3110 Micro Econ Lecture 2 (Chp 9, 10, 11)

Producer Surplus

Producer surplus is the extra return that producers earn by making transactions at the market price over and above what they would earn if nothing were produced. It is illustrated by the size of the area below the market price and above the supply curve.

Marginal Revenue and Elasticity

*The elasticity of demand is defined as the percentage change in quantity demanded that results from a 1% change in prices.* If elasticity is < -1, MR > 0 If elasticity = -1, MR = 0 If elasticity > -1, MR < 0

Firm, Profit Maximizing Firm

A firm is a combination of production technology, entrepreneur, an inputs used (labor L and capital K). Profit Maximizing Firm: a firm chooses both its inputs and its ouputs with the sole goal of maximizing economic profits, the difference between total revenues and its total economic costs.

Economic Profit

A firm sells some level of output, q, at a market price of p per unit. Total Revenue = p(q) * q where the selling price the firm receives might be impacted by how much it sells. Economic profits = pi(q) = R(q) - C(q) Both revenues and costs depend on the quantity produced, and so does economic profits.

Concavity of the profit function

Concavity of the profit function requires that the production function itself is concave. Diminishing marginal productivity for each input is not enough to ensure increasing marginal costs, because there are interactions between the inputs; this cross productivity must be relatively small.

Chapter 10 Summary:

1) A firm that wishes to minimize the economic costs of producing a particular level of output should choose that input combination for which the RTS of the production function = w/r (the ratio of the input's rental prices) 2) Repeated application of this minimization procedure yields the firm's expansion path. Bc this expansion path shows how input usage expands with the level of output, it also shows the relationship between output level and total cost. That relationship is summarized by the total cost function, C(w,r,q) which shows production costs as a function of output levels and input prices. 3) The firm's average cost (AC) and marginal cost (MC) functions can be derived directly from the total cost function. 4) All cost curves are drawn on the assumption that the input prices are held constant. When input prices change, the cost curves will shift to new positions. The extent of the shifts will be determined by the overall importance of the input whose price has changed and by the ease with which the firm may substitute one input for another. Technical progress will also shift curves. 5) Input demand functions can be derived from the firm's total const function through partial differentiation. These input demand functions will depend on the quantity of output that the firm chooses to produce and are therefore called "contingent" demand functions. 6) In the short run, the firm may not be able to vary some inputs. It can then alter its level of production on by changing its employment of variable inputs. In doing so, it may have to use nonoptimal, higher cost input combinations that it would choose if it were possible to vary all inputs. Usually k is held constant, and only labor can be varied.

3 Simple Production Functions

1) A linear production function describes two inputs that are perfect substitutes, where there's a constant returns to scale, and where elasticity of substitution is infinite. For example, one ATM (K) may consistently substitute for two human bank tellers (L) each day: q = 2K + L. 2) A fixed proportions function describes two inputs that are perfect compliments, with a constant returns to scale, where elasticity of substitution is 0. For example, one supermarket checker may be assigned to one register. Production is set by the minimum of the number of checkers and registers; q = min (L,K). Or one check to 8 self check registers: q = min(8L,K) 3)A cobb douglas production function has the normal convex shape of an isoquant, can exhibit any degree of returns to scale (but we assume constant returns to scale), where elasticity of substitution = 1. The input with the larger exponent adds more to output per unit of input.

Properties of a profit function:

1) Homogeneity: a doubling of all the prices in the profit function will precisely double profits 2) Profit functions are non decreasing in output price, P 3) Profit functions are non increasing in prices r and w 4) Profit functions are convex in output prices

