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**ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip**

Consumer and Firm Behavior: The Work-Leisure Decision and Profit Maximization Chapter 4 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**Static vs Dynamic Decision Making**

In this and the next chapters, we are considering static decision making, i.e., planning over a single period. From chapter 6 to 9, we are going to discuss dynamic decision making, i.e., planning over more one period. Chapter 4 first recalls what you’ve learnt in the last semester: the micro behavior of a representative consumer and a representative firm. Chapter 5 then assembles these in a macro model in order to address some important macro issues. The role of government is also introduced. 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**Objectives of the Representative Consumer & the Representative Firm**

Representative Consumer: To maximize utility subject to budget (and time) constraint by allocating time between work and leisure; Representative Firm: To maximize profits subject to technological constraint by deciding how much labor to be hired. 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**Assumptions of the Model**

1) Two Goods: Consumption good, which is an aggregation of all consumer goods in the economy. Leisure, which is any time spent other than working in the market. e.g. Recreational activities, sleep and household work. 2) One Consumer: All consumers are identical in terms of preferences, ability, time constraint and budget constraint. Then, the economy will behave as if there were only one consumer, one that we refer to as the representative consumer. 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**Assumptions of the Model**

3) Price-Taking: The representative consumer is a price-taker, i.e., he takes all market prices as given, and acts as if his actions had no effect on those prices. 4) No Money: The economy we’re considering is a barter economy, i.e., all trade involves barter exchanges of goods for goods in the absence of money. 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**The Representative Consumer’s Optimization Problem**

Objective: to make himself as well off as possible given the constraints he faces. Two Ingredients in this problem: Consumer’s preferences Consumer’s budget constraint 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip**

Preferences The preferences of the representative consumer is captured by the utility function, U(C , l) where C is the quantity of consumption, l is the quantity of leisure Any particular pair of consumption and leisure (C , l) is called a consumption bundle. For each consumption bundle, the utility function U assigns a real number so that different bundles can be ranked. 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip**

Preferences Consider two distinct bundles (C1 , l1) and (C2 , l2) (C1 , l1) is strictly preferred to (C2 , l2) if U(C1 , l1) > U(C2 , l2) Consumer is indifferent between the two bundles if U(C1 , l1) = U(C2 , l2) Assumptions on Preferences: 1) More is always preferred to less A consumer always prefers a consumption bundle that contains more consumption, more leisure, or both. 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**Preferences 2) Consumer prefers a more diversified consumption bundle.**

If the consumer is indifferent between (C1 , l1) and (C2 , l2), then some mixture of the two will be preferable to either one. Example: Consider a new bundle (C3 , l3), where C3 = C1 + (1 – )C2, l3 = l1 + (1 – )l2 and lies between 0 and 1 (a fraction), then U(C3 , l3) > U(C1 , l1) = U(C2 , l2) 3) Consumption and leisure are normal goods. A good is normal (inferior) for a consumer if the quantity of the good that he/she purchases increases (decreases) when income increases. 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**Graphical Representation of Preferences**

An indifference curve connects a set of points, with these points representing consumption bundles among which the consumer is indifferent. A family of indifference curves is called indifference map. 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**Properties of Indifference Curves**

Consider a consumption bundle B. Since a consumer prefers more to less, any bundle that is indifferent to B must lie within quadrant II and IV. Implication: An indifference curve slopes downward. Consumption IV I B III II Leisure 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**Properties of Indifference Curves**

Consider any two bundles A and B, since a consumer prefers a more diversified bundle C to either A or B, the set of bundles that are indifferent to A and B must lie below the straight line AB. Implication: An indifference curve is convex, that is bowed-in toward the origin. Consumption A C B Leisure 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**Graphical Representation of Preferences**

Consumption, C A B C1 I2 D C2 I1 l1 l2 Leisure, l 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**Marginal Rate of Substitution**

Marginal rate of substitution of leisure for consumption (MRSl,C) is the rate at which the consumer is just willing to substitute leisure for consumption good. It is also minus the slope of the indifference curve. Convexity of indifference curve is equivalent to Diminishing marginal rate of substitution. (compared slope at A and slope at B) Consumption, C Slope = MRSl,C A C1 D B C2 I2 I1 l1 l2 Leisure, l 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**Marginal Rate of Substitution**

