Microeconomics: Chapter 10 The Price-Taking Firm

PART THREE THE COMPETITIVE MODEL
Chapter 10 The Price-Taking Firm
A price-taking firm chooses its actions under the assumption that it cannot influence the prices of the output that it sells or the inputs that it buys.

We examine the behavior of price-taking firms because:
1. Many firms are price-takers – no single cherry farmer can influence the price of cherries, nor the price of tractors.
2. We can summarize a price-taking firm's behavior entirely in terms of its supply curves in product markets and its input demand curve in factor markets.
3. It is particularly easy to analyze a price-taking firm's input and output choices.

Supply in Product Markets
Two Rules for Profit Maximization
1. Marginal Output Rule: If the firm does not shut down, then it should produce at an output level where marginal revenue is equal to marginal cost.
2. Shut-Down Rule: If for every choice of output level the firm's average revenue is less than its average economic cost, then the firm should shut down.

A price-taking firm's marginal revenue is always equal to the price that it takes as given – it is perfectly elastic at that price (horizontal line). The demand, marginal revenue (MR) and average revenue (AR) curves are all equal to the price.

Marginal Output Rule for a Price-Taker: If a firm takes the price of its output as given, then unless the firm shuts down entirely, it should produce output at a level where the price is equal to marginal cost.

Shut-Down Rule For a Price Taker: If the firm takes the price of its output as given and this price is less than average economic cost for every output level, then the firm should shut down.

The Long and the Short of the Matter
The short run is a period over which only one factor is variable. The long run is a period over which all factors are variable. These conditions result in different cost curves and different outcomes for the marginal output and shut-down rules.

The Firm's Short-Run Supply Curve
A price-taking firm's short-run supply curve coincides with the vertical axis at prices less than the minimum of its short run average variable cost. The short-run supply curve coincides with the firm's short run marginal cost curve where it is above the firm's short-run average variable cost curve.

Note that a firm may produce a product even when it is suffering short-run accounting losses – when revenues are less than its short-run total costs (including fixed and variable inputs).

Deriving a Firm’s Long-Run Supply Curve
A price-taking firm's long-run supply curve coincides with the vertical axis at prices less than the minimum of its long-run average cost. The long-run supply curve coincides with the firm's long-run marginal cost curve where it is above the firm's long-run average cost curve.

Comparison of the Firm’s Short- and Long-Run Supply Curves
1. In the short run, some factors are fixed and expenditures on these fixed factors are not short-run economic costs. In the long run, all factors are variable and expenditures on them are economic costs.
2. In the long run, the firm has opportunities to substitute one factor for another. In the short run, it does not.

Because the firm has more opportunities to make adjustments, the long-run supply curve is more price elastic than is the short-run supply curve.

Long-run marginal cost is equal to short-run marginal cost when the level of capital is fixed at its long-run equilibrium – i.e. the firm uses the same input combination in both the short and long-runs. (In this case, they have the same marginal physical product of labor, and the marginal cost – w/MPPL – is the same.)

If the output price changes after the long-term plans are made (after a factory is built, for example), then the firm will have to adjust its output level, using the short-run curve (which will not include the fixed cost of the already-built-factory).

10.2 Factor Demand
A firm's demand for an input is known as derived demand because it depends on, or is derived from, the demand for the firm's output.

Short-Run Factor Demand
The firm should hire an input just up to the amount at which the marginal benefit to the firm is equal to the marginal cost.

The Marginal Benefit of an Input
The marginal revenue product is the change in revenue due to the sale of the additional output contributed by the hiring of one additional factor.

To find the marginal revenue product of an input, simply multiply its marginal physical product by the marginal revenue associated with each unit of input. For labor, this would be: MRPL=MPPL*MR

For a price-taking firm, a factors' marginal revenue product is equal to its marginal physical product multiplied by the price of output. MRPL=MPPL*p
(Where p is the price of the output)

The Marginal Cost of an Input
The marginal revenue cost is the increase in the firm's total expenditures on inputs when it hires one more unit of a factor.

For a price taker in an input market, the marginal factor cost is equal to the price of a factor.

The Profit-Maximizing Input Level
Factor Hiring Rule: A profit-maximizing firm should hire a factor up to the point at which its marginal revenue product is equal to the marginal factor cost (MRP=MFC)

The marginal output rule provides one way of calculating the profit-maximizing output level. But the factor hiring rule provides another.
Factor Hiring Rule:  Firm should hire input up to level where MRP=MFC;
MRP=MPP*MR, so MPP*MR=MFC

Marginal Output Rule: Firm should choose an output level at which MR=MC;
MC=MFC/MPP, so MR=MFC/MPP

Factor Hiring Rule for a Price-Taker: For a price-taking firm, marginal revenue product is equal to the marginal physical product times the output price (p), and marginal factor cost is equal to the input price (w). Therefore, a firm that is a price-taker in both the factor market and the output market maximizes its profit by hiring a factor up to the point at which marginal physical product times the output price is equal to the input price, or MPP*p=w

The short-run derived demand curve for a firm that is a price-taker in the market for the variable input coincides with the firm's marginal revenue product curve for that factor.

The output effect is the change in quantity demanded of an input due to a change in the firm's output level that results from an increase in the input's price. (For example, a higher input price leads to a lower output and therefore less of the input being demanded – so the input demand curve slopes downward.)

Long-Run Factor Demand
In the long-run, in addition to the output effect, we must consider the factor substitution effect. The factor substitution effect is the reduction in the quantity demanded of an input that results from the firm's substituting away from that factor when its price rises.

Factor Substitution Effect: When the price of one factor rises relative to the other, the firm substitutes away from the factor whose price has risen and replaces it with the factor whose relative price has fallen (as long as it is possible for the firm to substitute one input for the other in production).

Output Effect: In the long run, the output effect may be either positive or negative. (e.g. A manufacturer is planning to scrap its current factory and build a new factory in order to increase its production level. If the increase in output is substantial, then the firm may choose to build a highly automated factory requiring almost no workers. Thus, even though output has increased, the quantity of labor hired will go down.)

Factor Substitution and Output Effect Simultaneously
When the price of a factor rises, the quantity of that input demanded falls – the sum of the factor substitution effect and the output effect is always negative. A profit-maximizing firm would never increase its use of an input in response to an increase in that input's price.

Algebraic Approach – Analyzing Long-Run Factor Demand
Applying the factor hiring rule to each input, the firm's quantities of labor and capital must satisfy: p*MPPL=w    and    p*MPPK=r
Divide the first equation by the second to get: MPPL/MPPK=w/r
This tells us that the marginal physical product of each input should be proportional to its marginal factor cost at the margin.

Investment and the Demand of Capital
The firm should invest in a project if and only if it has a nonnegative net present value.

The higher the interest rate, the lower the present value of any given machine, and the fewer the number of machines that will have a positive net present value. Hence, as the interest rate rises, the number of machines demanded falls, all else equal.

Microeconomics: Chapter 9 Cost

Chapter 9 Cost
This chapter focuses on how the firm chooses its optimal input combination for a given level of output.

9.1 Cost in the Short Run
The short-run economic cost is the minimum level of total expenditures (measured in opportunity-cost terms) needed to produce a given amount of output in the short run.

The short-run variable cost is an abbreviated name for the short-run total economic cost. Economic cost and variable cost are synonyms because if the firm cannot vary the level of expenditure, then it is sunk and is not an economic cost.

