Utility

The basic assumption of demand theory is that consumers maximize their utility (i.e., satisfaction) subject to the constraints of their budgets.  Economists systematize utility as a function of the quantities of goods consumed.   In the interests of realism, the utility function should have two properties: (1) Total utility increases as consumption of either or both goods increases, and (2) marginal utility, defined as the extra satisfaction derived from each additional unit, decreases.  The utility function might have a peak ("bliss point') beyond which additional consumption of some goods actually reduces utility--an economic good becomes a "bad" in other words--but rational consumers should never consume to this point.

Here's a quadratic utility function for beers (B): U = 15B - B².  The total utility (red) and marginal utility (blue) schedules are graphed below:  This consumer maximizes the evening's total utility at 7.5 beers, where total utility peaks and marginal utility equals zero.  Here the marginal utility values are the increments in total utility.

Beers	Utility	Marginal  Utility
0	0	--
1	14	14
2	26	12
3	36	10
4	44	8
5	50	6
6	54	4
7	56	2
8	56	0
9	54	-2

The single-good problem is rather trivial, since consumers actually have a large selection of goods they can purchase, and we are often interested in analyzing the trade-offs consumers make between multiple goods.  The graph below shows the utility levels some individual consumer might obtain from various combinations of two goods, Spam and widgets, say.  This is the Cobb-Douglas function U = 10X^0.4Y^0.4.  The general form of these functions is U = aX^bY^c. where a, b and c are positive constants, b + c < 1, and X and Y are quantities of different goods.
 

• They are downward-sloping, since we are dealing with two goods rather than a good and a "bad."
• They do not intersect.  Each combination of goods generates a unique value of U.  Ppreferences are transitive, so that if consumption bundle A is preferred to bundle B, and B is preferred to C, then A must be preferred to C.
• They are convex to the origin.  This convexity results from the fact that the marginal utility schedule for each good declines as consumption of that good increases.  Moving northwest along any indifference curve, as the consumer trades away more widgets for Spam, the marginal utility of widgets rises and the marginal utility of Spam falls.  The slope of the indifference curve at any point is the negative of the ratio of the marginal utilities of the two goods, -MU(widgets)/MU(Spam).  This ratio, the marginal rate of substitution (MRS), is the rate at which the consumer is just willing to trade one good for the other.

The consumer's purchases are naturally constrained by a money budget.  A budget line can be superimposed on the indifference mapping, showing all combinations of goods that exhaust the budget.  Suppose widgets cost $1.50 each, cans of Spam cost $3.00 each, and the consumer's total budget (M) is $12.00.  The end-points of the budget line are easily calculated: if the consumer spent all $12 on widgets, she could buy $12/$1.50 = 8; or if she spent all $12 on Spam, she could buy $12/$3 = 4 cans.  And since the prices are constant, all just-affordable combinations of widgets and Spam lie on a straight line between these points.  The slope of the budget line is the price ratio P(widgets)/P(Spam).  This ratio, the marginal rate of transformation (MRT), is the rate at which the consumer can trade widgets for Spam in the market.

The consumer can only afford combinations of widgets and Spam that lie on (or else southwest of) her budget line.  The affordable portion of the indifference mapping is the region southwest of the budget line.  Graphically, the consumer maximizes utility by climbing as high as she can on the affordable portion of the utility surface.  The consumption bundle that provides maximum affordable utility is defined by the single point of tangency between the budget line and the highest attainable indifference curve.  On the lower of the two budget lines shown above, the utility-maximizing consumption bundle is (4 widgets, 2 Spams), which yields a total utility of 50.

Now suppose the price of widgets drops to $1.  This price reduction rotates the entire budget line outward to a new end-point on the horizontal axis at 12 widgets.  Now the utility maximizing consumption bundle is now (6 widgets; 2 Spams), which yields a total utility of 60.

The tangency of the indifference curve and the budget line implies that the slopes of these schedules are the same at the optimal consumption point.  In other words, the utility-maximizing consumers buys a consumption bundle where the marginal rate of (preference) substitution MU(widgets)/MU/(Spam) equals the marginal rate of (market) transformation P(widgets)/P(Spam):

We can rearrange this to obtain a more common-sense result:

which, in plain English, means that, to maximize utility, this consumer should allocate her budget between widgets and Spam so that the extra satisfaction per dollar spent on widgets equals the extra satisfaction per dollar spent on Spam.

