true/false

1. a firm uses only two inputs to produce its output. these inputs are perfect substitutes. this firm could have increasing returns to scale, constant returns to scale, or decreasing returns to scale.

2. a firm has the production function (f(x,y) = x^{1.40}y^1). this firm has diminishing marginal products for factor (x).

3. a firm has two variable factors and a production function (f(x_1,x_2) = (2x_1 + 4x_2)^{1/2}). the marginal rate of technical substitution between (x_1) and (x_2) is constant.

4. if there are constant returns to scale, then doubling the amount of any input will exactly double the amount of output.

5. if the production function is (f(x,y) = \min\{2x+y, x+2y\}), then there are constant returns to scale.

6. it is possible to have decreasing marginal products for all inputs, and yet have increasing returns to scale.

Question

true/false

1. a firm uses only two inputs to produce its output. these inputs are perfect substitutes. this firm could have increasing returns to scale, constant returns to scale, or decreasing returns to scale.

2. a firm has the production function (f(x,y) = x^{1.40}y^1). this firm has diminishing marginal products for factor (x).

3. a firm has two variable factors and a production function (f(x_1,x_2) = (2x_1 + 4x_2)^{1/2}). the marginal rate of technical substitution between (x_1) and (x_2) is constant.

4. if there are constant returns to scale, then doubling the amount of any input will exactly double the amount of output.

5. if the production function is (f(x,y) = \min\{2x+y, x+2y\}), then there are constant returns to scale.

6. it is possible to have decreasing marginal products for all inputs, and yet have increasing returns to scale.

Accepted Answer

{ "quality": "clear", "question_count": 6, "skills_matched": [ "step_cot", "table_formatter", "explain_then_answer", "tutor" ], "needs_drawing": false, "concepts_used": [ "Production Functions", "Returns to Scale", "Technical Rate of Substitution", "Diminishing Marginal Product" ], "new_concepts": [], "current_concepts": [ "Production Functions", "Returns to Scale", "Technical Rate of Substitution", "Diminishing Marginal Product" ] } ## Analyze perfect substitutes and returns to scale Using the **Production Functions** and **Returns to Scale** knowledge points: Perfect substitutes mean the production function has the form $f(x_1, x_2) = g(a x_1 + b x_2)$ for constants $a, b > 0$. - If $g(u) = u^2$, then $f(tx_1, tx_2) = (a(tx_1) + b(tx_2))^2 = t^2 f(x_1, x_2)$, which exhibits increasing returns to scale for $t > 1$. - If $g(u) = u$, then $f(tx_1, tx_2) = t f(x_1, x_2)$, which exhibits constant returns to scale. - If $g(u) = u^{1/2}$, then $f(tx_1, tx_2) = t^{1/2} f(x_1, x_2)$, which exhibits decreasing returns to scale. Thus, a firm with perfect substitute inputs can exhibit any of these returns to scale. Statement 1 is True. ## Analyze marginal product of factor x Using the **Diminishing Marginal Product** and **Partial Derivatives** knowledge points: The production function is $f(x, y) = x^{1.40} y^1$. The marginal product of $x$ is the first partial derivative: $$MP_x = \frac{\partial f}{\partial x} = 1.40 x^{0.40} y$$ To check for diminishing marginal product, we find the second partial derivative with respect to $x$: $$\frac{\partial^2 f}{\partial x^2} = 0.56 x^{-0.60} y$$ Since $x, y > 0$, the second derivative is positive ($\frac{\partial^2 f}{\partial x^2} > 0$), meaning the marginal product of $x$ is increasing, not diminishing. Statement 2 is False. ## Analyze technical rate of substitution Using the **Technical Rate of Substitution** knowledge point: The production function is $f(x_1, x_2) = (2x_1 + 4x_2)^{1/2}$. The marginal rate of technical substitution (MRTS) is: $$MRTS = \frac{\partial f / \partial x_1}{\partial f / \partial x_2} = \frac{\frac{1}{2}(2x_1 + 4x_2)^{-1/2} \cdot 2}{\frac{1}{2}(2x_1 + 4x_2)^{-1/2} \cdot 4} = \frac{2}{4} = \frac{1}{2}$$ Since the MRTS is a constant value ($1/2$) and does not depend on $x_1$ or $x_2$, it is constant. Statement 3 is True. ## Analyze definition of constant returns to scale Using the **Returns to Scale** knowledge point: Constant returns to scale means that scaling *all* inputs by a factor $t$ scales the output by exactly $t$: $f(t x_1, t x_2) = t f(x_1, x_2)$. Doubling only *one* input (e.g., $f(2x_1, x_2)$) does not generally double output unless the other input is not used or is 0. Thus, doubling the amount of *any* single input does not double output. Statement 4 is False. ## Analyze returns to scale of minimum function Using the **Returns to Scale** knowledge point: The production function is $f(x, y) = \min\{2x + y, x + 2y\}$. Let us scale both inputs by $t > 0$: $$f(tx, ty) = \min\{2(tx) + (ty), (tx) + 2(ty)\} = \min\{t(2x + y), t(x + 2y)\} = t \min\{2x + y, x + 2y\} = t f(x, y)$$ Since $f(tx, ty) = t f(x, y)$, the function exhibits constant returns to scale. Statement 5 is True. ## Analyze relationship between marginal products and returns to scale Using the **Diminishing Marginal Product** and **Returns to Scale** knowledge points: Consider a Cobb-Douglas production function $f(x, y) = x^a y^b$ with $a, b > 0$. - The marginal products are diminishing if $a < 1$ and $b < 1$. - The returns to scale are increasing if $a + b > 1$. If we choose $a = 0.6$ and $b = 0.6$, both inputs have diminishing marginal products because $0.6 < 1$. However, $a + b = 1.2 > 1$, which means the production function exhibits increasing returns to scale. Thus, it is possible. Statement 6 is True. | No. | Answer | |---|---| | 1 | True | | 2 | False | | 3 | True | | 4 | False | | 5 | True | | 6 | True | { "subject": "Business", "question_type": "Multi-part", "knowledge_point": [ "Business", "Microeconomics", "Production Theory" ] }

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Analyze perfect substitutes and returns to scale

Analyze marginal product of factor x

Analyze technical rate of substitution

Analyze definition of constant returns to scale

Analyze returns to scale of minimum function

Analyze relationship between marginal products and returns to scale

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