To find costs in the short run, follow these steps:
1. Draw the isoquant associated with the output level whose short-run variable cost we are trying to find. (The x0-isoquant.)
2. Label the level at which capital is fixed in the short run, say Kf.
3. Read across from this fixed level of capital to find the point on the x0-isoquant associated with Kf robots. (Point a.)
4. Multiply the quantity of labor found in step (3), La, by the wage rate to get the short-run variable cost, VCSR(x0)

Properties of Short-Run Costs
Variable Cost: When the amount of capital is fixed at Kf, the firm must employ La units of labor to produce x0 units of output. Hence, the short-run variable cost of producing x0 units of output is w*La, when w is the wage per unit of labor. The short-run variable cost curve must slope upwards. This is because a firm operating on its production function must hire more inputs in order to produce more output, and more inputs cost more money.

Marginal Cost: The short-run marginal cost is the change in the short-run variable cost due to the production of one more unit of output. The marginal cost is derived from the variable cost, and the variable cost is derived from the production function.

The cost of one additional unit of the variable input is known as the marginal factor cost (MFC) of that input.

The short-run marginal cost = (additional amount of the variable input needed to produce one more unit of output: 1/MPPL)*(marginal factor cost of the variable input: MFC)
MCSR=MFCL/MPPL

Whenever a firm is a price taker in an input market, the marginal factor cost is equal to the price of the factor. (For labor, this is the wage, w)
MFCL=w therefore: MCSR=w/MPPL

This tells us that the higher the marginal product of labor, the lower the marginal cost of output. If a single new worker can produce a lot of output, then not many new workers are needed, and the extra output does not cost very much to make.

If there are diminishing returns to labor, increasingly large increments of labor are needed to produce additional output. As the quantity of labor employed rises, the marginal physical product of labor falls. (MPPL falls when MCSR rises). So when the production function exhibits a diminishing marginal product of labor, the short-run marginal cost curve is upward sloping. (And the MPPL curve slopes downward)

When the marginal physical product of labor is constant at m, each additional unit of output requires 1/m additional units of labor. Because each laborer receives a wage of w, short-run total cost rises by w/m. Hence the short-run marginal cost is w/m at every output level.

Average Cost
Short-run average variable cost is the short-run variable cost divided by the number of units being producted.

The short-run average total cost is the short-run total cost divided by the number of units being produced.

The short-run average fixed cost is the short-run fixed cost divided by the number of units produced.

The Relationship between Short-Run Marginal Cost and Short-Run Average Variable Cost
Given that marginal and average variable cost curves are derived from the same variable cost curve, they are related in these ways:
1. Whenever marginal cost is below average variable cost, average variable cost falls.
2. Whenever marginal cost is above average variable cost, average variable cost rises.
3. The short-run marginal cost curve crosses the short-run average variable cost curve at the point where average variable cost is at a minimum.

9.2 Cost in the Long Run
Since all factors are variable in the long run, the (explicit and imputed) expenditures on all factors are economic costs in the long run.

Since the levels of more than one factor can be varied, it may be possible for the firm to substitute quantities of one factor for another.

Graphical Analysis
To maximize profit, the firm must choose the cheapest combination of inputs that can produce the desired level of output. It must make an economically efficient input choice – an input combination is economically efficient when it has the lowest opportunity cost of those input combinations that can be used to produce the desired output.

Isocost Lines: An Isocost line is the line representing all input combinations that cost the firm the same amount. An isocost map is the whole family of isocost lines that exists for a given set of factor prices.

Finding the Economically Efficient Input Mix: The firm minimizes the total cost of producing a given number of outputs by using the input bundle that is on the isoquant representing that level of output and on the lowest possible isocost line.

Algebraic Interpretation
The isocost line and isoquant line have the same slope where they are tangent (at the equilibrium). The slope of an isocost line is equal to the factor price ratio, w/r, in absolute value. By definition, the negative of the slope of the isoquant is the marginal rate of technical substitution between capital and labor. So, at the point of tangency: MRTS=w/r

Also, along an isoquant, MRTS=MPPL/MPPK, so we can say: MPPL/MPPK=w/r

A firm that takes factor prices as given should operate at a point where, at the margin, the inputs’ marginal physical products are proportional to their prices.

Comparative Statics
Factor Prices
How does a change in the price of one factor affect the firm’s long term decisions? When the price of labor rises, the isocost lines pivot inward. To produce a given level of output, the firm substitutes away from the factor whose price has risen. Total cost must always rise in response to an increase in the price of one of the factors that the firm uses in production.

Technology
As a result of technological improvement, the firm can continue to produce the same amount of output while using fewer inputs. The isoquant shifts inward, and the total cost falls.

The Nature of Output
A firm may decide to improve the quality of its output. This change shifts the isoquant outward – it takes more inputs to produce the same amount of outputs.

Output Level
The expansion path of a firm is the long-run set of least-cost input levels traced out as the level of output changes, all else equal.

Summary of Comparative Statics Analysis
1. Sketch the equilibrium that holds before there are any changes in the underlying market conditions
2. Given a change in underlying market conditions (Such as factor price change or a shift in technology), determine whether the isoquant map or the isocost map is affected.
3. Ascertain the direction of the shift in the affected set of curves
4. Find the new equilibrium by locating the tangency between the relevant isoquant curve and the relevant isocost line.

Deriving the Long-Run Total Cost Curve
The long-run total cost is the minimal level of total expenditures (measured in opportunity-cost terms) needed to produce a given amount of output in the long run. To find the entire long-run total cost schedule, just follow these steps:
1. Choose an output level.
2. Find the optimal input combination by taking the tangency of the isoquant and isocost lines.
3. Compute the cost of this input combination by multiplying the price of each input times the quantity used and then adding up the dollar total.
4. Graph this point.
5. Repeat steps (1) through (4) for each output level.

Properties of Long-Run Costs
Long-Run Marginal Cost: The long-run marginal cost is the change in long-run total cost due to the production of one more unit of output.

Long-Run Average Cost: The long-run average cost is the total cost divided by the number of units being produced.

Economies of Scale: When long-run average costs fall as output rises, costs are said to exhibit economies of scale. When the production function exhibits increasing returns to scale, the long-run total cost function exhibits economies of scale. The average cost curve is downward sloping.

When the production function exhibits constant returns to scale, long-run average cost remains constant as the level of output changes. The average cost curve is flat.

When long-run average costs rise with the output level, costs are said to exhibit diseconomies of scale. When the production function exhibits decreasing returns to scale, the long-run total cost function exhibits diseconomies of scale. The average cost curve is upward sloping.

Economies of Scope: When it is cheaper to produce two products together in one firm instead of separately in two specialized firms, costs are said to exhibit economies of scope. Economies of scale and scope both arise when a fixed facility can be “shared” to produce a large number of units of output, either of a single product, or a mix of products.

Long-Run Costs Compared to Short-Run Costs
1. In the short run, fixed factors are fixed. Since these fixed factors have no alternative uses, any expenditures made by the firm on these factors are sunk expenditures and thus are not economic (opportunity) costs. In the long run, however, all input levels are variable and there are no sunk expenditures; hence, everything counts as an economic cost. Because more things get counted as costs, this effect raises long-run economic cost relative to short-run economic cost.

2. Since the levels of more than one factor can be varied in the long run, it may be possible to substitute quantities of one factor for another. For example, in the long run but not the short run, National Motors can adjust the number of robots that it uses on its assembly line. This increased flexibility tends to lower the cost of producing a given amount of output.