Properties of Cost Functions

1) Homogeneity: the total cost functions are homogeneous of degree 1 in the input prices, meaning that if input prices double, the cost of producing any given output level will double. 2) Total cost functions are non decreasing in q, r, and w: Because cost functions are derived from a cost minimization process, any decrease in costs from an increase in one of the functions arguments would lead to a contradiction. If an increase in output causes total costs to decrease, it must only be because that firm was not minimizing costs in the first place. 3) Total cost functions are concave in input prices: A pseudo cost curve is linear, because cost will increase as input prices r and w increase. HOWEVER, this doesn't account for an actual cost function: if labor or capital prices increase, there would be input substitution. SO, actual cost functions fall below the pseudo linear cost function, in a concave way. AKA *total costs increases at a decreasing rate for any given input, because a firm will start switching away from an input as its price increases*

Producer Choice (Cost) General Info:

1) If a firm wishes to minimize its long run costs at a given production level (aka for a given isoquant), the production function is our constraint in a cost minimization problem. We must pick the inputs so that the lowest cost line touches the isoquant. 2) We have input prices w (for L) and r (for K) 3) Cost function or line that represents spending = wL + rK 4) Tangency condition represents the isoquant's MRT = isocost line, combined with the production function at some definite production level q0: MPl / MPk = w/r But, instead of doing the tangency condition, do the equivalent by equating the marginal values (the "last dollar rule"), which is derived by rearranging the tangency rule into: MPl/w = MPk/r 5) Lagrange multiplier set up: wL + rK + lamda(q0 - f(L,K) 6)Exogenous variables for cost optimization: input prices and production aka: w, r, q0 7) The expansion path is important in a cost function. The changing levels of q serve as the constraint in the cost minimization problem. The fact that the cost function is the result of a series of optimization problems means that it represents the best that a company can do, given some set of input prices and its production technology. Inefficient firms could certainly produce at higher costs, but cost efficiency is our base case.

Input Substitution

A change in the price of an input will cause the firm to alter its input mix.

Returns to Scale

A way to characterize a production function is to describe its returns to scale, which describes how output responds to increases in all inputs together. If the production function is given by q = f(K,L), and if all inputs are multiplied by the same positive constant t( where t > 1), then we classify the resturns to scale of the production function be: Constant if f(tK,tL) = tq -> output increases by same % as increase in input Decreasing if f(tK,tL) = tq < tq -> output increases by a smaller % than increase in input....What can a firm do if facing decreasing returns to scale?? Possibly determine the most cost efficient scale, then replicate. Increasing if f(tK, tL) = tq > tq -> output increases by larger % than increase in input

Elasticity of Substitution

Another important characteristic of the production function is how "easy" it is to substitute one input for another. Along an isoquant, the RTS will decrease as K/L decreases. Now, defining some parameter that measures this degree of responsiveness: If RTS does not change at all for changes in k/l, we might say that substitution is easy because the ratio of the marginal productivities of the two inputs doesn't change as the input mix changes. If the RTS changes rapidly for small changes in k/l, we might say that substitution is difficult, because minor variations in the input mix will have a substantial effect on the inputs relative productivities. Elasticity of Substitution: percent change in (k/l) ________________________________ percent change in RTS

Graphical analysis of average cost and marginal cost

Average cost will cross marginal cost at average cost's lowest point. Or, if total cost curve exhibits constant returns to scale (total costs are proportional to ouput level), then AC = MC b/c both will be a horizontal, straight line.

Chapter 10: Cost Functions

Costs are incurred by a firm for the inputs it needs to produce a given output. Costs are defined as implicit costs + explicit costs, which equals to the economic cost: The economic cost of any input is the payment required to keep that input in its present employment. Equivalently, the economic cost of an input is the gain that input would receive in its best alternative employment (think of opportunity cost). Assumptions of a cost function: 1) assume there are only two inputs: homogeneous labor, and homogeneous capital 2) Inputs are hired in perfectly competitive markets, and thus can buy or sell all the labor or capital services they want at the prevailing rental rates (r = capital costs and w = labor costs).

Profit maximization and the margin

Decisions are made in a "marginal" way. Variables are adjusted until it is impossible to increase profits further; this involves looking at the incremental or marginal profit obtainable from producing one more unit of output, or at the additional profit available from hiring one more laborer. As long as this incremental profit is positive, the extra output will be produced or the extra laborer will be hired.

