Chapter 19 Profit Maximization
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Transcript of Chapter 19 Profit Maximization
• 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 trade-off 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