These act in opposite directions on the economic cost curves, so in some instances short-run variable cost of producing a given level of output may be greater than the long-run total cost, while in other cases the relationship will be the reverse. However, short-run total cost is always at least as large as long-run total cost.

Microeconomics: Chapter 8 Technology and Production

Chapter 8 Technology and Production
8.1 Technology
The Production Function
The production function is a schedule that shows the highest level of output the firm can produce from a given combination of inputs. The highest total amount of output the firm can produce given the amount of its outputs is also called the total product of L and K (labor and capital).

Isoquants
An isoquant is a curve showing all of the input combinations that yield the same level of output. An isoquant map is the collection of all isoquants corresponding to a given production function.

How Many Inputs?
More than two inputs are used to produce output. However, all of the key insights that we gain from the two-factor case hold generally.

What Is Output?
Output can be something tangible – bikes, cars, etc., or something non-tangible – education, health services, etc.

The Decision-Making Horizon
The production function tells us whether a given input combination is capable of producing a particular amount of output, but it does not tell us whether the firm can actually obtain the input combination under consideration. For this, we need to consider the time horizon.

A variable factor is a factor whose level can be varied over the relevant planning horizon. A fixed factor is a factor whose level cannot be varied over the relevant planning horizon.

The short run is defined as a time period over which only one of the firm’s inputs is variable, while all the others are fixed. The long run is defined as a period of time long enough that all of the factors are variable and none is fixed. The actual time periods for the short and long run vary depending on the product.

8.2 Properties of the Production Function
Marginal Physical Product (MPP)
The marginal physical product (MPP) of an input is the extra amount of output that can be produced when the firm uses one additional unit of input. If using an additional ∆L units of labor raises output by ∆X, the marginal physical product of labor: MPPL=∆X/∆L

Increasing Marginal Returns
A technology exhibits increasing marginal returns when the marginal physical product of an input rises as the amount of the input used increases. The slope of the total product curve is the marginal physical product of labor. When the firm’s technology exhibits increasing returns to labor, the total product curve gets steeper as we increase the quantity of labor, and the marginal physical product curve is upward sloping.

Constant Marginal Returns
A technology exhibits constant marginal returns when there is a range of input levels over which the marginal product of a factor remains unchanged as the amount of the factor increases. When the technology exhibits constant marginal returns, the marginal physical product of labor is unaffected by changes in the amount of labor employed, and the marginal physical product curve is flat.

Diminishing Marginal Returns
A technology exhibits diminishing marginal returns when the marginal product of an input falls as the amount of the input used increases. When a technology exhibits diminishing marginal returns, the marginal physical product curve falls as the quantity of the input rises, but the curve remains positive. Since the height of the marginal product curve is the slope of the total product curve, the total product increases as the number of workers rises, but at a decreasing rate (the curve becomes flatter as the amount of labor rises).

A Changing Pattern of Marginal Returns
Perhaps the most common situation is that initially there are increasing marginal returns, then a range of constant marginal returns, followed by a range of diminishing marginal returns. In this case, the MPP curve slopes up, is horizontal, and then slopes down.

Marginal Rate of Technical Substitution
The marginal rate of substitution is the rate at which the available technology allows the substitution of one factor for another. It is -1 times the slope of the isoquant. This is a rate of -∆K/∆L. (This is completely analogous to the rate of substitution for the household.)

Two Polar Cases of Factor Substitution
Case I: Perfect Substitutes: Whenever a technology has a constant marginal rate of technical substitution between two inputs, the resulting isoquants are straight lines. Since gasoline and gasohol can always be substituted for each other at the same rate, they are said to be perfect substitutes in this use. Perfect substitutes are two inputs that have a constant marginal rate of technical substitution of one for the other.

Case II: No Factor Substitution: When two products must be used together in a constant proportion, no factor substitution is possible, and the isoquants are right angles that are placed along a ray from the origin whose slope is equal to the proportion.

The Relationship between MPP and MRTS
If a firm hires ∆L additional workers, then the marginal physical product of labor (MPPL) times ∆L gives the total output rise. The firm can now reduce the number of robots it has until the amount of output lost offsets the gain from the extra labor. If the firm employs ∆K fewer robots, the reduction in the total output is MPPK * ∆K. This gives:   MPPL*∆L=MPPK*∆K which can be rewritten as -∆K/∆L=MPPL/MPPK
Since -∆K/∆L is the marginal rate of substitution, then: MRTS=MPPL/MPPK

The marginal rate of technical substitution between two inputs is equal to the ratio of the marginal physical products of the inputs.

A technology exhibits a diminishing marginal rate of technical substitution when the rate at which one factor can be substituted for the other falls as the amount of the first factor rises.

Returns to Scale
The degree of returns to scale is the rate at which the amount of output increases as the firm increases all of its inputs proportionately.

Constant Returns to Scale: A technology such that a proportional increase in all input levels leads to output growth in the same proportion. (e.g. Almond Yummies production states that chocolate and almonds have to be used in the exact proportion of 4 ot 1. If the firm doubles the usage of both inputs, it doubles its output.)

Increasing Returns to Scale: A technology such that a proportional increase in all input levels leads to greater than proportionate output growth. (e.g. Because Boeing workers learn through experience, the amount of labor needed to manufacture an aircraft falls dramatically as the number of planes produced rises.)

Decreasing Returns to Scale: A technology such that a proportional increase in all input levels leads to less than proportional output growth. (Some argue that if the firm can simply replicate itelf, then there would never be a reason to observe decreasing returns to scale.)

Graphing Returns to Scale
The degree of returns to scale can be seen graphically using isoquants. A ray drawn from the origin shows all of those input combinations that involve the use of capital in the same proportion. If doubling the each of the inputs results in more than double the output (shown by the isoquant), then there are increasing returns to scale (isoquants seem closer together). If doubling the inputs results in a point on an isoquant representing double the original output, then the technology is subject to constant returns to scale. If doubling the inputs results in a point on an isoquant representing less than double the output, then the technology is subject to diminishing returns to scale (and isoquants seem farther apart).

Marginal Returns and Returns to Scale
The shape of a marginal physical product (MPP) curve reflects the impact of a change in the amount of a single factor, while the degree of returns to scale concerns the impact of a simultaneous change in all of the factors in proportion.

For a production function that allows no substitution, such as “Almond Yummies,” the recipe requires 4 almonds and 1 ounce of chocolate. If the firm currently has 20 almonds, then the marginal physical product of chocolate is equal to one if C is less than 5. If C is greater than or equal to 5, an additional ounce of chocolate has no effect on what the firm can produce – at this point the marginal physical product of chocolate is zero. Since the MPP of chocolate goes from 1 to zero, we might think it has diminishing marginal returns. However, we know from our earlier discussion that this product exhibits constant returns to scale.

Microeconomics: Chapter 7 Consumer Choice

PART TWO: THE FIRM
Chapter 7 The Firm and Its Goals
7.1 What Do Firms Do?
1. What should the firm produce?
2. How should the firm produce its output?
3. How much should the firm sell, and at what price?
4. How should the firm promote its product?

Why Do Firms Exist?
Transaction Costs are the costs of conducting an economic exchange between two parties. The theory of transaction costs predicts that economic exchange will tend to be organized in ways that minimize the costs of those exchanges. In market-based relationships, people have reputations and information, and trust can be built.