Diminishing Marginal Productivity

Diminishing marginal productivity is dependent on more than just how much of an input is used. It's true, that for example, labor cannot be added indefinately to a given field (while keeping everything else fixed) without eventually experiencing deterioration in productivity. This is shown on the second order partial derivatives.... Fkk < 0 for high enough k Fll < 0 for high enough L However, changes in the marginal productivity over time depend not only on how labor input is growing, but also on changes in other inputs, such as capital. We must be concerned with Flk. (combined second derivative..) Flk > 0 in most cases, thus declining labor productivity as both L and K incrase. ...Malthus' population theory being incorrect is what helps us understand. The population has continued to grow, but it didn't result in lower labor productivity because of changes in other factors (increase in capital inputs and technology).

Firm's Expansion Path

If input costs w and r remain constant, we can follow the cost minimization process for each level of input with the firm's expansion path. The expansion path records how input expands as output expands while holding the prices of the inputs constant. Expansion paths don't always need to be a straight line; the use of some inputs may increase faster than others as output expands, and which of these inputs expand more rapidly will depend on the shape of the production function. It's reasonable to assume that the expansion paths will be positively sloped, bc successively higher output levels will require more of both inputs.

Relationship Between Profit Maximization and Cost Minimization

Economic profits = Total Revenues - Total Costs Supposing the firm takes market price for its total output as a given, then profit can be written as: Profit = marketPrice * quantity - wage*L - r*Capital = price*f(K,L) - wl - wk That formula shows that economic profits obtained are a function of the amount of capital and labor is employed.

Chapter 11: Profit Maximization

Firms will minimize costs for any level of output they choose; they choose that level of output through profit maximization.

Marginal Rate of Technical Substitution (RTS)

For smooth isoquants, we measure the substitution between inputs, which is the slop of the isoquant, marginal rate of technical substitution (RTS). (This RTS is the same as the MRS of the utility indifference curve). RTS = MPl / MPk RTS shows the rate at which, having added a unit of labor, capital can be decreased while holding output constant along an isoquant. Remember, output is held constant as L is substituted for K.

Cost Minimizing Input Choices

Goal: minimize costs subject to a production constraint This is a constrained minimization problem. So, to minimize cost of producing a given level of output, a firm should choose a point on the isoquant where: *RTS = w/r aka, equating the rate at which k can be traded for l in production to the rate at which they can be traded in the marketplace*

Average Physical Product (AP)

Labor productivity often means average productivity. If an industry has experienced productivity increases, then output per unit of labor input has increased. Average product of labor: APl = output/labor input = q/L = f(k,L) / L Notice how APl also depends on the level of capital used. APk = q/K = F(L,k)/K

Summary:

In chapter 11, we studied the supply decision of a profit maximizing firm. Our general goal was to show how such a firm responds to price signals from the marketplace. In addressing that question, we developed a number of analytical results. 1) To maximize profits, the firm should choose to produce that output level for which marginal revenue (the revenue from selling one more unit) is equal to marginal cost (the cost of producing one more unit)...MR = MC 2) If a firm is a price taker, then its output decisions do not affect the price of its output; thus, marginal revenue is given by this price. If the firm faces a downward sloping demand for its output, however, then it can sell more only at a lower price. In this case, marginal revenue will be less than price, and may even be negative. 3) Marginal revenue and the price elasticity of demand are related. 4) The supply curve for a price taking, profit maximizing firm is given by the positively sloped portion of its marginal cost curve above the point of minimum average variable cost (AVC). If price falls below minimum AVC, the firm's profit maximizing choice is to shut down and produce nothing. 5) The firm's reactions to change in the various prices it faces can be studied through use of its profit function; profit = (p,w,r). This function shows the maximum profits that the firm can achieve given the price for its output, the prices of its input, and its production technology. Differentiation with respect to market price will get us the supply function, whereas differentiation with respect to any input price yields (the negative of) the demand function for that input. 6) Short run changes in market price results in changes to the firm's short run profitability. These can be measured graphically by changes in the size of producer surplus. The profit function can also be used to calculate changes in producer surplus. 7) Profit maximization provides a theory of the firm's derived demand for inputs. The firm will hire any input up to the point at which its MRPl/MRPk = w/r Increases in the price of an input will induce substitution and output effects that cause the firm to reduce hiring of that input.

