Maths for Economics (ECO1005)
*small increments formula*
*Δz=(dz/dx)Δx+(dz/dy)Δy* -the accuracy of the formula increases as Δx and Δy become smaller and smaller
Given the consumption function c(y,t) where y=income and t=taxes, derive the impact of changes in income on consumption, assuming that taxes are a function of income.
-*total differential*: dc=(dc/dy)dy+(dc/dt)dt -then: dc/dy=(dc/dy)+(dc/dt)(dt/dy) -that is, the Δ in y has both a *direct effect dc/dy* and an *indirect effect*, working through income tax changes, on consumption
PED example
For Qa=150-2Pa-3Pb, the PED when Pa=10 and Pb=10 is: Qa=150-2(10)-3(10)=100 Ea=(dQa/dPa)x(Pa/Qa)=(-2)(10/100) = *-0.2*
df/dxi meaning
the derivative with respect to xi, holding all other elements of x constant
*own price elasticity of demand*
*Ea=(dQa/dPa)x(Pa/Qa)*
why is fxx>0, fyy>0 at a critical point not sufficient for a maximum? why is fxx<0, fyy<0 at a critical point not sufficient for a minimum?
*This alone does not confirm that the function is a minimum/maximum in ALL directions, which it must be
procedure of total differentiation
*1) start from the total differential* *2) divide through by the differential dy* -the term *(dz/dx)(dx/dy)* is the *INDIRECT EFFECT of y on z (as it first affects x, which then affects z) -the term *(dz/dy) is the *DIRECT EFFECT* of y on z
*first order conditions of Lagrangian*
*FOCs*: dL/dxi=fi-λgi=0 --> λ=fi/gi dL/dj=fj-λgj=0 --> λ=fj/gj THUS: the tangency requirement is: fi/gi=fj/gj
*minimising expenditure to achieve a certain level of utility example*
*Lagrangian* (L) =(expenditure)-λ(utility) -find the FOCs of dL/dx1, dL/dx2, and dL/dλ -solve the *compensated demand functions*
what does it mean when FOC lead to critical points in the profit function at outputs where *MC in each plant=MR in the output market*? (problems with one TR and two TC functions)
*MC must be the same in both plants*
*marginal rate of technical substitution*
*Q=f(L,K): production function depending on two factors L and K* *MRTS(L,K)=MP(L)/MP(K)* -along an isoquant, the MRTS shows the rate at which one input (ex. capital or labour) may be substituted for another, while maintaining the same level of output -*MRTS(L,K) measures how much K has to decrease if L increases by 1 extra unit, in order to keep production constant (and vice versa)* -it shows the relation between inputs, and the trade-offs amongst them, without changing the level of total output -MRTS can also be seen as (minus) the slope of the isoquant at the point in question
types of income elasticity
Ey>0 --> *normal* Ey<0 --> *inferior*
univariate optimisation
*To find the stationary points of y=f(x):* 1) Solve equation f'(x)=0 (first order condition) and you will find the stationary points, x=a 2) Check whether each point is a local minimum, maximum, or point of inflection *Second derivative test:* -if f''(a)>0, then f(x) has a minimum at x=a -if f''(a)<0, then f(x) has a maximum at x=a -if f''(a)=0, then the point cannot be classified using the available information
*marginal rate of substitution*
*U(X,Y): Utility depending on two goods X and Y* *MRS(X,Y)=MU(X)/MU(Y)* -*MRS(X,Y) of X for Y is the amount of Y for which a consumer is willing to exchange X locally -the MRS is different at each point along the indifference curve, thus it is important to keep 'locally' in the definition* -for example, if the MRS(X,Y)=2, then the consumer will be willing to give up 2 units of Y to obtain 1 additional unit of X -MRS(X,Y) corresponds to minus the slope of the indifference curve
univariate vs. multivariate
*Univariate* = curve in 2D-space *Multivariate* = surface in 3D-space
*compensated demand functions*
*a hypothetical demand function in which the consumer's income is adjusted as the price changes so that the consumer's utility remains at the same level*
what increases the accuracy of the smalled increments formula?
*changes in x and y should be small or the function should be close to being linear*
what does a Cobb-Douglas function whose powers sum up to be less than 1 imply?