Economic Profit
The firm’s goal is to maximize its “economic profit.” Total revenue is the sum of the payments that the firm receives from the sale of its output. The total economic cost is the firm’s total expenditures on the inputs used to produce output, where expenditures are measured in terms of opportunity cost. Economic profit is total revenue minus total economic cost.
Economic profit = Total revenue – Total economic cost

The imputed cost is the opportunity cost incurred when the owner of a factor employs the factor in one use rather than in its best alternative use. E.g. An owner may not pay himself a salary, but his time is worth something (it’s opportunity cost), and this is part of the Total economic cost – since it is implicit, it is an imputed cost.

A sunk expenditure is a factor expenditure that, once made, cannot be recovered. (If a factory already has a paid-for 2-year lease, and cannot be sub-leased, then the opportunity cost during this time would be zero – it can’t be used for any other purpose – it is considered a sunk expenditure.)

User Cost of Capital
The user cost of capital is the opportunity cost that an owner incurs as a consequence of owning and using an asset. This includes both the economic depreciation and the foregone interest.  E.g. If the $8000 was not spent on buying an item, it would bring in about $560 in one year – assuming interest is 7%.  If an item costs 8,000 and after one year could be sold for $1500, then the user cost of capital includes both the $6500 in depreciation, as well as the $560 in foregone interest. The total user cost of capital would be $7,060.

Depreciation is the fall in value of an asset over a defined period of time. (If an item is bought for $8000 and sold for $1500, then depreciation was $6500.)

Apple Computer Forgets How to Price Memory
Apple ordered millions of 1 megabit memory chips at a price of $38 per chip. Then the price of chips on the market fell to $23 per chip, before their inventory was gone.  They had to decide whether to use the price they had paid or the current market price. The economic cost of the chips was $23 – if Apple did not use the chips for its own machines, this is the price at which the chips could be sold to other computer companies (best alternative use). Instead, apple executives chose to use the original purchase price - $38, and their profit plummeted, since they were stuck with millions of unsold chips.

7.2 The Firm as Supplier: The Profit-Maximizing Level of Output
Total Revenue Curve
The total revenue of the firm is equal to the number of units of output sold times the price per unit. The market demand curve tells us how much output can be sold in total by all of the firms in the industry at a given price.

The firm-specific demand curve is a schedule showing the quantity of a single firm’s output demanded for any price charged by that particular firm. Using market research, this information can be determined, and a firm can figure out what price it must choose to sell the number of units it wants to sell. The firm-specific demand curve contains all of the information the firm needs to calculate its total revenue function (price * quantity).

The total revenue curve, R, relates the total revenue earned by the firm (y axis) to the quantity of output produced and sold (x axis). [Image page 207]

Total Economic Cost Curve
The total economic cost curve is the schedule showing the relationship between a firm’s output level and the resulting level of total economic cost. This requires that we hold factor (input) prices constant, hold technological possibilities constant, and also hold product characteristics constant.

The total economic cost curve, C, relates the total cost incurred by the firm (y axis) to the quantity of output produced (x axis).

Maximizing Profit
The profit function is the algebraic or geographical relationship between a firm’s output level and its resulting profit level. To maximize its profit, a firm should produce at the output level where the total revenue curve is the greatest distance above the total economic cost curve. This distance can be graphed separately, as the profit function curve. The highest point on this curve shows where profit is maximized.

The Optimal Output Level for an Active Firm
The optimal level of output can be found by looking at the highest point on the profit curve. It can also be found by looking at how profit changes as output changes.

Marginal revenue (MR) is the change in revenue due to the sale of one more unit of output. The marginal cost is the change in total cost due to the production of one more unit of output.

The marginal output rule states that if the firm does not shut down, then it should produce output at a level where marginal revenue is equal to marginal cost. (Although the precise shapes of the marginal revenue and marginal cost curves depend on the particulars of the market in which the firm operates, the rule that the firm should produce at a point where marginal revenue is equal to marginal cost is valid for any profit-maximizing firm.)

The average revenue is the firm’s total revenue divided by the number of units produced. The average economic cost is the firm’s total economic cost divided by the number of units produced.

The Shut-Down Decision
The shut-down rule states that if for every choice of output level the firm’s average revenue is less than its average economic cost, the firm should shut down. (It is important to use economic cost – sometimes it may be better to operate at an accounting loss, rather than shutdown completely – this happens when there are sunk costs – costs like a lease on a building that cannot be sublet that are accounting costs, but not economic costs.)

Refractories: Staying in Business to Lose Money
New technology was developed, and producers of refractories (special bricks) found that the revenues from producing bricks failed to cover the costs of the special clay formed into bricks, the gas and oil used to fire the kilns, and the large capital costs of the plants used to produce the bricks. However, the large expenditures on specialized plants were sunk because the plants had no alternative uses. The managers correctly recognized that the costs of the plants were sunk and did not let that influence their shut-down decision. The average economic costs – the clay and fuel – were below the price of bricks.

East Germany: Shutting Down on Economy
When East Germany reunified, there were many state firms that had to be brought into the free-market economy. For many people, the best alternative use of their labor was unemployment. The East German wages were set at artificially high wages, making the firms unprofitable. However, from the perspective of maximizing Germany’s profit from these firms, they should be kept in business.

7.3 Do Firms Really Maximize Profits?
1) You might argue people don’t think about producing one “marginal” gallon of paint. However, the size of the margin may be larger than one unit, and at some level, it seems likely that this is taken into account.

2) Some argue that firms don’t think in terms of a marginal choice rule or a shut-down rule. However, the validity of the theory of the firm does not require that firms think of themselves as acting in the terms described by the theory. What is essential is that their actual behavior fits the pattern predicted by the theory.

3) Some argue that perfect profit maximization is not possible. However, models still gives us a good idea of tendancies in firms’ behavior.

4) Some argue that a firm is a collection of people, not one decision maker, so it can’t make rational decisions. This is dealt with next.

The Divorce of Ownership and Control
Most large companies are owned by a large number of stockholders, who have a claim on the firm’s profit (and therefore want the firm to maximize profit). However, they are run by managers. The managers may want to maximize things other than profit, such as leisure, amenity, or sales (maximizing total revenue rather than marginal revenue). Stockholders and managers are an example of a principle-agent relationship – an economic relationship in which one party, the principle, hires a second party, the agent, to perform some task on the first party’s behalf.

Control Mechanisms
Internal Control Mechanisms
Internal control mechanisms are a means of controlling a firm’s managers involving solely the owners and managers of the firm itself. These include the corporate governance scheme, which includes the rules and institutions that specify the responsibilities of the managers and set up means for the shareholders to monitor the managers and to replace them if necessary. It could also include the proxy fight, which is an election in which shareholders vote on key corporate decisions or, in some cases, choose to fire the incumbent managers. However, stockholder apathy often takes place – this is an example of free riding. Free riding is said to take place when a person refrains from taking a costly action because he or she knows that someone else will undertake it (which allows the free rider to reap the benefits without bearing the costs.) A third internal control mechanism is performance-based compensation, which is a system for compensating a firm’s manager under which the amount paid to the manager depends on how well the firm does.

