Transcript • Chapter 19 Technology • First understand the technology constraint

• Chapter 19 Technology
• First understand the technology constraint
of a firm. Later we will talk about
constraints imposed by consumers and
firm’s competitors (the demand curve
faced by the firm, the market structure)
• Inputs: labor and capital
• Inputs and outputs measured in flow units,
i.e., how many units of labor per week,
how many units of output per week, etc.
• Consider the case of one input (x) and
one output (y). To describe the tech
constraint of a firm, list all the
technologically feasible ways to produce
a given amount of outputs.
• The set of all combinations of inputs and
outputs that comprise a technologically
feasible way to produce is called a
production set.
• A production function measures the
maximum possible output that you can
get from a given amount of input.
• Isoquant is another way to express the
production function. It is a set of all
possible combinations of inputs that are
just sufficient to produce a given amount
of output. Isoquant looks very much like
indifference curve, but you cannot label it
arbitrarily, neither can you do any
monotonic transformation of the label.
• Some useful examples of production
function. Two inputs, x1 and x2.
• Fixed proportion (perfect complement):
f(x1,x2)=min{x1,x2}
• Perfect substitutes: f(x1,x2)=x1+x2
• Cobb-Douglas f(x1,x2)=A(x1)a(x2)b, cannot
normalize to a+b=1 arbitrarily
• Some often-used assumptions on the
production function
• Monotonicity: if you increase the amount
of at least one input, you produce at least
as much output as before
• Monotonicity holds because of free
disposal, that is, can free dispose of any
extra inputs
• Convexity: if y=f(x1,x2)=f(z1,z2), then
f(tx1+(1-t)z1,tx2+(1-t)z2)y for any t[0,1]
• Some terms often used to describe the
production function
• Marginal product: operate at (x1,x2),
increase a bit of x1 and hold x2, how much
more y can we get per additional unit of
x1?
• Marginal product of factor 1:
MP1(x1,x2)=∆y/∆x1=(f(x1+ ∆x1,x2)f(x1,x2))/∆x1 (it is a rate, just like MU)
• Marginal rate of technical substitution
factor 1 for factor 2: operate at (x1,x2),
increase a bit of x1 and hold y, how much
less x2 can you use? Measures the tradeoff between two inputs in production
• MRTS1,2(x1,x2)=∆x2/∆x1=?
• ∆y=MP1(x1,x2)∆x1+MP2(x1,x2)∆x2=0
• MRTS1,2(x1,x2)=∆x2/∆x1=MP1(x1,x2)/MP2(x1,x2) (it is a slope, just
like MRS)
• Law of diminishing marginal product:
holding all other inputs fixed, if we
increase one input, the marginal product
of that input becomes smaller and smaller
(diminishing MU)
• Diminishing MRTS: the slope of an
isoquant decreases in absolute value as
we increase x1 (diminishing MRS)
• Short run: at least one factor of
production is fixed
• Long run: all factors of production can be
varied
• Can also plot the short run production
function y=f(x1,k)
• Returns to scale: if we use twice as much
of each input, how much output will we
get?
• constant returns to scale (CRS): for all
t>0, f(tx1,tx2)=tf(x1,x2)
• Idea is if we double the inputs, we can
just set two plants and so we can double
the outputs
• Increasing returns to scale (IRS): for all
t>1, f(tx1,tx2)>tf(x1,x2)
• Decreasing returns to scale (DRS): for all
t>1, f(tx1,tx2)<tf(x1,x2)
• MP, MRTS, returns to scale