The MRS at A is larger (in terms of absolute magnitude) than the MRS at B. As we increase l and reduce C, i.e. moving from A to B along I1, the consumer needs to be compensated more in terms of l to give up another unit of C. The consumer requires this extra consumption because of a preference for diversity. Consumption, C Slope = MRSl,C A C1 D B C2 I2 I1 l1 l2 Leisure, l 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**Marginal Rate of Substitution**

Mathematical Derivations: Suppose indifference curve I1 represents the utility level , Totally differentiate this with respect to C and l gives Consumption, C Slope = MRSl,C A C1 D B C2 I2 I1 l1 l2 Leisure, l 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**Constraints faced by The Representative Consumer**

Two constraints: Time constraint for l Budget constraint for C The time constraint for the consumer is given by l + Ns = h where h is the total number of hours available (e.g., 24 hours a day), l is the leisure time and Ns is the time spent working (or labor supply). 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip**

Budget Constraint Sources of income: 1) Real wage income, wNs w is the real wage, i.e., the price of one unit of labor time in terms of consumption goods (the numeraire). 2) Real dividend income, Since the firms are owned by the representative consumer, any profits made by firms are distributed to him as dividends. 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip**

Budget Constraint Taxation T: A lump-sum tax, i.e. a tax that does not depend on the actions of the economic agent who is being taxed. Real Disposable Income = wNs + – T The consumer first receives income and pays taxes in terms of consumption goods, and then decides on how much to consume out of the disposable income. All disposable income is consumed, i.e. C = wNs + – T = w(h – l) + – T 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip**

Budget Constraint Reasons: Since the consumer only lives for one period, there is no incentive to save anything. Since more is preferred to less, any wastage is not optimal. The consumer’s budget constraint can be written as C + wl = wh + – T RHS = Total implicit real disposable income LHS = Implicit real expenditure on consumption goods and leisure Note: w can also be interpreted as the market price, or the opportunity cost, of leisure time. 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**Graphical Representation of the Budget Constraint**

C = –wl + (wh + – T) thus slope = –w. The vertical intercept, wh + – T, is the maximum consumption that can be achieved when the consumer consumes no leisure. Case 1: < T Consumption, C A wh + – T C = –wl + wh + – T B h + ( – T)/w h Leisure, l 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**Graphical Representation of the Budget Constraint**

Case 2: > T The consumer can still enjoy C = – T > 0 even if he chooses not to work. When C = 0, l = h + ( – T)/w, but it is not feasible as the maximum time can only be h When l = h, C = – T Consumption, C A C = –wl + wh + – T – T B D h Leisure, l 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**Graphical Representation of the Budget Constraint**

The budget constraint tells us what consumption bundles are feasible to consume given the market real wage (w), dividend income () and taxes (T). The consumption bundles within the shaded regions and on the budget constraint, are feasible. Thus the shaded region together with the budget constraint is called the feasible set. Consumption, C A Not Feasible Feasible B D h Leisure, l 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**Consumer Optimization**

The representative consumer is assumed to be rational, i.e. he always chooses the best feasible consumption bundle, or the optimal consumption bundle. “Best” in the sense that it lies on the highest possible indifference curve. “Feasible” in the sense that it lies within the feasible set. 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip**

Graphical Solution Suppose > T. Claim: H is the optimal consumption bundle. Reasons: Any bundle inside the budget constraint is not optimal (compare J to F). B is preferred to any point on BD. For any point on AB, the consumer can always improve by moving closer to H. Consumption, C A F H J I2 I1 E – T B D h Leisure, l 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**Mathematical Solution**

The consumer tries to solve the following constrained optimization problem max U(C , l) C , l subject to C = w(h – l) + – T and C 0, h l 0. Lagrangian L = U(C , l) + [w(h – l) + – T – C] where is the Lagrangian multiplier. 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**Mathematical Solution**

We assume that an interior solution can be obtained. This means choosing C = 0, l = h or l = 0 are not optimal (so that we can ignore the last two constraints). Formally, we can impose the restrictions: Uc(0 , l) = ∞ and Ul(C , 0) = ∞ For any C and l, to guarantee an interior solution. First-order (Necessary) conditions (FOCs): Obtained by differentiating the Lagrangian with respect to C, l and . (Recall: Lagrangian equation L = U(C , l) + [w(h – l) + – T – C] Uc(C , l) = , Ul(C , l) = w, w(h – l) + – T – C = 0. 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**Mathematical Solution**