External Control Mechanisms
External control mechanisms are control mechanisms that involve people outside of the firm. An example of an external control mechanism is the hostile takeover. If a firm is not maximizing profit, someone may be able to buy the firm for a relatively low price, put it under good management (which will maximize profit) and then sell it for a high stock price. Even the threat of hostile take-over may incentivize a company to maximize profit. Firms may also be pressured to maximize profit if they face strong product market competition. Another form of external control is discipline from capital suppliers – they may not be able to get loans unless they can show they are making good decisions.

7.4 Profit Maximization over Time and under Uncertainty
Intertemporal Choice
How should management take into account the fact that the expenditures and receipts (costs and revenues) occur at different dates? (e.g. A 777 is designed and developed, with no revenue expected from it until many years later.)

Stockholder Myopia?
Some argue that stockholders tend to hold stock in a given firm for only a short period of time, so that they don’t care about the firm’s long-term prospects. However, this is not logical. Stockholders receive dividend payments from the firm. These are distributions of a firm’s profit that are paid to the owners of the firm’s stock. The stockholder also gets money based on the difference between the price of the stock when they buy it, and the price of the stock when they sell it. (This is the capital gain or capital loss – the difference between the buying and selling prices, depending on whether the selling price is higher or lower than the buying price.) So a stockholder that sells his stock after one year receives the present value of the dividend of the stock, plus the present value of the price he sells it for, minus the price that he bought it for.
Income from one year of stock = d1997/(1+i)+p1998(1+i)-p1997

A stockholder wants the firm to maximize the present value of all future profits even if he or she plans to sell the stock soon. It is not in stockholder’s interest for management to by myopic.

Decisions under Uncertainty
A manager with an undiversified human capital portfolio will take the riskiness of his choices into account while well-diversified stockholders are concerned only with the expected returns. From the owner’s point of view, managers will be biased against risky projects.

Policy Implications of Expected Profit Maximization
Congressman Henry Waxman of California proposed a “consumer protection law” that would force firms that make “excessive profits” from anti-AIDS drugs to lower their prices. However, AIDS activists argued that the bill would harm AIDS victims. Since drugs developed for relatively small populations usually produce low profits, and the firms are penalized if they develop a profitable drug, then they will no longer undertake these projects – the expected value will always be low.

Gold Mines, Shutdown, and Option Value
This year, the price of gold is low, but next year, it could be low or high. In the first year, the mining firm must decide whether to keep the mine open or to close it. In the second year, the firm again confronts the decision of whether to have the mine open or closed, but now it knows the second-year price of gold. A decision tree can be drawn to help decide what action should be taken in the first year, given the uncertain conditions.

Microeconomics: Chapter 6 Choice Under Uncertainty

Chapter 6 Choice Under Uncertainty
6.1 Gambles and Contingent Commodities
The outcome of an uncertain situation is referred to as a state of the world. Contingent commodities are commodities whose level depends on which state of the world occurs.

Budget Constraint
The budget constraint for contingent commodities is graphed on an axis with consumption in one state of the world on the horizontal axis, and consumption if the state of the world is not that event on the vertical axis (the two conditions must be complete). The endowment point of the budget constraint is the point where no bet is placed. In general, the slope of the budget constraint is –N/H, where H is the change in consumption if the contingency on the horizontal axis occurs, and N is the change in consumption if the contingency on the vertical axis occurs.

Probability and Expected Value
The probability of a given state of the world is a measure of the likelihood that it occurs.

The expected value of X is the value of X that occurs “on average.” The expected value is found by weighting the value of X in each state of the world by the probability of that state of the world occurring.   E[X]=p*X1+(1-p)*X2

The Fair Odds Line
A gamble is actuarially fair if the expected monetary gain equals zero. The budget constraint that reflects the opportunities presented by an actuarially fair gamble is called the fair odds line. The slope of the fair odds line is minus the probability of the contingency on the horizontal axis divided by the probability of the contingency on the vertical axis. The expected value of consumption is equal at every bundle on a fair odds line.

The odds of two events occurring is the ratio of their probabilities. If the probability of rain today is 1/5 and the probability of no rain is 4/5, then the odds of rain are 1 to 4. If the probability of one event is p and the probability of a second event is (p-1), the odds of the first event are p/(1-p).

Preferences
Indifference lines are drawn based on preferences in answer to the question, “which bundle of contingent commodities would you prefer?” The shape of the indifference curves depend on the individuals attitude towards risk.

1. Risk Averse: A risk averse person will not bet when offered an actuarially fair gamble. A 45 degree line traces out the locus of all possible certain consumption levels. This is referred to as the certainty line. For a risk-averse individual, the slope of each indifference curve as it intersects the certainty line is minus the odds in favor of the event on the horizontal axis. (i.e. It is tangent to the fair odds line at the 45 degree line.) The curves are bowed inward to the origin.

2. Risk Loving: A risk loving person is an individual who prefers an uncertain prospect with a particular expected value to a certainty with the same expected value. The curves are also tangent to the fair odds line whenever the certainty and a fair odds line intersect, but they are bowed outward to the origin.

3. Risk Neutral: A person is said to be risk neutral if he is indifferent among all alternatives with the same expected value. The indifference curve for a risk-neutral person coincides with the fair odds line. The indifference map consists of parallel lines. The slope of each is minus the odds.

Equilibrium
As always, the equilibrium is the intersection of the budget constraint and indifference curve. When confronted with an actuarially fair bet, a risk-averse individual maximizes utility at the endowment point – i.e. the bet is not accepted. A risk-loving individual has indifference curves that are bowed outward from the origin. For this person the individual is the point at which he places all of his money on an actuarially unfair bet.

6.2 Some Applications of Contingent Commodities
Risk Premia
In general, more people are risk averse than risk loving. An important piece of evidence for this effect is that riskier assets tend to pay higher rates of return than safe assets. (e.g. A government bond vs. a risky company bond.)

A risk premium is the extra return on an investment to compensate for risk. The risk premium is the distance between the utility curve that goes through the risky bet, and the utility curve representing the safe level of utility. [pg. 174]

The Role of Diversification
Diversification is the process of buying several assets in order to reduce risk. Buying multiple risky assets can help reduce risk. (e.g. You bet $10 that it will rain and also bet $10 that it will not rain – the outcome of each bet contains risk, but together, there is no risk.)

Tax Evasion
Tax evasion is the failure to pay taxes that are legally due. The decision to take the risk to evade taxes can be analyzed using these uncertainty methods. The decision to evade taxes depends on the tax rate, the penalty rate, and the likelihood of getting audited (p).

Along the 45 degree ray from the origin the MRS of each indifference curve is p(1-p), where p is the probability of an audit.  On the other hand, the slope of the budget constraint is minus the marginal tax rate (t) divided by the penalty rate (f). The equilibrium is where the indifference curve hits the budget constraint line – this equilibrium tells whether the individual will pay full taxes or not.

Designing Policy toward Evasion
A risk averse person will choose not to accept an actuarially fair gamble. If the penalty rate (f) and the probability of an audit (p) are set so that the expected gain from getting away with underreporting a dollar of income equals the expected loss from getting caught, then a risk-averse person will not underreport any income. Mathmatically, this means:
p*f=(1-p)*t rearranged: f=((1-p)/p)*t

6.3 Insurance
A risk-averse person would not accept an actuarially fair gamble, but sometimes they don’t have a choice – some risks just inherently exist. Risk averse people will pay to get rid of the risks they have to face.

