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\begin{document}
\title{Hirshleifer Ch. 12}
\author[1]{Clara Jace}%
\affil[1]{George Mason University}%
\vspace{-1em}
\date{\today}
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\section*{12.1 Production and Factor Employment with a Single Variable
Input}
{\label{486345}}
Production functions generally take the form
\begin{quote}
\(q \equiv \phi (a, b, c, ...)\)
\end{quote}
That states that the output quantity (q) depends in some specified way
upon the amounts a, b, and c\ldots{} of the resource inputs A, B, and
C\ldots{}
Suppose all inputs are held fixed except for the amount of A. Then we
can write the function as
\begin{quote}
\(q \equiv q(a)\)
\end{quote}
These production functions are often governed by the \emph{Law of
Diminishing Returns:}
\emph{If the amount a of input A increases, with other inputs held
fixed, the rate of increase of Total Product (q) - that is, the marginal
product - eventually begins to fall. This is the point of diminishing
marginal returns. As the input amount increases further, Average Product
also begins to fall. This is the point of diminishing average returns.
And as use of input A rises further, even Total Product may fall (the
case of extreme overabundance hindering production). This would be the
point of diminishing total returns.~}
\subsection*{From Production Function to Cost
Function}
{\label{914250}}
Cost functions reflect the underlying production function with the
prices of inputs, either renting or purchasing. To begin, suppose the
firm is a price-taker (no monopsony power). For any given combination of
inputs, the Total Cost will be
\begin{quote}
\(C \equiv h_a a + h_b b + h_c c + ...\)
\end{quote}
When some factors are held fixed, perhaps all except for A, the Total
Cost can be divided into a fixed component and a variable component
\begin{quote}
\(C \equiv F + V \equiv F + h_a (q)\)
\end{quote}
This states that the Total Cost depends on the fixed cost, the
hire-price, and the production function of A. We can measure the
marginal cost using this equation
\begin{quote}
\(MC \equiv \frac{\Delta C}{\Delta q} = \frac{h_a \Delta a}{\Delta q} = \frac{h_a}{\Delta q / \Delta a} = \frac{h_a}{mp_a}\)
\end{quote}
Here is is clear that the MC necessarily increases as MP decreases. This
explains why the marginal cost curve must eventually rise. We can show
the relationship between average variable cost and average product as
such
\begin{quote}
\(AVC \equiv \frac{V}{q} = \frac{h_a a}{q} = \frac{h_a}{q / a} = \frac{h_a}{ap_a}\)
\end{quote}
As the average product falls, the average cost increases. The
relationship holds even after adding in fixed costs.~
\subsection*{The Firm's Demand for a Single
Variable}
{\label{288862}}
The production function is a technological relation between inputs and
outputs. In deciding upon the amounts of factors to employ, firms must
decide concerning factor prices as well. The firm faces a horizontal
product supply curve at the market price of the input. In deciding
whether or not to employ an additional unit of A, the firm must balance
the hire-price with the benefit gained - both additional physical output
and what that output generates as revenue.
Combining the product price and the marginal product leads to the
concept of Value of the Marginal Product:
\begin{quote}
\(vmp_a \equiv P * mp_a\)
\end{quote}
This curve is the firm's demand for the input, taking the same shape as
the marginal product curve, just shifted up to account for the price.
The factor employment condition for a price-taking firm will thus be
\begin{quote}
\(vmp_a = h_a\)
\end{quote}
Monopolists will use the marginal revenue product -- the marginal
revenue times the physical marginal product.~
\section*{12.2~ Production and Factor Employment with Several
Variables}
{\label{951446}}
With more than one factor to be considered, the graphs of the production
function becomes multi-dimensional. Now there are ``families'' of total
product curves, marginal product curves, and the other relevant
information. They all follow the same shape (for the Law of Diminishing
Returns applies) but now all must be taken into account. The classic
example of this is the Cobb-Douglas production function
\begin{quote}
\(q = \kappa a^\alpha b^\beta\)
\end{quote}
If the sum of the exponents exceeds 1, there will be increasing returns
to scale since the doubling of both inputs more than doubles output.
Exactly equaling 1 is constant returns to scale and decreasing returns
to scale would be less than 1.~
\subsection*{Factor Balance and Factor
Employment}
{\label{921352}}
The question of factor balance asks what are the best input proportions
at any given level of cost or output. This is a preliminary to asking
about factor employment -- the actual amounts of inputs to hire at any
given set of factor hire-prices.~~
Unlike consumers, the firm can decide its budget constraint, or how much
cost to incur. These lines will be intersected by the Scale Expansion
Path (SEP) that shows the best combination of inputs at each level of
cost
\par\null\par\null\par\null\par\null\par\null\par\null
Recall that the consumer's optimum condition is expressed as equality
between the product price ratio~\(\frac{P_x}{P_y}\) (absolute slope of
the consumer's budget line) and the marginal rate of substitution in
consumption (MRSc, the absolute slope of the indifference curve). MRSc
was defined as the ratio at which a person was just willing to
substitute a small amount of Y for a small amount of X in the
consumption basket, leaving the consumer at the same level of utility or
indifference.
Correspondingly here, the marginal rate of substitution in production,
MRSQ, is defined as the mount of input B that can be substituted for a
small change in input A
\begin{quote}
\(MRS_Q \equiv - \frac{\Delta b}{\Delta a} |_q \equiv \frac{mp_a}{mp_b}\)
\end{quote}
The slope of the isocost line is~\(- \frac{h_a}{h_b}\) and the tangency
condition is
\begin{quote}
\(\frac{mp_a}{mp_b} = \frac{h_a}{h_b}\)
\end{quote}
Leading us to the Factor Balance Equation:
\begin{quote}
\(\frac{mp_a}{h_a} = \frac{mp_b}{h_b}\)
\end{quote}
Inputs are balanced when the marginal products per dollar are equal for
all resources employed. This gives us the Scale Expansion Path.~
\subsection*{The Firm's Demand for
Inputs}
{\label{829323}}
Since some inputs are used jointly, the firm's demand for them will be
interrelated. When a firm maximizes profit by equating marginal cost to
marginal revenue, it automatically satisfied the Factor Employment
Conditions:
\begin{quote}
\(mrp_a = h_a\)
\(mrp_b = h_b\)
\end{quote}
Two inputs are complementary if increased use of one raises the marginal
product of the other. They are anticomplementary is vice versa (close
substitutes). These differences give rise to differing elasticities for
the demand curves, which are always flatter than the marginal revenue
product curves.~
\section*{12.3~ The Industry's Demand for
Inputs}
{\label{917780}}
After a fall in hire-price, industry-wide output increases and so
product price falls - thus lessening the firms' incentive to hire more
of the cheapened input A. This product-price effect makes the industry
demand curve steeper than the simple aggregate~ of the individual demand
curves for the factor. On the other hand, the entry-exit effect cuts in
the opposite direction. A fall in hire-price increases firms' profits,
inducing new firms to enter and thereby flattening the industry demand
curve for input A.~
Two contesting models, the competitive model and the monopsonist model,
have been used in analyzing~ minimum-wage laws. If the competitive model
is applicable, so that firms are price-takers in the input market for
labor, a legally imposed minimum wage higher than the previous
equilibrium will raise wages for those workers who remain employed but
disemploy others. If the monopsony model applies, the effects are mixed.
Over a certain range it is even possible that wages adn employment may
both increase. However, there is much evidence accumulated by economists
in favor of the competitive model.~
\par\null\par\null
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