Demand curve

Demand and supply curves.

The demand curve is the graphical representation of the mathematical relationship between the maximum amount of a certain good or service that a consumer would be willing to buy and its price.

The demand curve, along with the supply curve, is one of the theoretical analysis tools used since neoclassical economics to predict price determination. The intersection point between both curves is known as the equilibrium between supply and demand.

Introduction

The demand curve is a useful construct for predicting the possible or probable effect of certain economic situations on the frequent consumption of goods. The demand curve is often spoken of as a really existing object, although in reality it is an abstract object whose existence is derived from concrete mathematical assumptions that are sometimes only approximately true. In addition, the demand curve and its properties depend on consumers presenting perfect rationality, merchandise being infinitely divisible, and another series of assumptions, which have been criticized. However, even with the limitations that the above abstractions may impose, the demand curve is a useful theoretical construct for understanding the qualitative behavior of markets, and in many cases it is an empirically adequate description.

From a mathematical point of view the "curva" of demand of a consumer or a market with n goods or products, is a dimension hypersurface n in space ${displaystyle scriptstyle mathbb {R} _{+}^{2n+1}}$. If for a specific situation both the available rent and the prices of n-1 products (ceteris paribus) then can project such hyper-surface over ${displaystyle scriptstyle mathbb {R} _{+}^{2}}}$ to genuinely build a demand curve proper.

The Demand Curve

Factors that determine demand

It should be remembered that the factors that determine the demand for a good are its price, the price of other goods, the personal income of the consumer and also the preferences or tastes of individuals. The displacements along the demand curve express the variation in the quantity demanded due to the price effect, assuming that the other factors remain constant.

For a given consumer, who consumes n different goods, the demand of this consumer for a certain product P will depend not only on disposable income and his preferences but also of the price of the n-1 products that make up your shopping basket, only when considering the assumption of ceteris paribus for the markets of the other n i>-1 products and income will result in a demand curve for P solely dependent on the price of the product P.

Price demand curve

The price demand curve normally has a downward trajectory that shows how, as the price rises, the consumption of the product decreases. Exceptionally, there are some goods, called Giffen goods, for which the price demand curve is not decreasing. A Giffen good can only exist in a market with other substitutable goods.

Shift of the demand curve

When the demand curve shifts to the right, it explains an increase in demand due to a change in a factor other than price, and when the curve shifts to the left, this shows a decrease in demand also due to the change in a factor other than price.

Shifts in the demand curve may be due to:

• The increase in the claimant population of the good.
• Changes in prospects for future prices.
• Changes in consumer preferences.
• The increase in the available income of some consumers.
• If the demand for a particular good is being considered P regardless of the rest, the alteration of the price of any of the other goods can result in a shift in the good P.

The Demand Curve and Equilibrium

For the equilibrium point between supply and demand to be unique, there are several characteristics that the demand curve must meet:

• Decreasing - Requires that the elasticity to the price is negative for the entire domain of the function.
• Continuity - It depends on the infinite divisibility of good.
• Referral - It depends on the structure of the indifference curves.
• Full preferences - The consumer has to know which of the following cases applies to him: X/2005Y or X~Y. He wants to understand that one prefers X, or rather Y, or cares about those options.
• Consumer Rationality - If the consumer prefers X to Y, and prefers Y to Z, he has to prefer X to Z.

Deduction of a consumer demand

Under certain idealized mathematical assumptions the existence of a "curve" of demand for a rational consumer can be demonstrated for which "curves" of continuous indifference can be defined. In a market with n goods available the "curva" of demand as the "curves" are hypersurfaces of n dimensions, and not a curve as it happens in a market of a single good that is neither complementary nor substitute for other goods. It is usually assumed that an idealized rational consumer knows beforehand the available income and plans its consumption for a certain period of time choosing to consume in it an amount that maximizes its "satisfaction" and at the same time fulfills the budgetary restriction that the cost of the consumed amounts does not exceed the available income. Mathematically that implies finding the maximum utility (1) on a certain set ${displaystyle scriptstyle Sigma }$ (which is the compatible set (2) with the budget restriction:

(1)${displaystyle max _{(Q_{1},dotsQ_{n})in sigma }f_{U}(Q_{1},dotsQ_{n})}$

(2)${displaystyle sigma ={(Q_{1},dotsQ_{n})in mathbb {R} ^{n}IND P_{1}Q_{1}dots +P_{n}Q_{n}{n}leq Y}}}$

Under certain reasonable conditions on utility function ${displaystyle scriptstyle f_{U}}$ It can be shown that the previous problem admits a unique solution for a level of income and a set of given prices and therefore defines a function or "curve" of demand.