Fair Insurance
The premium of the insurance policy is the price of obtaining insurance coverage.  Actuarially fair insurance is when the premium equals the expected payout by the insurance provider. (The probability of the event happening times the cost if it does happen.) For an actuarially fair insurance policy, the premium for $1 worth of insurance is simply the probability of the “bad” state of the world occurring. (e.g. if the probability of an injury is 1/5 then a person can buy $1 of actuarially fair insurance for $0.20.)

Budget Constraint with Fair Insurance
The budget constraint shows the trade-offs between the consumption if the event occurs and the consumption if the event doesn’t occur. The budget line associated with actuarially fair insurance is a straight line that passes through the endowment point and whose slope is –p/(1-p).

Preferences
The shape of the indifference curves depend on a person’s tastes. If a person is risk averse, the indifference curves exihibit a diminishing marginal rate of substitution (bowed toward the origin), and on points along a 45 degree from the origin, the marginal rate of substitution is p/(1-p).

Equilibrium Amount of Fair Insurance
When insurance is actuarially fair, the equilibrium occurs where the consumption in the case of the “bad” event is equal to the consumption in the case of “good” event. The individual is fully insured in the sense that consumption is the same in every state of the world.

Just as a capital market allows an individual to spread consumption over different periods of time, an insurance market allows an individual to spread consumption over different states of the world.

The Demand for “Unfair” Insurance
Most real-world insurance policies are “unfair” in the sense that the premium exceeds the expected monetary benefit. This is because insurance firms need some margin to cover operating costs.

Changing the Premium
When the insurance policy is actuarially unfair, even a risk-averse individual purchases less than full insurance. Intuitively, when the premium is higher than the expected payoff, it is rational for an individual to assume some of the risk in return for reducing payments to the insurance provider.

Changing the Probability of a Lawsuit
The probability of an accident occurring does not affect the budget constraint. However, it does affect the indifference curves. When the probability of a lawsuit (or an accident) increases, tastes for the contingent commodities change. The indifference curves become steeper.

The Importance of Insurance
There are many examples of insurance that are not purchased through formal contracts. For example, a person might buy a safer car even though it’s more expensive as insurance against an accident.

6.4 Decision Making with Many Uncertain Outcomes: von Neumann-Morgenstern Utility
Decision Trees
A decision tree is a schematic representation of a choice problem that shows the possible outcome and how they are related to current actions. This allows us to look at more complicated situations. A decision node is a point in a decision tree where an individual faces a decision, with branches coming out of it representing the available choices. At chance nodes, the outcome is a random variable. Terminal nodes show the values of final outcomes. (An example would be a person decides to go to business school or law school (decision node), and then they either make partner in a firm or fail, and Wall Street is either bullish or bearish (chance nodes), and the result of these gives all of the various salaries that could be achieved (terminal nodes).

Utility Functions for Uncertain Situations
The decision tree shows the link between actions and consequences, but it does not evaluate these consequences. To do this, we need a utility function that takes into account the utility of consumption in each state of the world weighted by the probability of that state of the world occurring.

A von Neumann-Morgenstern utility function is a utility function where the utility associated with some uncertain event is the expected value of the utilities of each of the possible outcomes.

Decision trees in conjunction with von Neumann-Morgenstern utility functions can be used to break up complex problems into simple components, by following these steps:
1. Sketch a decision tree to keep track of all the outcomes. Each time a choice has to be made, draw a decision node with one branch for each alternative. Uncertainty is captured by having “Nature make the choice” at a chance node.
2. Evaluate the utility function to find the utility of each outcome at each terminal node.
3. Find the expected utility associated with each option.
4. Compare the expected utilities of the various options. Choose the one that has the highest expected utility.

Sequential Decisions
Sequential decisions are solved using backward induction. At each step along the way, choose the optimal action for continuing. (Work backwards in making decisions.)

Applying von Neumann-Morgenstern Utility Functions: The Value of Information
When a particular piece of information has no effect on an individual’s actions, it has no economic value to that individual. (e.g. A person can decide to rent a house or put money in a savings account, the house may or may not have a radon problem. If it does, they make no money from renting. We could find the expected value of choosing to rent or put money in a savings account. Then we could re-evaluate – if the person were able to complete a radon test first, and then decide whether to rent or put money in savings, then the expected value may be different. The difference in expected value is the value of the information.)

Microeconomics: Chapter 5 The Household as a Supplier

Chapter 5 The Household as a Supplier
5.1 Labor Supply
For most households, labor is the single most important source of income.

Budget Constraint and Indifference Curves
For analyzing supply, a budget constraint is usually drawn between leisure and consumption. If an individual can work as many hours as he or she wants at a wage rate of w1, then the individual’s budget constraint between leisure and consumption is a straight line whose slope is –w1. At a given point, leisure is the number of hours between the origin and that point on the horizontal axis, and the number of hours is the space between that point on the horizontal axis and the horizontal intercept.

The horizontal intercept of the budget constraint line between leisure and consumption is the time endowment (T). This is the upper limit of time that can be dedicated to labor or leisure by an individual in a specified period.

The equation for the budget constraint is consumption (c) equal to wage per hour (w) times the number of hours worked (T-n), where n is the number of hours at leisure.
(5.1) c=w*(T-n)   can rewrite as: c+w*n=w*T

The amount of money an individual would have if he worked every available hour is his value of the time endowment, this is w*T.

To characterize tastes, we use a set of convex indifference curves between leisure and consumption. The optimal amount of labor to supply is defined by a tangency between the budget constrain and an indifference curve. (Image pg. 124)

Comparative Statics and the Consumption-Leisure Model
Depending on the slope of the indifference curves, a decrease in wages can either increase or decrease the hours of labor supplied. The substitution effect and income effect work in opposite directions, and the result depends on which effect dominates.

If the substitution effect dominates, labor supply decreases when wage decreases. This is consistent with the statement “Now that my wage rate has fallen, it’s really not worth it for me to work as much as I used to.”

If the income effect dominates, labor supply increases when wage decreases. This is consistent with the statement, “Now that my wage rate has fallen, I have to work more to maintain my standard of living.”
[Images page 125.]

Labor Supply Curve
The labor supply curve is a schedule showing the relationship between the quantity of labor supplied and the wage rate, all else equal. The supply of labor curve is found by determining the utility-maximizing number of hours associated with each wage rate (this can be found by looking at various budget constraint possibilities and their associated indifference curves).

The supply curve will slope upward if the substitution effect dominates the income effect. If the income effect dominates that substitution effect, then the supply curve will slope downward. It is possible that the substitution effect will dominate at low wage rates and the income effect will dominate at high wage rates. This results in a labor supply that bends back on itself (starts downward sloping, then becomes upward sloping.)

The Work Incentives of AFDC
The Aid to Families with Dependent Children (AFDC) program provided cash to families with dependent children and an absent or incarcerated parent. The program reduced the grant by one dollar for each dollar of earnings – it was argued this might reduce incentives to work. The budget constraint line for a person on this program is kinked. It has a normal slope down to the level of AFDC, and then becomes horizontal. On that horizontal section, when a person increases work by one hour, they increase their wage, but also decreases the AFDC payment, so the total income is still the same. Based on this result, we can see that a rational person may work for wages higher than the AFDC payment, but would never work any hours if they were on the horizontal section of the line (wage is less than or equal to the AFDC payment). This is because on this line, they can increase leisure “for free,” or in other terms, their wages are being taxed 100 percent along this area.

Producer Surplus
The producer surplus is the amount of income an individual receives in excess of what she would require in order to supply a given number of units of a factor. The area above the supply curve and below the wage rate is the producer surplus.

