Exercises Ehk
13.1. Let v p m be the indirect utility function of a representative consumer, and let ir p be the profit function of a representative firm. Let welfare as a function of price be given by v p ir p . Show that the competitive price minimizes this function. Can you explain why the equilibrium price minimizes this welfare measure rather than maximizes it 13.2. Show that the integral of the supply function between po and pi gives the change in profits when price changes from p0 to p . 13.3. An...
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Hence, if firm 2 believes that firm 1 will respond in this way, then firm 2 will not expect to profit from violating its quota. The nature of firm l's punishment can be most easily seen by thinking about the case of asymmetric market shares. Suppose that firm 1 produces twice as much output as firm 2 in the cartel equilibrium. Then it has to threaten to punish any deviations from the cartel output by producing twice as much as its rival. On the other hand, firm 2 has to only threaten to produce...
Properties of demand and supply functions
The functions that give the optimal choices of inputs and outputs as a function of the prices are known as the factor demand and output supply functions. The fact that these functions are the solutions to a maximization problem of a specific form, the profit maximization problem, will imply certain restrictions on the behavior of the demand and supply functions. For example, it is easy to see that if we multiply all of the prices by some positive number t, the vector of factor inputs that...
Duality in consumption
We have seen how one can recover an indirect utility function from observed demand functions by solving the integrability equations. Here we see how to solve for the direct utility function. The answer exhibits quite nicely the duality between direct and indirect utility functions. It is most convenient to describe the calculations in terms of the normalized indirect utility function, where we have prices divided by income so that expenditure is identically one. Thus the normalized indirect...
Specification of technology
Suppose the firm has n possible goods to serve as inputs and or outputs. If a firm uses yJ units of a good j as an input and produces y of the good as an output, then the net output of good j is given by yj y j j. If the net output of a good j is positive, then the firm is producing more of good j than it uses as an input if the net output is negative, then the firm is using more of good j than it produces. A production plan is simply a list of net outputs of various goods. We can represent a...
Uniqueness of the expected utility function
We have now shown that there exists an expected utility function u C R. Of course, any monotonie transformation of u will also be a utility function that describes the consumer's choice behavior. But will such a monotonic transform preserve the expected utility property Does the construction described above characterize expected utility functions in any way It is not hard to see that, if u - is an expected utility function describing some consumer, then so is v - au - c where a gt 0 that is,...
Exercises
1.1. True or false If V y is a convex set, then the associated production 1.2. What is the elasticity of substitution for the general CES technology y a Xj a-ix lP when ai a2 1.3. Define the output elasticity of a factor i to be If x xfx , what is the output elasticity of each factor 1.4. If e x is the elasticity of scale and et x is the output elasticity of factor i, show that e x e x - 1.5. What is the elasticity of scale of the CES technology, f x 1,2 2 1.6. True or false A differentiate...
Quasilinear utility
Suppose that there exists a monotonic transformation of utility that has the form U so,xi, ,xk x0 u xi, , xk . Note that the utility function is linear in one of the goods, but possibly nonlinear in the other goods. For this reason we call this a quasilinear utility function. In this section we will focus on the special case where k 1, so that the utility function takes the form xo xi , although everything that we say will work if there are an arbitrary number of goods. We will assume that u xi...
Comparative statics using the profit function
At the beginning of this chapter we proved that the profit function must satisfy certain properties. We have just seen that the net supply functions are the derivatives of the profit function. It is of interest to see what the properties of the profit function imply about the properties of the net supply functions. Let us examine the properties one by one. First, the profit function is a monotonic function of the prices. Hence, the partial derivative of 7r p with respect to price i will be...
The Sign Of Dxi Dwj
3.1. A competitive profit-maximizing firm has a profit function n wi, W2 lt fii wi lt f gt 2 1 2 . The price of output is normalized to be 1. a What do we know about the first and second derivatives of the functions lt f gt i wi 7 b If i u gt i,u gt 2 is the factor demand function for factor i, what is the sign of dxi dwj c Let f xi,x2 be the production function that generated the profit function of this form. What can we say about the form of this production function Hint look at the...
dyp
9.7. Consider the utility function u xi,z2,z x z z . Is this utility function weakly separable in z2, z3 What is the subutility function for the z-good consumption What are the conditional demands for the z-goods, given the expenditure on the z-goods, mz 9.8. Two goods are available, x and y. The consumer's demand function for the x-good is given by In x a bp cm, where p is the price of the x-good relative to the y-good, and m is money income divided by the price of the y-good. a What equation...