Isoquant Maps

Isoquants are the level sets for production functions. An isoquant shows those combinations of K and L that can produce a given level of output (q0). Mathematically, f(k,l) = q0 There are infinitely many isoquant curves, and each represents a different level of output. Presumably, using more of each of the inputs will permit output to increase. The shape of an isoquant is more important than the shape of an indifference curve because the labeling on the curves has meaning. On the Y axes: capital K per period On the X axes: labor L per period

Reasons for a diminishing RTS

Isoquants have a negative slope, and convex curves. RTS is diminishing. For high ratios of k to l, the RTS is a large positive number, indicating that a great deal of capital can be given up if one more unit of labor becomes available. On the other hand, when a lot of labor is already used, RTS is slow, as only a small amount of capital can be traded. Remember; marginal productivity of an input depends on the level of both inputs. So we assume diminishing RTS not only from the assumption of diminishing marginal productivity, but also condisdering Fll and Fkk are assumed to be negative.

Economies of Scope

It may be less expensive to produce goods jointly than separately. Economies of scope often occur because fixed costs are spread over more types and units of production. AKA, starbucks may choose to sell coffee and CD's bc same fixed costs of building expenses, etc.

Marginal Revenue

Marginal revenue is the change in total revenue R resulting from a change in output q: aka marginal revenue is the derivative of total revenue

Profit Maximization and Input Demand

New profit equation: profit = P * F(K,L) - (rK + wL) So, profit maximizing becomes about choosing the appropriate levels of capital and labor input. Maximization condition: MRPl / MRPk = w/r which is another verision of MR = MC Hence, profit maximization requires that the firm hires each input up to the point at which its marginal revenus product is equal to its market price...notice how profit maximization also implies cost minimization because RTS = w/r

Profit Functions

Profit = total revenue - total cost = p*q - c = pf(K,L) - rK - wL variables L, K, and q are endogenous because they're in the firm's control. They choose the levels of these inputs to maximize profits. Prices w, r, and p are exogenous factors that they must accept as parameters. The firm's profit function shows its maximal profits as a function of the prices that firm faces

Marginal Cost

Remember that marginal cost is the cost of producing one more unit of output. It's the only cost concept based on calculus. MC assumes that the firm is "small" in the market for labor, so w does not vary with L The equation MC = w/MPl relates the cost measure to the physical measures MPl. As MPl falls, marginal costs increase, if we hold w constant.

Short Run Costs

SC = rK + wL, where we hold capital fixed. Short run fixed costs are costs associated with inputs that cannot be varied in the short run. Short run variable costs are costs of those inputs that can be varied to change the firm's output level. Short run fixed costs must always be paid, even if output is 0. Short run variable costs can be avoided by producing more or less, or nothing at all. Short run Average cost: SAC = total costs / total output = SC/q Short run Margincal Cost: SMC = change in total costs / change in output = partial derivative where SAC and SMC are defined for a specified level of capital input. short run total cost curve = SC (r,w,q,K)`

Average and Marginal Cost Functions

The average cost function (AC) is found by computing total costs per unit of output: average cost = AC(r,w,q) = C(r,w,q) / q The marginal cost function (MC) is found by computing the change in total costs for a change in output produced: marginal cost = MC(r,w,q) = derivative of cost curve ^^in perfect competition, the short run MC is = supply curve

Shifts in Cost Curves

The cost curves show the relationship between costs and quantity produced under the assumption that all other factors are held constant. Meaning, input prices and level of technology do not change. If they do change, the cost curve will shift.