Unemployment Insurance
The concept of consumer surplus can be used to help understand unemployment insurance. A person who initially works 2000 hours at $20 per hour does not need $40,000 to compensate him for his job loss. Instead, he should be compensated equal to the area of his producer surplus. Although unemployment forces the person to consume more leisure than he would prefer, he does place some value on that leisure, and this should be taken into account in determining the net cost to him of being unemployed.

The Supply of Labor to Occupations
The total number of hours supplied to a given occupation is the total numbers of hours in the economy times the proportion of hours deveoted to this profession.

Market Supply Curve of Labor
The market supply curve of labor is a schedule showing the aggregate quantity of labor that all individuals in the market are willing to supply at each wage rate, all else equal. This can be found by summing horizontally he supply curves of labor for all individuals.

Occupational Choice
People are concerned both about the wages they make as well as the nonmonetary aspect of their jobs (clean, safe jobs are preferred to dangerous, dirty jobs). The wage premium paid to compensate workers for taking jobs with undesirable characteristics is the compensating differential.

For a low-skilled individual, the available set of combinations of safety and wage rates lies below that for a high-skilled individual. However, within their constraints, both prefer a higher-paying and safer job. [Image page 136]

5.2 Capital Supply
Real capital are those things made to aid future production, such as drill presses, factory buildings, office desks, and computers. These are purchased using money borrowed from households’ savings.

Financial capital is the money that is lent to firms to purchase or rent real capital (this comes from people’s savings).

Life-Cycle Model
The life-cycle model is a model that says people’s consumption and saving decisions during the given year are the outcome of a decision-making process that takes into account their lifetime economic circumstances.

Intertemporal Budget Constraint
The intertemporal budget constraint is the budget constraint in the life-cycle model, showing the trade-off between consumption levels at different periods. It has current consumption on the horizontal axis, and future consumption on the vertical axis, so that the budget constraint shows the trade-off between current and future consumption. It has a slope of –(1+i) and must pass through the endowment point.

The endowment point is the feasible consumption bundle if the individual makes no trades with the market. In the life-cycle model, it is the bundle an individual can consume if he or she neither saves nor borrows. This is the point (I0,I1), where I0 is his current income and I1 is his future income.

If the person decides to save some of his money, he increases future consumption by (1+i)*S (where i is the interest rate, and S is the number of dollars he decides to save).

If a person decides to borrow money to increase future consumption, he can increase current consumption by B, and reduce future consumption by (1+i)*B.

The present value of the endowment Is the maximum level of present consumption that can be obtained, given the endowment. (This is the horizontal intercept of the intertemportal budget constraint). It is equal to I0+I1/(1+i) where I0 is his current income and I1 is his future income.

Intertemporal Indifference Map
Indifference curves showing the preferences for future as opposed to present consumption have a marginal rate of substitution that is called the marginal rate of time preference. For an ‘impatient’ individual, the marginal rate of time preference exceeds one when future consumption equals present consumption; that is, the negative of the slope is greater than one around the 45 degree line.

Equilibrium in the Life-Cycle Model
The equilibrium is the point where the intertemporal budget constraint is tangent to the indifference curve. If this point is to the left of the endowment point, the person will be a saver. If this point is to the right of the endowment point, this person will be a borrower.

Comparative Statics with the Life-Cycle Model
Saving and Interest Rates
A change in the interest rate can either lower or increase saving. The effect depends on whether the substitution effect or income effect dominates.
1. The Substitution Effect: When the interest rate decreases, the opportunity cost of present consumption decreases due to the reduction in the amount of future consumption that is sacrificed for each dollar of present consumption. This tends to increase present consumption, and therefore reduce saving.
2. Income Effect: If you are a saver, when the interest rate decreases, you become poorer because the people to whom you lend pay you less money. Because present consumption is a normal good, this decreases in your income tends to lower present consumption, and therefore increase saving.

Supply of Saving
The market supply curve of saving is the aggregate amount of saving that all individuals are willing to supply at each interest rate, all else equal. It is found by using a horizontal summation of the individuals supply curves.

The Taxation of Interest Income
Some argued that lowering the tax rates on interest would induce a great surge in saving, others argue that there won’t be much effect. When interest income is taxed at rate t, the effective interest rate is (1-t)/i. Hence, the opportunity cost of reducing current consumption by a dollar is [1+(1-t)i] of future consumption. This is minus the slope of the after-tax budget constraint to the left of the endowment point (on the saving section of the line). To the right of the endowment point (on the borrowing section of the line), the slope is still –(1+i). So the budget constraint is a kinked line. Therefore, for individuals who were borrowers before the tax was imposed, the system has no effect at all. If people were savers before the tax, we cannot predict whether the change will cause them to save more or less (see table above). This analysis tells us that the effect of interest taxes on the supply of saving is ambiguous.

5.3 More on Present Value
Fundamental Formulas
The present value is the amount of money you would be willing to pay today for the right to receive a given amount of money at a specified date in the future.

This depends on the discount rate. The discount rate is the interest rate used in the computation of present value.

In general, when the annual interest rate is I, the present value of a promise to pay $M in T years is:
MT/(1+i)T

If there is an income stream, rather than a one-time payment, then we add the present value of each year’s payment: PV=M0+M1/(1+i)+M2/(1+i)2+…+MT/(1+i)T

Present Value in Action
Michael Wittkowski’s Lottery Price
Michael Wittkowski won $40 million in the Illinois State Lottery. The prize was to be distributed in 20 equal installments, with $2 million paid each year. Using the equation for present value, we can see that with an interest rate of about 12 percent, this prize was only worth about $16 million.

“Truth-in-Lending” Laws
Truth in lending laws require mortgage providers to tell borrowers the sum of all the payments that will have to be made over the life of the loan. However, these calculations do not take into account present value. Without this information, it may be difficult to tell the true cost of a 15 or 30-year loan, as well as the difference between the two.

Perpetuities
A perpetuity is a stream of income that lasts forever. To find the value of a perpetuity that pays $M annually, we need to determine how much money would have to be insvested now at an annual rate of i percent to obtain $M each year. The present value of a perpetuity is: PV=M/i

Though few payments are really perpetuities, it can be a good approximation for a stream of income that comes in over a finite number of years (such as 20).

5.4 Human Capital
Human capital are the investments that individuals make in education, training, and health care that rise their productive capacity.

Human Capital as the Only Asset
The human capital production function is the relationship between investments in human capital and future gains in income.

When the individual does not have access to financial capital markets, optimal human capital investment is determined at the tangency of an indifference curve and the human capital production function.

Human and Physical Capital
When the individual has opportunities to borrow and lend at the market interest rate, the optimal amount of human capital investment is defined by a tangency between the human capital production function and a straight line whose slope is –(1+i). This occurs regardless of preferences between present and future consumption.

The separation theorem states that the existence of markets allows a person to separate his production decision from his consumption decision.

If the individual can borrow and lend at the going interest rate, he should make whatever human capital investment maximizes the present value of his endowment. Once the present value is maximized, particular preferences determine how much to consume in the present and how much in the future.

Microeconomics: Chapter 4 Price Changes and Consumer Welfare

Chapter 4 Price Changes and Consumer Welfare
4.1 Income and Substitution Effects
The Law of Demand is the notion that demand curves slope downward – when the price of a commodity goes up, the quantity demanded goes down, ceteris paribus.

