Economic Models:

Basic Mathematical Tools applied in economics

Simplified representations of reality play a crucial role in economics.

Why models?

Models in EconomicsA model is a simplified representation of a real situation that is used to better understand real-life situations.Create a real but simplified economySimulate an economy on a computerEx.: Tax models, money models

The other things equal assumption means that all other relevant factors remain unchanged.

DemandA relation showing how much of a good consumers are willing and able to buy at each possible price during a given period of time, other things held constant

Law of DemandA decrease in the price of a good, all other things held constant, will cause an increase in the quantity demanded of the good.An increase in the price of a good, all other things held constant, will cause a decrease in the quantity demanded of the good.

The Demand CurveThe demand curve slopes downward because of the law of demand

Functional RelationshipsRelationship between two variables, for e.g. price and output sold, expressed in various waysTable or graphUse of equations Quantity sold depends on the price, in other words quantity sold is a function of price.

P is the independent value and Q is the dependent value

Linear Demand FunctionQX = a0 + a1PX + a2N + a3I + a4PY + a5TPXQXIntercept: a0 + a2N + a3I + a4PY + a5T = 200Slope: QX/PX = a1 =-5

Non-linear functions For e.g. Total RevenueTR=PQ

Marginal Revenue (Slope of Total revenue)

MR=Change in TR associated with change in Q

Tabular form Representation

QP=100-10QTR=100Q-10Q2ARMR01000-0190909090280160807037021070504602406030550250501064024040-10

Graphical Representation & Concept of Slope & CurvatureTRTRQOABC

Slope of TR Curve at a particular point represents MR at a particular output, i.e., change in TR for an infinitesimal change in output levelImplication of slope for any variable implies marginal value of the same variableCurvature depends on changes in slope or changes in marginal valueChanges in Slope

Changes in CurvatureLinear Curve Marginal value constant, no change in curvatureCurve Convex to the origin Marginal value (Slope) changing at an increasing rateCurve Concave to the origin Marginal value ( Slope) changing at a decreasing rate

Average and MarginalGraphically Average value can be derived from the total value curve. Average at a point on the Total value curve is equal to the slope of the ray from the origin to that particular pointTo increase (decrease) the average value, Average value should be less (more) than the Marginal valueAverage Value constant implies its equality with Marginal Revenue

Find out from Total Cost, Average, & Marginal CostAC = TC/QMC = TC/Q

Average Cost (AC)AC = TC/Q

Total, Average, and Marginal CostAC = TC/QMC = TC/Q

Total, Average, and Marginal Cost

Optimization Techniques In Economics different optimization techniques as a solution to decision making problemsOptimization implies either a variable is maximized or minimized whichever is required for efficiency purposes, subject to different constraints imposed on other variablesE.g. Profit Maximization, Cost Minimization, Revenue Maximization, Output MaximizationA problem of maxima & minima requires the help of differential calculus

Profit Maximization

Profit Maximization

Marginal Analysis to profit maximizationMarginal Analysis requirement for profit Maximization, Marginal Revenue = Marginal Cost (MR) (MC)Marginal Value represents slope of Total value curves,Thus slopes of TR &TC should be equal

Conditions of Profit MaximizationMR=MC is a necessary condition for Maximization, not a sufficient one as this condition also hold for loss maximizationSufficient condition requires that reaching a point of maximization, profit should start declining with any further rise in output, i.e. Slope of TC should rise & Slope of TR must fall after reaching the point of Maximization, Change in MC>Change in MR*Case Study to be discussed: An alleged blunder in the stealth bombers design

Concept of the DerivativeThe derivative of Y with respect to X is equal to the limit of the ratio Y/X as X approaches zero.

Rules of DifferentiationConstant Function Rule: The derivative of a constant, Y = f(X) = a, is zero for all values of a (the constant).

Rules of DifferentiationPower Function Rule: The derivative of a power function, where a and b are constants, is defined as follows.

Rules of DifferentiationSum-and-Differences Rule: The derivative of the sum or difference of two functions U and V, is defined as follows.

Rules of DifferentiationProduct Rule: The derivative of the product of two functions U and V, is defined as follows.

Rules of DifferentiationQuotient Rule: The derivative of the ratio of two functions U and V, is defined as follows.

Rules of DifferentiationChain Rule: The derivative of a function that is a function of X is defined as follows.

Using derivatives to solve max and min problemsOptimization With CalculusTo optimize Y = f (X):First Order Condition: Find X such that dY/dX = 0Second Order Condition:A. If d2Y/dX2 > 0, then Y is a minimum.ORB. If d2Y/dX2 < 0, then Y is a maximum.

CENTRAL POINTThe dependent variable is maximized when its marginal value shifts from positive to negative, and vice versa

The Profit-maximizing ruleProfit(p) = TR TCAt maximum profit dp/dQ = dTR/dQ - dTC/dQ = 0So,dTR/dQ = dTC/dQ (1st.O.C.)==> MR = MCd2TR/ dQ2 = d2TC/dQ2 (2nd O.C.) ==> dMR/dQ < dMC/dQ

This meansslope of MC is greater than slope of MR function

Constrained OptimizationTo optimize a function given a single constraint, imbed the constraint in the function and optimize as previously defined

**Note to the instructor: Mentioning Ceteris Paribus, the Latin word for other things equal might be interesting for some students. *********************