Decreasing character

It can also be demonstrated without requiring it a priori that if the utility functions are different and convex then the demand function will be fulfilled. ${displaystyle scriptstyle mathbf {Q} =f_{D}(mathbf {P}}}}}$ is "decreasing" at the price or more exactly than:

${displaystyle left({frac {partial Q_{i}}{partial P_{i}}}}}}}{P_{j}[neq P_{i}}}}}}}}$

It is said that two goods A and B are complementary when it is true that:

${displaystyle left({frac {partial Q_{B}}}{partial P_{A}}}}}}{P_{B}}}leq 0}$

While for goods A and B that are substitutes it would be fulfilled that:

${displaystyle left({frac {partial Q_{B}}{partial P_{A}}}}}}{P_{B}}}geq 0}$

Existence of the demand curve

The existence and uniqueness of the demand curve under the above conditions can be proved from the implicit function theorem. To prove this, it may be necessary to pose a problem of conditioned extremes, using the method of Lagrange multipliers. To do this, the auxiliary function is defined:

${displaystyle Phi (Q_{1},dotsQ_{n};P_{1,}dotsP_{n};lambda)=f_{U}(Q_{1},dotsQ_{n})-lambda (Y-sum _{k}P_{k}$

The previous function has a relative maximum when the utility reaches a maximum, which implies that the following relations between marginal utilities are fulfilled:

${displaystyle {frac {partial Phi }{partial Q_{i}}={frac {partial f_{U}}}{partial Q_{i}}}}}}-lambda P_{i}=0}$

From previous relationships you can clear one of them ${displaystyle lambda ,}$ For example:

${displaystyle lambda ={frac {1}{P_{i}}}}{frac {partial f_{U}}{partial Q_{i}}}}}}}}}}$

Now we define a function ${displaystyle mathbf {F} }$ to which to apply the theorem of the implicit function:

${displaystyle F_{i}(Q_{1},dotsQ_{n}; P_{1},dotsP_{n}):={frac {1}{P_{i}}{frac}{partial f_{U}}{partial f_{U}}{$

${displaystyle F_{n}(Q_{1},dotsQ_{n};P_{1},dotsP_{n};Y):=Y-sum _{k=1}{n}{n}P_{k}Q_{k}}}$

It is easy to check if the utility function is strictly convex:

${displaystyle mathbf {F}:{mathbb {R} ^{+}{n}times {mathbb {R}{+}{n}{n}{n}{times mathbb {R} tomathbb {R} ^n}},quad mathbf {bf}{bf}{$

The theorem of the implicit function applied to the previous function implies that there is a function ${displaystyle f_{D},}$ such that:

${displaystyle mathbf {Q} ^{*}=f_{D}(mathbf {P};$

Function ${displaystyle f_{D}(cdotcdot)}$ is precisely the function that gives demand curve for the vector of prices and the available income of the consumer.

Slope of the Demand Curve

If some basic assumptions about the utility functions considered above are included, it can be shown that the demand curve slopes downward, or more exactly than for any good:

${displaystyle {frac {partial Q_{i}}{partial P_{i}}}leq 0qquad forall i}$

The above equations are interpreted as that the claimed amounts of an asset should decrease by increasing the price of the asset, keeping everything else equal (i.e., maintaining the level of income and the price of the rest of the goods). The previous magnitudes are precisely the terms of the Jacobin matrix that makes the function differential ${displaystyle scriptstyle mathbf {Q} ^{*}=f_{D}(mathbf {P}; y)}$ I mean:

$♪$

By the chain rule of functions of several variables and the implicit function theorem, the above Jacobian matrix can be expressed as a product of Jacobian matrices:

(♪)${displaystyle D{f_{D}}=D_{mathbf {P} }mathbf {Q} ^{*}=-(D_{mathbf {Q} ^{*} ^mathbf {F})^{-1}D_{mathbf {P} }mathbf {F}}$

In general, the previous expression is very complicated for a totally general utility function. For a separable utility function:

${displaystyle f_{U}(Q_{1},Q_{2},dotsQ_{n})=f_{1}(Q_{1})+f_{2}(Q_{2})+dots +f_{n}(Q_{n})=sum _{i=1}{nf_{i}$

The expression (*), is a more easily calculable result for example for the first good:

(**)$## ##############################################################################################################################################################################################################################################################$

If we assume that marginal utility is strictly decreasing:

${displaystyle f''_{i}(Q_{i})leq 0,qquad forall iin {1,dotsn}}}$

Then yes ${displaystyle scriptstyle n}$ is pair the numerator is positive and the denominator (**) is positive and the negative numerator, if it is odd the numerator is negative and the positive denominator, and therefore in all cases under the previous condition the expression becomes negative, and therefore it is proved in that case that the demand curve has negative slope.

Example: Market for two goods

In this section we consider applying the theory of the previous sections to a market for two goods. In this case, the utility function and the budget constraint will be given by:

${displaystyle U:=F_{U}(Q_{1},Q_{2};Y),qquad P_{1}Q_{1}Q_{2}Q_{2}=Y}$

The Jacobian matrix of quantities versus prices will be given by:

$## ###########################################################################$

Being ${displaystyle scriptstyle D vis0}$ a negative amount, assuming that the utility function is non-decreasing and convex have the conditions:

${displaystyle partial _{Q_{i}}F_{U}={frac {partial F_{U}{partial Q_{i}}}}}geq 0,qquad partial _{Q_{i}}{2}F_{U}={frac {partial ^{2F_{U}{o}{partial}}{$

Under these assumptions it is shown that the "curve" Demand has a negative slope at all its points since:

${displaystyle {frac {partial Q_{i}}{partial P_{i}=}underbrace {-{frac {1}{DP_{1}}}{geq}{underbrace {left({cH00FFFF}{cHFFFF}{cHFFFF}{cHFF}}{cHFF}{cHFFFF}}{cHFF}}{cHFF}{cHFFFFFFFFFF}}}}{cHFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF}}}}}{cHFFFF}}{cHFF}{cHFFFFFFFFFFFFFF}{cHFF}{cHFF}{cHFF}}{cHFFFFFFFFFFFF}{cHFF}{cHFF}}{cHFF}}{cHFF}}{cHFFFFFF}}{cH$