*decreasing returns to scale* (THERE IS A MAXIMUM)
*implicit differentiation of a bivariate function*
*dy/dx=-(dz/dx)/(dz/dy)*
*implicit differentiation of a multivariate function*
*dy/dxi=-(dz/dxi)/(dz/dy)* (for each i=1,...,n) -we have a function z=f(y,x1,x2,...,xn)=0, and we are interested in how y depends on x1,x2,...,xn -the explicit function y=g(x1,x2,...,xn) is only implicit in f(y,x1,x2,...,xn)=0
when are the necessary conditions for a maximum met (in problems with one TR and two TC functions)?
*if slope of MR<slope of MC curve for both plants*
what does a Cobb-Douglas function whose powers sum up to be greater than 1 imply?
*increasing returns to scale* (NO MAXIMUM)
what is the change in utility when both x1 and x2 change by a small quantity? (using *smalled increments formula*)
*ΔU≈U1Δx1+U2Δx2*
for y=f(x), if x changes by a small amount Δx...
*Δy≈(dy/dx)Δx* -approximation is *exact* if f(x) is linear -if not, the accuracy of the approximation increases as Δx becomes smaller and smaller -*As Δx->0, dy=(dy/dx)(dx)
more on the interpretation of λ
*λ* *= measures the change in the objective function brought about by a small increase in the constraint at the optimum* L=f(x1,x2,...,xn)+λ[c-g(x1,x2,...,xn)] -if *c* increases by a small quantity *d*, then the objective function increased by *λxd* -if λ*<0, the maximum z decreases as we relax the constraint -λ* is equal to the rate of change in the maximal value of the objective function as the constraint is relaxed
*isoquant* (with production function)
-*Isoquant*=a curve in K,L space which represents a locus of input combinations which all produce the same level of output -dK/dL=-(df/dL)/(df/dK)=-MP(L)/MP(K)=*-MRTS(L,K) --> *the slope of the isoquant*
*a two good consumer 'dual' example*
-a consumer chooses a bundle of goods to *maximise their Cobb-Douglas utility* (objective function) subject to a *budget constraint* -*Lagrangian*: L=(utility)-λ(budjet constraint) -find the FOCs of dL/dx1, dL/dx2, and dL/dλ -divide first equation by the second, and plug this into the third equation to find the *demand functions*
*indifference curves*
-a particular level of utility is determined by the utility function: *U=f(S,F)*, where S=shelter and F=food --> {(S,F): f(S,F)=U} --> f(S,F)-U=0 -assume S and F functionally related, but the explicit relationship between them cannot be found -*the indifference curve gives all bundles of S and F between which the consumer is indifferent as they all give the same utility, U* -the equation of the indifference curve is f(S,F)=U, which is an implicit function dS/dF=-(df/dF)/(df/dS)=-MU(F)/MU(S)=*-MRS(F,S)* --> *the slope of the indifference curve*
how do we differentiate an implicit function?
-an explicit function can often be written in explicit form (ex. x^2+2y=3 --> y=(3-x^2)/2 -sometimes expressing an implicit function is explicit form is hard to do, and you may end up with very complicated explicit expressions -sometimes an implicit function does not define an explicit function
second order condition of constrained optimisation
-confirmation of a maximum or minimum depends on the *Hessian matrix* of second derivatives of the Lagrangian -if U is linear, all its second order partial derivatives are zero, and we have the following simple rule: *MAXIMUM IF:* 2(U1)(U2)(Z12)>(U1^2)(Z22)+(U2^2)(Z11) *MINIMUM IF:* 2(U1)(U2)(Z12)<(U1^2)(Z22)+(U2^2)(Z11)
example of substitution method
-consider maxf(x1,x2) subject to g(x1,x2)=0 -total differential df=f1dx1+f2dx2 --> at the optimum, df=0 for any dx1 and dx2, so we have: *dx2/dx1=-(f1)/(f2)* -on the constraint we have g1dx1+g2dx2=0, so we have: *dx2/dx1=-(g1)/(g2)* -THUS: *f1/f2=g1/g2* -*we are looking for TANGENCY between the constraint and the function*
*examples of optimization with constraints*
-consumers maximise utility U(x1,x2,...,xn), while their expenditure is constrained at the level of their income -firms minimise cost, while maintaining a set level of output
finding PED/XED/YED when given demand function and prices (and income)
-first find Q1 (plug given values into demand function) -next, find the required partial derivatives -now you can find the elasticities
