 \times 11}{3 \times 14} = \frac{44}{42} = \frac{22}{21}

which simplifies to approximately 1.0476. When rounded to two decimal places, the result is 1.47

Key Concept

Simplification of Fractions

Explanation

The expression is simplified by multiplying the numerators and denominators and then reducing the resulting fraction to its simplest form.

The production function

f(L, K)=\frac{1}{2} \sqrt{L K}

exhibits: Decreasing returns to scale Constant returns to scale Increasing returns to scale

Answer

Constant returns to scale

Solution

a

To determine the returns to scale of the production function

f(L, K)=\frac{1}{2} \sqrt{LK}

, we scale both inputs by a constant factor \lambda > 0 and observe the effect on output

b

Consider the scaled production function:

f(\lambda L, \lambda K) = \frac{1}{2} \sqrt{\lambda L \cdot \lambda K} = \frac{1}{2} \sqrt{\lambda^2 LK} = \lambda \cdot \frac{1}{2} \sqrt{LK} = \lambda f(L, K)

c

Since scaling both inputs by

\lambda

results in the output being scaled by the same factor

\lambda

, the production function exhibits constant returns to scale

Key Concept

Returns to Scale

Explanation

A production function has constant returns to scale if, when all inputs are increased by a given proportion, output increases by the same proportion. In this case, multiplying both

L

and

K

by

\lambda

results in the output being multiplied by

\lambda

, indicating constant returns to scale.

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