Question

Administrative Intensity A study defined administrative intensity as the number of administrative personnel $A$ per production personnel $X,$ or $I=A / X .$ To seek the optimum administrative intensity for an industry, the following Cobb- Douglas production function was defined: $Q=C S^{\sigma} A^{\alpha} K^{\beta} X^{\gamma}$ where $S$ is the number of distinct occupational specialties repre- sented in the labor force, $\sigma$ is the marginal relative productivity of specialization, $K$ is the total productive capital, and $C$ is a constant of proportionality. Source: Administrative Science Quarterly. (a) Unlike the production functions described in this chapter, for which $\alpha+\beta+\gamma=1,$ the study assumed that in- creasing cumulative control losses result in decreasing re- turns to scale, so that $\alpha+\beta+\gamma=1-\lambda$ $\lambda$ is the control loss parameter. The study also defined $k=K / X$ as the capital intensity per worker. Show that the production function can then be rewritten as $Q=C S^{\sigma} I^{\alpha} k^{\beta} X^{1-\lambda}$ (b) The study showed that the profit $P$ can be written as $P=(p-r) Q-W_{0} S-W_{1} A-F$ where $p$ is the market price per unit, $r$ is the production cost per unit, $W_{0}$ is the average wage per production worker, $W_{1}$ is the average salary per administrative worker, and $F$ is the fixed costs. Letting $A=X I$ in the equation for $P,$ show that $\partial P / \partial I=0$ (which leads to the maximum profit) when the administrative intensity is given by $I=\left(\frac{\alpha(p-r) C S^{\sigma} k^{\beta}}{W_{1} X^{\lambda}}\right)^{1 /(1-\alpha)}$ c) Marginal productivity of administration is given by $\partial Q / \partial A$ . Show that this can be written as $\frac{\partial Q}{\partial A}=\alpha C S^{\sigma} A^{\alpha-1} k^{\beta} X^{1-\lambda-\alpha}$ $$\begin{array}{c}{\text { (d) If } x \text { and } y \text { are functions of } z, \text { and } f \text { is a function of } x, y, \text { and }} \\ {z, \text { the total differential for three variables (to be discussed }} \\ {\text { later in this chapter) can be converted to the chain rule for }} \\ {\text { three variables: }} \\ {\frac{d w}{d z}=f_{x}(x, y, z) \frac{d x}{d z}+f_{y}(x, y, z) \frac{d y}{d z}+f_{z}(x, y, z)}\end{array}$$ $$\begin{array}{c}{\text { which gives the total rate of change of } w \text { with respect to } z} \\ {\text { Use this result plus the result from part (c) to show that the }} \\ {\text { total rate of change of the marginal productivity of adminis- }} \\ {\text { tration with respect to the number of workers can be given as }} \\ {\frac{d}{d X}\left(\frac{\partial Q}{\partial A}\right)=\left(\frac{\partial Q}{\partial A}\right)\left[\frac{\sigma}{S}\left(\frac{d S}{d X}\right)+\frac{\beta}{k}\left(\frac{d k}{d X}\right)-\frac{\lambda}{X}\right]}\end{array}$$ when $S$ and $k$ are both functions of $X$

Name: Problem 54, Chapter 9: Calculus with Applications
Uploaded: 2019-04-21T08:38:08-08:00
Duration: 24 min 15 s
Channel: Sid Wan