From the FOCs, we obtain At H, where an indifference curve is just tangent to the budget constraint, the above equality holds. If MRS > w (e.g. at F), the consumer would be better off by increasing l and reducing C, thus moving closer to H. Consumption, C A F H J I2 I1 E – T B D h Leisure, l 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip**

Comparative Statics To determine how C and l changes when any of , T and w changes. Recall the FOC of the consumer’s problem, which can be written as Ul(C , l) – wUc(C , l) = (1) From the budget constraint, w(h – l) + – T – C = (2) The two form a system of equations in terms of C and l (endogenous variables). 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**Comparative Statics Totally differentiate the two equations**

–dC – wdl + (h – l)dw + d – dT = from (2) [Ucl – wUcc]dC + [Ull – wUcl]dl – Ucdw = from (1) In matrix form, Determinant of the bordered Hessian matrix A is = –Ull + 2wUcl – w2Ucc Strict quasiconcavity of U > 0. A 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip**

1) Changes in and/or T Using Cramer’s Rule, we get The assumption that consumption and leisure are normal goods is equivalent to the conditions –Ull + wUcl > 0 and Ucl – wUcc > 0. 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**1) Changes in and/or T Graphical Illustration**

Consider a net increase in – T. Since w (slope) remains the same, the budget constraint makes a parallel shift (from AB to FJ). Since disposable income , while prices remain the same, there is only a pure income effect on the consumer’s choices. The new optimal consumption bundle is K, where both C and l (normal goods). Consumption, C F A K C2 H C1 I2 J I1 B D l1 l2 h Leisure, l 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**1) Changes in and/or T Graphical Illustration**

Remark: The increase in consumption (C2 – C1) is less than the increase in nonwage income (distance AF). Since the consumer is working less (leisure ), wage income . This will offset part of the consumption increase. Consumption, C F A K C2 H C1 I2 J I1 B D l1 l2 h Leisure, l 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip**

2) Changes in w Using Cramer’s rule, C is normal good –Ull + wUcl > 0, together with > 0 and Uc > 0 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip**

2) Changes in w However, we cannot determine the effect of a change in w on l. Reason: It depends on the relative magnitude of the opposing income and substitution effects. Substitution effect: w Opportunity cost of leisure (l becomes more expensive relative to C) Demand for leisure Income effect: w Wage income Demand for leisure (normal good) 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**2) Changes in w Graphical Illustration**

Suppose > T and w . The budget constraint shifts from ABD to EBD (with a steeper slope). This shows a special case in which leisure remains unaffected. Pure substitution effect: Movement from F to O (on the same indifferent curve). Pure income effect: Movement from O to H. Both income and substitution effects act to C. Consumption, C I1 I2 E J A H C2 O C1 F B K D l1 h Leisure, l 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**2) Changes in w Graphical Illustration**

Labor supply curve which specifies how much labor the consumer wishes to supply given any real wage. Algebraically, the labor supply curve is Ns(w) = h – l(w), where l(w) is the demand function for leisure. Substitution effect > Income effect Upward sloping labor supply curve Net in ( – T) Upward shift in labor supply curve Real Wage, w Ns Employment, N 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**Example: C and l are perfect complements**

Suppose the consumer’s utility function can be represented by U(C , l) = min{C , al}. (Leontief Function) where a is a positive constant. Note that more is not always preferred to less. The consumer can be better off only if he receives more of both goods. Thus, it is always optimal to choose C = al. Consumption, C C = al A I2 E I1 F B D h Leisure, l 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**Example: C and l are perfect complements**

Combining C = al and the budget constraint gives In this case, This is because with perfect complements, there are no substitution effects. Thus leisure as real wages . 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**The Representative Firm**

The firm owns productive capital and hires labor to produce consumption goods. Production technology is captured by the production function, which describes the technological possibilities for converting factor inputs (capital K and labor Nd) into outputs Y. Y = zF(K , Nd) where z is total factor productivity. z both K and Nd will be more productive. 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**Assumptions on Production Function**

Production function exhibits constant returns to scale (or homogenous of degree one). For any x > 0, xY = zF(xK , xNd). If all factor inputs are changed by a factor x, then output changes by the same factor x. In this case, a perfectly competitive economy with numerous small firms will behave in exactly the same way as one with a single representative firm (same level of efficiency). Increasing return to scale: zF(xK , xNd) > xzF(K , Nd). Decreasing return to scale: zF(xK , xNd) < xzF(K , Nd). 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**Assumptions on Production Function**