A price increase changes the feasible set on the budget constraint diagram. When the price of one good increases, the intercept for that good changes, and the budget line pivots, from B1 to B2. As a consequence, the opportunity cost of that good increases (line B2 is steeper than line B1). In addition, the consumer is poorer in the sense that some bundles are no longer available. The net effect of a price increase is the effect on quantity demanded of the substitution effect and the income effect.

The substitution effect is the effect of a price change on the quantity demanded due exclusively to the fact that its relative price has changed. It is always negative when a price inceases.

The income effect of a price increase is the effect on quantity demanded due exclusively to the fact that the consumer’s real income has fallen.

Finding the Substitution and Income Effect
The substitution effect is found by looking at how the quantity of goods changes when the price goes up (the budget line shifts) and simultaneously income adjusted so the level of utility is the same (the new line is parallel to the pivoted budget line, but shifted to be tangent to the original indifference curve). The change from this point to the actual new equilibrium is the income effect of the change – it is the effect when income falls, but relative prices stay the same.

Graphical Representation
The Substitution and Income Effect: Normal Good
A price increase leads to both a substitution effect and an income effect. The substitution effect is the change in quantity demanded induced by the higher price while keeping the individual at his initial utility level. It is represented by the movement from e1 to ec . The income effect is the change in quantity demanded induced by the change in income, and is represented by the movement from ec to e2. Distance C on the vertical axis is known as the compensating variation of the price change.
[Image on pg. 96]

The Substitution and Income Effect: Inferior Good
When income increases, ceteris paribus, the quantity demanded of sugar goes down. Hence, the income effect of a price increase increases quantity demanded, from xc’ to x2’. Nevertheless, quantity demanded falls when price increases, because the substitution effect (from x1’ to xc’) outweighs the income effect. [Image on pg. 98]

The Substitution and Income Effect: Giffen Good
A Giffen good Is a commodity (which is always inferior) whose demand curve slopes upward.

If quantity demanded increases in response to an increase in price, then the good is not only inferior, it is a Giffen good. The huge income effect, which increases quantity demanded outweighs the substitution effect, which decreases quantity demanded. [Image on pg. 98]

The compensated response to a price change is the change in quantity demanded resulting from changing the price while simultaneously compensating the individual with income; the substitution effect.

The uncompensated response is the observed change in quantity demanded in response to a change in the price.

Algebraic Representation
Observed response = Substitution effect + Income Effect

∆x/∆p=(∆x/∆p)Compensated+ Income effect

The Slutsky Equation is a decomposition of the effect of a price change on quantity demanded into a substitution effect and an income effect.
∆x/∆p=(∆x/∆p)Comp-x1*(∆x/∆I)

4.2 Compensating and Equivalent Variations
The compensating variation is the amount of money by which an individual would need to be compensated for a price change to remain at his initial level of utility. (e.g. After the price of sugar increases, how much money does Samson need to bring him back to his initial level of utility?) The compensating variation is the same as the vertical distance between the two parallel budget lines when computing the substitution effect.

The equivalent variation is a change in income that is equivalent in its effect on utility to a change in the price of a commodity. (e.g. Given Samson’s bundle before the price increases, how much money would you have to take away from the individual to reduce his welfare as much as the price increase does?) To find the equivalent variation, take the budget constraint associated with the initial prices (B1), and shift it in a parallel fashion until it is tangent to the new indifference curve. The vertical distance between the two curves is the equivalent variation.

Comparing the CV and EV
The compensating variation and the equivalent variation are not equal. The two measures are different because they evaluate the welfare change in terms of different sets of relative prices. The compensating variation determines how much income, given the new prices required to compensate for the price increase (The two lines being compared have the slope created by the new prices). The equivalent variation determines how much income would have to be taken away at the original prices to do equivalent damage to the consumer’s welfare as does the price change. (The two lines being compared have the orginal slope)

4.3 Applying Compensating and Equivalent Variations
Evaluating Price Subsidies
Sometimes the government provides subsidized prices, for example on housing.  Using compensating and equivalent variations, it is possible to examine how the subsidy affects the budget constraints of the individual. As long as indifference curves have the usual shape, any subsidy that changes relative prices is inefficient in the sense that the value to the recipient is less than the cost to the government.

President Carter’s Gasoline Tax
An argument was made to impose a tax on gasoline and rebate the procedes back to the gasoline consumers. These methods of compensating and equivalent variations help to tell whether this plan is theoretically possible.

Because the gasoline tax raises its opportunity cost in terms of all other goods, it is in the consumer’s interest to consume less gasoline than he did initially – even after the rebate.

The tax would therefore succeed in reducing gasoline consumption (substitution effect), but it would make gasoline consumers worse off than before the tax (the rebate was less than the compensating variation of the price increase).

4.4 Consumer Surplus
The Demand Curve as a Marginal Valuation Schedule
The marginal value of consuming a unit of a commodity is the price on the associated point of the demand curve. In this case, the marginal value of the first unit is 50 cents (A), of the second unit 34 cents (B), and the third unit, 26 cents (C). These amounts correspond to areas under the demand curve A, B, and C, respectively.

When the demand curve is smooth, we can still interpret the vertical distance from the horizontal axis to the curve as the marginal value placed on the corresponding unit. The area under the curve between two levels of consumption represents the total value placed on those units of consumption.

The demand curve can be thought of as a marginal variation schedule – for each unit of consumption, it shows the value that the consumer attaches to the additional (i.e. marginal) unit.

Prices and Consumer Surplus
Consumer surplus is the difference between what a consumer is willing to pay and what she has to pay. This is also referred to as the Marshallian consumer surplus.

The consumer surplus associated with the ability to purchase as many units of a commodity as you want at the going price is the area under the demand curve and above the going place (usually a triangle).

A two-part tariff is a pricing system under which a consumer first pays a lump sum for the right to purchase a good, and then pays a price for each unit of the good actually purchased. An example is the access fee for telephone service, which indicates that the consumer surplus exceeds the fee for service.

Effect of Price Changes on Consumer Surplus
When the price of a commodity changes from p1 to p2, the area behind the demand curve and between the two prices is a dollar measure of the resulting change in welfare.

Application of Consumer Surplus: Analysis of a Trade Quota
A trade quota is a restriction on the quantity of some commodity that can be imported into the country.

According to Tarr and Morke, the quota on Japanese cars decreased the quantity purchased from 2.69 to 1.91 million, and raised the price per car from $4,573 to $4,967. As a consequence, consumer surplus fell by $908 million, of which $753 million went to Japanese producers in the form of quota rents, and which $155 million was a deadweight loss.

Deadweight loss is the pure waste induced by an increase in the price above the efficient level.

“Exact” Consumer Surplus and the Compensated Demand Curve
 An exact marginal valuation schedule must eliminate the “income effects” that are embodied in the demand curve (the value that an individual puts on an additional unit of a good may depend on the amount that he has already spent on previous units of the good.)

Compensated Demand Curve
A compensated demand curve is a schedule that shows how the quantity demanded varies with the price, assuming that, as price changes, consumers are compensated with enough income to keep them at their initial utility level.

The process of creating a compensated demand curve involves repeating these steps:
1. Pivot the budget line in (for price increases) or out (for price decreases)
2. Shift the new budget line away from the origin (for price increases) or toward the origin (for price decreases) in a parallel fashion until it is just tangent to the original indifference curve.
3. Record the new point of tangency on a second graph.