   Administrative Intensity A study defined administrative intensity as the number of administrative personnel $A$ per production personnel $X,$ or $I=A / X .$ To seek the optimum administrative intensity for an industry, the following Cobb-
Douglas production function was defined: $Q=C S^{\sigma} A^{\alpha} K^{\beta} X^{\gamma}$
where $S$ is the number of distinct occupational specialties repre-
sented in the labor force, $\sigma$ is the marginal relative productivity of
specialization, $K$ is the total productive capital, and $C$ is a constant
of proportionality. Source: Administrative Science Quarterly.
(a) Unlike the production functions described in this chapter,
for which $\alpha+\beta+\gamma=1,$ the study assumed that in-
creasing cumulative control losses result in decreasing re-
turns to scale, so that
$\alpha+\beta+\gamma=1-\lambda$
$\lambda$ is the control loss parameter. The study also
defined $k=K / X$ as the capital intensity per worker. Show
that the production function can then be rewritten as
$Q=C S^{\sigma} I^{\alpha} k^{\beta} X^{1-\lambda}$
(b) The study showed that the profit $P$ can be written as
$P=(p-r) Q-W_{0} S-W_{1} A-F$
where $p$ is the market price per unit, $r$ is the production cost
per unit, $W_{0}$ is the average wage per production worker, $W_{1}$
is the average salary per administrative worker, and $F$ is
the fixed costs. Letting $A=X I$ in the equation for $P,$ show
that $\partial P / \partial I=0$ (which leads to the maximum profit) when
the administrative intensity is given by
$I=\left(\frac{\alpha(p-r) C S^{\sigma} k^{\beta}}{W_{1} X^{\lambda}}\right)^{1 /(1-\alpha)}$
c) Marginal productivity of administration is given by $\partial Q / \partial A$ .
Show that this can be written as
$\frac{\partial Q}{\partial A}=\alpha C S^{\sigma} A^{\alpha-1} k^{\beta} X^{1-\lambda-\alpha}$
$$\begin{array}{c}{\text { (d) If } x \text { and } y \text { are functions of } z, \text { and } f \text { is a function of } x, y, \text { and }} \\ {z, \text { the total differential for three variables (to be discussed }} \\ {\text { later in this chapter) can be converted to the chain rule for }} \\ {\text { three variables: }} \\ {\frac{d w}{d z}=f_{x}(x, y, z) \frac{d x}{d z}+f_{y}(x, y, z) \frac{d y}{d z}+f_{z}(x, y, z)}\end{array}$$
$$\begin{array}{c}{\text { which gives the total rate of change of } w \text { with respect to } z} \\ {\text { Use this result plus the result from part (c) to show that the }} \\ {\text { total rate of change of the marginal productivity of adminis- }} \\ {\text { tration with respect to the number of workers can be given as }} \\ {\frac{d}{d X}\left(\frac{\partial Q}{\partial A}\right)=\left(\frac{\partial Q}{\partial A}\right)\left[\frac{\sigma}{S}\left(\frac{d S}{d X}\right)+\frac{\beta}{k}\left(\frac{d k}{d X}\right)-\frac{\lambda}{X}\right]}\end{array}$$
when $S$ and $k$ are both functions of $X$

Calculus with Applications

Margaret L. Lial •… 11th Edition

Chapter 9, Problem 54

Instant Answer

Solved by Expert Sid Wan

04/21/2019

Step 1

We want to rewrite the production function in terms of these new variables. We can substitute $A=IX$ and $K=kX$ into the production function to get \begin{align*} Q &= C S^{\sigma} (IX)^{\alpha} (kX)^{\beta} X^{\gamma} \\ &= C S^{\sigma} I^{\alpha} k^{\beta} Show more…