constrained optimization by substitution
-rewrite "subject to" equation in terms of a single variable -substitute into objective function -differentiate and optimize
*Lagrangian*
-suppose we want to choose values of (x1,x2,...xn) to maximise f(x1,x2,...,xn) subject to g(x1,x2,...xn)=C -we create a *new variable λ* and use it to set out our problem as an *unconstrained* optimisation *FOCs*: dL/dxi=fi-λgi=0 (for i=1,...,n) PLUS dL/dλ=c-g(x1,x2,...,xn)=0
*interpretation of λ*
-the value of λ at the optimal solution tells us the effect on the optimised value of the function at a slight relaxation of the constraint -when something in the equation changes, you can predict the function to change by λ*Δ
constrained optimization with n variables and r constraints
-we need *n>r*, otherwise we don't have the degrees of freedom to satisfy all r constraints -we need the r constraints to be compatible (we need it to be possible to satisfy all r constraints at the same time)
partial differentiation
-when taking a partial derivative, the other variables are held constant -partial derivative, with respect to x1, is the slope of the tangent to the surface of x1 -dQ/dxi = slope in the xi direction
marginal product (MP) of a product
-ΔQ/Δxi = the change in production due to a one unit change in the factor xi -MP of factor xi=*limit*
*total differential*
As Δx->0 and Δy->0... *dz=(dz/dx)dx+(dz/dy)dy* (will only be exact if linear in x and y) *Interpretation:* Small Δz = small Δx times the rate of change in z with respect to x (y constant) PLUS small Δy times the rate of change in z with respect to y (x constant)
types of cross price elasticity
Ecross>0 --> *substitutes* Ecross<0 --> *complements* Ecross=0 --> *independent*
*total derivatives vs. partial derivatives*
For f(x1, x2,...,xn): -*Partial derivative*: df/dx1 assumes all other x's are constants as xi varies -Calculation of the *total derivative* of f with respect to xi does not assume that the other arguments are constant while xi varies. Instead, it allows for the other arguments to vary as xi varies.
when x and y in z=f(x,y) can be expressed as a function of single variable (ex. time t)
In order to find dz/dt, we can employ the total derivative rather than substituting t for both x and y, which may: 1) be messy 2) obscure relevant information *Procedure:* dz=(dz/dx)dx+(dz/dy)dy *dz/dt=(dz/dx)(dx/dt)+(dz/dy)(dy/dt)*
λ in a utility maximization problem
L=U(x1,x2)+λ(I-p1x1-p2x2) -in a utility maximization problem, λ* measures the MARGINAL UTILITY OF INCOME (I) (the rate of increase in maximized utility as income is increased)
Young's theorem
if the function is well-behaved (has continuous first and second order partial derivatives), the the order of differentiation does not matter
*cross price elasticity of demand*
measure of the response in demand for good A to a change in the price of good B *Ecross=(dQa/dPb)x(Pb/Qa)*
*implicit function*
when the dependent variable cannot be separated like sin(x+e^y)=3y -neither x or y are given explicitly as a function of other variables -you would have to solve this in order to get the explicit function
*explicit function*
where the dependent variable can be separated y=f(x) z=f(x,y)
bivariate optimisation
z=f(x,y) *FOC:* Stationary (or critical) points x0,y0 are found by solving the two equations: fx=0, fy=0 *SOC:* -f has a *MINIMUM* at x0,y0 if: *fxx>0, fyy>0* and *fxxfyy-fxy^2>0* -f has a *MAXIMUM* at x0,y0 if: *fxx<0, fyy<0* and *fxxfyy-fxy^2>0* -f has a *SADDLE POINT* at x0,y0 if: *fxxfyy-fxy^2<0* -if at x0,y0: *fxxfyy-fxy^2=0*, then further investigation is needed
*total derivative: function of 2 variables example*
z=f(x,y) -finding the *total derivative* does not require x to remain constant as y varies and this allows for a postulated relationship between x and y -suppose x=g(y)--> *y* is the ultimate source of change affecting z through 2 channels: *1) DIRECTLY (via f)* *2) INDIRECTLY (via g, then f)* -Note: the partial derivative df/dy expresses the *direct effect alone!* -*The total derivative is dz/dy, where z=f(g(y),y)*