Positive marginal product of capital (MPK) and marginal product of labor (MPN). MPK (MPN) is the additional output that can be produced with one additional unit of capital (labor), holding constant the quantities of labor (capital). Fix the quantity of labor at N*, then the MPK at K* is the slope of the production function at point A. Output, Y Slope = MPK F(K , N*) A K* Capital Input, K 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**Assumptions on Production Function**

Algebraically, we assume that, FK(K , Nd) > 0 and FNd(K , Nd) > 0. Conceptually, this simply means: more inputs yield more output. Diminishing Marginal Product The declining MPK and MPN is equivalent to the concavity of the production function. Algebraically, this means FKK(K , Nd) < 0, and FNdNd(K , Nd) < 0. Implicitly, we assume that F(. , .) is twice differentiable. Marginal Product of Labor, MPN MPN Labor Input, Nd 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**Assumptions on Production Function**

MPN as K Algebraically, this means Increase in the quantity of machinery and equipment enhances the productivity of the workers. F(. , .) is quasiconcave. Marginal Product of Labor, MPN MPN2 MPN1 Labor Input, Nd 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**Cobb-Douglas Production Function**

Probably the most commonly used form of production function which satisfies all the above properties Y = zKa(Nd)b where 0 < a, b < 1. a + b = 1 Constant return to scale. a + b > (<) 1 Increasing (decreasing) return to scale. If there are profit-maximizing price-taking firms and a + b = 1, then a will be the share that capital receives of national income. 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**Changes in Total Factor Productivity (z)**

Changes in z is critical to our understanding of the causes of economic growth and business cycles (real business cycles theory). Effects of z : 1) Output for given values of K and Nd. 2) MPN for given value of K. Factors that would affect z: Technological innovation Weather Government regulations Price of energy Output, Y Z2F(K* , Nd) Z1F(K* , Nd) Labour Input, N Marginal Product of Labor, MPN MPN2 MPN1 Labor Input, Nd 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**Profit Maximization Problem**

Assume that capital K is fixed. Then the firm’s problem is to choose a quantity of Nd in order to maximize its profits. The representative firm is assumed to behave competitively, i.e. taking the real wage w as given. The problem can be stated as (choosing Nd) max = zF(K , Nd) – wNd Similar to the consumer’s problem, we assume FNd(K , 0) = ∞ and FNd(K , ∞) = 0 to ensure interior solution in the firm’s profit maximization problem. 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**Profit Maximization Problem**

The optimal condition (FOC) is z[∂F(K , Nd)/∂Nd] = w This states that it is optimal for the firm to hire workers up to a level in which the MPN equals the real wage. Graphically, the optimal quantity of labor N* is at A, where the slope of total revenue function is equal to the slope of the total variable cost function. The maximized profits * is given by the distance AB. Revenue, Variable Costs wNd zF(K , Nd) A B N* Labor Input, Nd 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**Profit Maximization Problem**

The FOC of the profit-maximization can also be interpreted as the firm’s demand curve for labor, for given values of z and K . The optimal condition (FOC) is MPN(K , N) = w. Diminishing MPN implies w and N are inversely related. Real Wage, w MPN or Labor Demand Curve Quantity of Labor Demanded, Nd 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**Comparative Statics Recall the FOC of the firm’s problem**

zFNd(K , Nd) = w Totally differentiate this gives zFNdNddNd – dw + FNddz + zFKNddK = 0. Thus, we obtain 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**Quasiconcavity A function f(x) is quasiconcave if**

f(x1) f(x2) f[x1 + (1 – )x2] f(x2) for any 1 0. f is strictly quasiconcave if f(x1) f(x2) f[x1 + (1 – )x2] > f(x2) for any 1 > > 0. Consider a strictly quasiconcave utility function U(C , l). Suppose x1 = (C1 , l1), x2 = (C2 , l2), then U(x1) = U(x2) U[x1 + (1 – )x2] > U(x1) = U(x2) Thus the indifference curves are strictly convex. 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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**ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip**

Quasiconcavity Strict quasiconcavity also implies that the bordered Hessian matrix of the utility function is negative definite, i.e., –Ull + 2wUcl – w2Ucc > 0 2017/3/31 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

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