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Administrative Intensity A study defined administrative intensity as the number of administrative personnel $A$ per production personnel $X,$ or $I=A / X .$ To seek the optimum administrative intensity for an industry, the following Cobb- Douglas production function was defined: $Q=C S^{\sigma} A^{\alpha} K^{\beta} X^{\gamma}$ where $S$ is the number of distinct occupational specialties repre- sented in the labor force, $\sigma$ is the marginal relative productivity of specialization, $K$ is the total productive capital, and $C$ is a constant of proportionality. Source: Administrative Science Quarterly. (a) Unlike the production functions described in this chapter, for which $\alpha+\beta+\gamma=1,$ the study assumed that in- creasing cumulative control losses result in decreasing re- turns to scale, so that $\alpha+\beta+\gamma=1-\lambda$ $\lambda$ is the control loss parameter. The study also defined $k=K / X$ as the capital intensity per worker. Show that the production function can then be rewritten as $Q=C S^{\sigma} I^{\alpha} k^{\beta} X^{1-\lambda}$ (b) The study showed that the profit $P$ can be written as $P=(p-r) Q-W_{0} S-W_{1} A-F$ where $p$ is the market price per unit, $r$ is the production cost per unit, $W_{0}$ is the average wage per production worker, $W_{1}$ is the average salary per administrative worker, and $F$ is the fixed costs. Letting $A=X I$ in the equation for $P,$ show that $\partial P / \partial I=0$ (which leads to the maximum profit) when the administrative intensity is given by $I=\left(\frac{\alpha(p-r) C S^{\sigma} k^{\beta}}{W_{1} X^{\lambda}}\right)^{1 /(1-\alpha)}$ c) Marginal productivity of administration is given by $\partial Q / \partial A$ . Show that this can be written as $\frac{\partial Q}{\partial A}=\alpha C S^{\sigma} A^{\alpha-1} k^{\beta} X^{1-\lambda-\alpha}$ $$\begin{array}{c}{\text { (d) If } x \text { and } y \text { are functions of } z, \text { and } f \text { is a function of } x, y, \text { and }} \\ {z, \text { the total differential for three variables (to be discussed }} \\ {\text { later in this chapter) can be converted to the chain rule for }} \\ {\text { three variables: }} \\ {\frac{d w}{d z}=f_{x}(x, y, z) \frac{d x}{d z}+f_{y}(x, y, z) \frac{d y}{d z}+f_{z}(x, y, z)}\end{array}$$ $$\begin{array}{c}{\text { which gives the total rate of change of } w \text { with respect to } z} \\ {\text { Use this result plus the result from part (c) to show that the }} \\ {\text { total rate of change of the marginal productivity of adminis- }} \\ {\text { tration with respect to the number of workers can be given as }} \\ {\frac{d}{d X}\left(\frac{\partial Q}{\partial A}\right)=\left(\frac{\partial Q}{\partial A}\right)\left[\frac{\sigma}{S}\left(\frac{d S}{d X}\right)+\frac{\beta}{k}\left(\frac{d k}{d X}\right)-\frac{\lambda}{X}\right]}\end{array}$$ when $S$ and $k$ are both functions of $X$

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Key Concepts

Cobb-Douglas Production Function

The Cobb-Douglas production function is a widely used mathematical representation of the relationship between inputs and output in production. It expresses output as a product of inputs raised to constant elasticities, which reflect the contribution of each factor. This form allows economists to analyze how varying the amount of each input affects total production and to study the substitution effects and scaling properties of the production process.

Returns to Scale

Returns to scale describe how output changes when all inputs are increased proportionally. In the context of a Cobb-Douglas production function, the sum of the exponents on the inputs indicates the type of returns to scale: if the sum equals one, there are constant returns to scale; if it is less than one, decreasing returns to scale are present. Decreasing returns to scale imply that proportional increases in all inputs lead to a less than proportional increase in output, reflecting inefficiencies such as increased coordination or control losses.

Profit Maximization

Profit maximization is a fundamental objective in economics, where firms aim to maximize the difference between total revenue and total costs. This process typically involves setting the derivative of the profit function with respect to a decision variable to zero to find the optimal level of that variable. It is essential for determining the most efficient levels of input use, as firms compare marginal benefits and marginal costs to achieve the highest possible profit.

Marginal Productivity

Marginal productivity refers to the additional output that is produced as a result of a one-unit increase in an input, holding all other inputs constant. This concept is key in resource allocation decisions because it helps in identifying the point at which the cost of an additional unit of input equals the revenue generated by the extra output. It is also used to analyze the efficiency of different production factors in contributing to the overall output.

Total Differentiation in Multivariate Functions

Total differentiation is a technique in calculus used to analyze the effect of changes in multiple independent variables on a function that depends on all of them. By applying the chain rule, total differentiation expresses the overall rate of change of the function as a sum of the partial derivatives with respect to each variable multiplied by the rate at which each variable changes. This method is crucial in economic analysis for understanding interdependencies between variables and the cumulative impact of simultaneous changes on outcomes such as marginal productivity.

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