Derivations of demand schedules

For any goods X₁ and X₂, a change in income or either price will shift the budget constraint line, and we can then trace out how the tangency point defining the just-affordable, utility-maximizing bundle (X₁*,X₂*) changes.  A reduction in the price of a good shifts the end-point of the budget line outward along the axis representing that good, since the consumer can now afford more of it; a price increase shifts the end-point inward toward the origin.  An increase in the consumer's income shifts both end-points of the budget line outward proportionately; a decrease in income shifts the budget line inward in parallel.

If we vary income or either price continuously, we can trace out a demand schedule.  A conventional demand schedule is the schedule of quantities of a good consumers are willing to buy at various prices of the good. In our two-good world, economic demands for X₁ and X₂ each depend on both prices P₁ and P₂ and the budget M.

If we vary the price of X₁, we can trace out a conventional own-price demand schedule for X₁.  Own-price demand schedules are always downward-sloping: the higher the price, the less people buy.

If we vary the price of the other good, X₂, we can trace out a cross-price demand schedule, for X₁ with respect to the price of X₂.  If the cross-price demand schedule is positively sloped, X₁ and X₂ are substitutes.  If the cross-price demand schedule is negatively sloped, X₁ and X₂ are complements.

If we vary the consumer's income, we can trace out an income demand schedule (Engel curve) for X₁.  Any good with a positively-sloped income-demand schedule is a normal good.  Some goods have negatively-sloped demand schedules, and are termed inferior goods.

Each of these demand functions can only be represented graphically if the other two arguments are held constant.  You should study these animated figures until you are clear on why the budget lines shift the way they do, and how the successive tangencies trace out the respective demand schedules

NOTE: It is important to distinguish between a shift in demand and a change in quantity demanded. Consider the (own-price) demand for X₁ with respect to P₁: a change in P₁ implies a movement along the demand schedule, while a change in P₂ or M (or any other variable besides P₁ which influences demand) implies a shift of the entire demand schedule.  To get clear on this, you may want to check out the answers to the micro thoery review quiz I give my FREC 444 students.

Elasticity

We can calculate elasticities of demand for either good with respect to either price or income.  Elasticity is a unit-free index of a function's responsiveness to a change in one of its arguments.  Specifically, elasticity is the percent change in an effect variable caused by a one-percent change in a causal variable.  For example, the ("own-price") elasticity of demand for X₁ with respect to P₁ is calculated as the percent change in quantity of X₁ demanded divided by the percent change in P₁.  The cross-price elasticity is the percent change in X₁ divided by the percent change in P₂, the price of the other good.  The income elasticity is the percent change in X₁ divided by the percent change in the budget M.

X₁ and X₂ are substitutes if the cross-price demands for X₁ with respect to P₂, and X₂ with respect to P₁, are positively sloped, so that the cross-price elasticities are positive too. X₁ and X₂ are complements if the cross price demands are negatively sloped and the cross-price elasticities are negative too.

X₁ is an inferior good if its Engel curve is negatively sloped and its income elasticity is negative. X₁ is a normal good if its Engel curve is positively sloped and income elasticity is positive. X₁ is a luxury good (luxury goods are a subset of normal goods) if its income elasticity is greater than one.

Consumer and producer surplus

A competitive market is characterized by full information, so everyone knows what price levels are and sellers can’t engage in price discrimination (charge different consumers different prices). A downward-sloping demand schedule implies that the WTP of most consumers is higher than the amount they actually have to pay. The aggregate amount consumers would theoretically be willing to pay for the equilibrium quantity X, above what they do pay, is called consumer surplus (CS). This measures the aggregate economic benefit realized by consumers in the market. Graphically, consumer surplus is the triangular area below the demand (WTP) schedule and above the market price. An increase (rightward shift) in supply implies a movement down the demand schedule: equilibrium price is lower for all consumers, and the area of the consumer surplus triangle is increased.

Similarly, an upward-sloping supply schedule implies that the willingness-to-sell (WTS) of most sellers is less than the market price they receive.  The aggregate amount that all sellers in the market collectively receive (P_eq x X_eq) above the minimum amounts they would be willing to accept is called producer surplus (PS).