Short Run Supply of a Price Taking firm

The firm's short run supply curve shows how much it will produce at various possible output prices. For a profit maximizing firm that takes the price of its output as given, this curve consists of the positively sloped segment of the firm's short run marginal cost above the point of minimum average variable cost. For prices below this level, the firm's profit maximizing decision is to shut down and produce no output.

Marginal Physical Product (MP)

The marginal physical product of an input is the additional output that can be produced by using one more unit of that input while holding all other inputs constant. Mathematically, it's the partial derivative. Marginal Physical Product of Capital = MPk Marginal Physical Product of Labor = MPl Ex: Consider a farmer hiring one more laborer to harvest the crop, all other inputs constant. 50 workers on a farm may produce 100 bushels of wheat per year, where as 51 workers, with the same land and equipment, can produce 102 bushels. The marginal physical product of the 51st worker is then 2 bushels per year. Marginal product is often assumed to be positive but declining: Fl > 0, Fk > 0; Fll < 0, Fkk < 0

Marginal Revenue Product

The marginal revenue product is the extra revenue a firm receives when it uses one more unit of an input. MRPl = p * MPl MRPk = p * MPk

Marginal Revenue for the price taker

The marginal revenue when a firm doesn't have any impact on price....the extra revenue obtained from selling one more unit is just the market price. A firm cannot sell all it wants at the prevailing market price; if it faces a downward sloping demand curve for its product, then more output can be sold ONLY by reducing the good's price. So, in general, MR will be different for different levels of q. If price does not change as quantity increases, marginal revenue will be equal to price. In this case, the firm is a price taker b/c its output decisions do not influence the price it receives. However, if price decreases as quantity increases, the marginal revenue will be less than price.

Chapter 9: Production Functions

The principal activity of any firm is to turn inputs into outputs, efficiently. A production function takes the form: q = f(k,L) where q = firm's output of a particular good during a period, k represents capital, and L represents labor This function shows the maximum amount of the good that can be produced using alternative combinations of inputs capital (k) and labor (L). The production function shows how the quantity of output depends on the quantity and interactions of inputs.

Cost Functions

The total cost function shows that, for any set of input costs and for any output level, the minimum total cost incurred by the firm is C = C(r,w,q) So, total costs increases as output, q, increases. We write the cost function this way because you trace out a cost function as you change q, while taking as given a particular set of prices. The changing levels of q serve as the constraint in the cost minimization problem.

Profit maximization Principle Rule:

To maximize economic profits, the firm should choose output q at which the marginal revenue is equal to marginal cost. That is, MR(q) = MC(q)

Constant Returns to Scale and Homogeneous and Homotheticity

When a production function exhibits constant return to scale, it meets the definition of "homogeneity", aka the production is homogeneous of degree 1, and the marginal products are homogeneous of degree 0. AKA the marginal productivity of any input depends only on the ratio of capital to labor input, not on the absolute levels of these inputs. A production function that exhibits constant returns to scale and is homogeneous of degree 0, is also homothetic; aka its isoquants will be radial expansions of one another (straight line expansion path). However, a function can also have a homothetic indifference curve map even if it doesn't exhibit constant returns to scales. You can determine this by determining whether or not RTS depends only on the ratio of k to l, and not on the scale of production. *A homothetic function's most important characteristic: scaling up or down (say by increasing budge) does not change the proportions of the inputs used.*

Substitution and output effects in input demand:

When the price of an input falls, two effects cause the quantity demanded of that input to rise: 1) the substitution effect causes any given output level to be produced using more of the input 2) the fall in costs causes more of the good to be sold, thereby creating an additional output effect that increases demand for the input... the output effect is different than the income effect in consumer choice. Consumers are constrained by their budget, but firms are not. Firms produce as much as the available demand allows. ______________________________________________________________________ If the wage rate falls, substitution and output effects cause more labor to be hired. It's not clear how capital usage will respond to a wage decrease....a fall in wage will cause a substitution away from capital, however the output effect will cause more capital to be demanded as part of the firm's increased production plan. Thus, substitution and output effects work in opposite directions, and no definite conclusion can be made.