Intermediate Micro HW June 3, 06 Leontief & Substitution An individual has Leontief preferences over goods x and x He starts ith income y and the to goods have respective prices p and p The price of good x increases to p Decompose this individual s change in demand into income and substitution effects Leontief means utility can be represented by ux, x ) = min αx, x } We kno that demand for perfect complements comes in lockstep individuals are only illing to purchase bundles along the corner of the indifference curves, hich is given by the line αx = x They then buy as many pairs of the goods as possible: p x + p x = p x + αp x = p + αp ) x = y y p +αp So x = p +αp, x = αx = When p increases to p, demand for good changes from p +αp to p +αp To decompose this into substitution and income effects, e first leave purchasing poer constant and look at the change in demand in the absence of income effects, then e change income to reflect the change in demand in the absence of relative price changes Substitution Effect If e simply rotate the ne budget line through the same corner of the ICs hich is in effect hat e do hen e rotate the demand curve through the originally demanded bundle), demand ill not change This is the hallmark of perfect complements As long as e can still purchase the same number of pairs, e ill Thus the substitution effect is 0 See Figure Income Effect Since the substitution effect is 0, the entire change in demand is due to the income effect
We ant to buy as many pairs as e can; the increase in the price of good has reduced our effective purchasing poer, thereby reducing the number of pairs e can afford It is for this reason and this reason alone that demand shifts Again, see Figure Figure : Progression of Demand Shift under Leontief CES Demand Another commonly used class of utility functions is CES utility, hich stands for constant elasticity of substitution In general they take the form: ux, x ) = θx + θ)x ) If e take θ =, these are equivalent to
ux, x ) = x + x ) Derive the demand for CES utility ith θ =, ie, find x p, p, y) and x p, p, y) For simplicity, I ll normalize p = Remember that demand is only defined relatively if e double all prices and income, the set of things e can afford is the same, and so demand ill be the same If e allo p to vary freely, e d replace all instances of p in hat follos ith p p, and all instances of y ith y p Essentially, e re defining all monetary units in terms of good to With that in mind, e proceed by setting the MRS equal to the price ratio hich is no just p ): u/ x u/ x = x x ) = p This gives us an expression for the ratio of goods that a CES individual ill choose as a function of prices We need to include information about income to pin don exact quantities To do this, e solve this expression for x and plug it into the budget constraint, hich ill yield the demand for x as a function of p and y: p x + x = + p From hich e determine x = y +p ) x = y, x = yp +p Decompose a change in the price of good from p to p into substitution and income effects No is hen things get messy The final demand for good is given by x = y p ) +p ) The intermediate demand for good e ll call x s This is the amount of good one e choose hen p is implemented, but e can still afford x and x That is, p x + x = p x + x = y p p p + p + This essentially gives us a ne income, y, hich e can plug into the demand for x For simplicity, e ll call the constant multiplying y in y λ, so y = λy: 3
x s = λy p p ) ) + Then the substitution effect is given by x s x and the income effect is given by x x s 3 Home Production Bunter consumes to goods in quantities x and x Good is a composite consumption good, and has price per unit Consumption ofx units of good requires tx units of time, so that t is the time cost per unit of good x For example, time must be spent preparing food in order to consume it Think of x as leisure time that is not spent orking and not spent fulflling the time-cost of consuming x Bunter has a total of time T available, and earns per unit of time spent orking All of time T is consumed either in x, orking, or fulfilling the time cost of consuming x Bunter s utility function is ux, x ) = x x 3 Solve Bunter s utility maximization problem to find his demand functions for goods and The problem of Bunter is the folloing: Max x x 3 } x,x,d,h st x h, tx = d, d + h + x = T here h is the time spent orking and d is the time spent preparing food The constraint x h says that the total amount spent on good cannot exceed income, and income is obtained by orking The constraint tx = d says that the time spent cooking depends on the quantity of good Bunter intends to consume The last constraint says that the total amout of time spent cooking, loafing and orking must be equal to the total time Bunter has The problem can be simplified in order to have only to choice variables and only one constraint In fact from the first constraint e get h = x to maximize utility Bunter spends all his income) and from the second constraint e get d = tx Plugging in the third constraint e obtain Max x x 3 } x,x st tx + x + x T 4
and finally st Max x x 3 } x,x + t x + x T Since e can eliminate all constraints, e must set up the Lagrangean: L = x x 3 + λ T + t ) x x Taking derivatives to get first order conditions L =x 3 λ = 0 x L =3x x λ + t = 0 x L + t =T λ x x = 0 From the first equations e get that x = 3 +t x Substituting into the third equation: T = + t 3 + t x + x = 4x x = T 4, x = 3 4 + t T Suppose t decreases For example, ne technologies maydecrease food preparation time What is the effect on the consumptionof good? What is the effect on the amount of time this person spends orking? Be precise Clearly reducing t, x increases Working time is calculated via h = x = 3 T 4 +t Increasing t reduces the time spent orking it s in the denominator) 3 Suppose the age increases What is the effect onthe amount of time this person spends orking, and on the amount ofleisure time, x she consumes? From the expression for h e observe that an increase in the age reduces time spent orking A change in age has no effect on leisure time, hich is a simple fraction of the total time available The levels of t and only affect the share of time allocated beteen orking and preparing food 5