True Or False If Good 1 Is A Normal Good The The Effect Of The Grant On His
8.1. Frank Fisher's expenditure function is e p,u . His demand function for jokes is x3 p, to , where p is vector of prices and m 2 gt 0 is his income. Show that jokes are a normal good for Frank if and only if d2e dpJdu gt 0. 8.2. Calculate the substitution matrix for the Cobb-Douglas demand system with two goods. Verify that the diagonal terms are negative and the cross-price effects are symmetric. 8.3. Suppose that a consumer has a linear demand function x ap bm c. Write down the...
Homothetic utility functions
A function f Rn gt R is homogeneous of degree 1 if ix i x for all t gt 0. A function x is homothetic if x g h x where g is a strictly increasing function and h is a function which is homogeneous of degree 1. See Chapter 26, page 482, for further discussion of the mathematical properties of such functions. Economists often find it useful to assume that utility functions are homogeneous or homothetic. In fact, there is little distinction between the two concepts in utility theory. A homothetic...
Exercises 1
2.1. Use the Kuhn-Tucker theorem to derive conditions for profit maximization and cost minimization that are valid even for boundary solutions, i.e., when some factor is not used. 2.2. Show that a profit-maximizing bundle will typically not exist for a technology that exhibits increasing returns to scale as long as there is some point that yields a positive profit. 2.3. Calculate explicitly the profit function for the technology y xa, for 0 lt a lt 1 and verify that it is homogeneous and convex...
Exercises Fsu
7.1. Consider preferences defined over the nonnegative orthant by xi, x2 gt - 2 i gt 2 2 if x2 lt 2 1 2 2- Do these preferences exhibit local nonsatiation If these are the only two consumption goods and the consumer faces positive prices, will the consumer spend all of his income Explain. 7.2. A consumer has a utility function u xi,x2 max a i, 2 . What is the consumer's demand function for good 1 What is his indirect utility function What is his expenditure function 7.3. A consumer has an...
cwi y wx
Since c w, y is the cheapest way to produce y, this function is always nonpositive. At w w , g w 0. Since this is a maximum value of lt w , its derivative must vanish Hence, the cost-minimizing input vector is just given by the vector of derivatives of the cost function with respect to the prices. I Since this proposition is important, we will suggest four different ways of proving it. First, the cost function is by definition equal to c w,y wx w, y . Differentiating this expression with...
The elasticity of substitution
The technical rate of substitution measures the slope of an isoquant. The elasticity of substitution measures the curvature of an isoquant. More specifically, the elasticity of substitution measures the percentage change in the factor ratio divided by the percentage change in the TRS, with output being held fixed. If we let Afe i be the change in the factor ratio and A TRS be the change in the technical rate of substitution, we can express this as This is a relatively natural measure of...
The Slutsky equation
We have seen that the Hicksian, or compensated demand curve, is formally the same as the conditional factor demand discussed in the theory of the firm. Hence it has all the same properties in particular, it has a symmetric, negative semidefinite substitution matrix. In the case of the firm, this sort of restriction was an observable restriction on firm behavior, since the output of the firm is an observable variable. In the case of the consumer, this sort of restriction does not appear to be of...
Homogeneous and homothetic technologies
A function x is homogeneous of degree k if x ife x for all t gt 0. The two most important degrees in economics are the zeroth and first degree.2 A zero-degree homogeneous function is one for which 2 However, it is sometimes thought that the Master SIji even more tx x , and a first-degree homogeneous function is one for which lt tf x . Comparing this definition to the definition of constant returns to scale, we see that a technology has constant returns to scale if and only if its production...
Convex technologies
Let us now consider what the input requirement set looks like if we want to produce 100 units of output. As a first step, we might argue that if we multiply the vectors 1,2 and 2,1 by 100, we should be able just to replicate what we were doing before and thereby produce 100 times as much. It is clear that not all production processes will necessarily allow for this kind of replication, but it seems to be plausible in many circumstances. If such replication is possible, then we can conclude that...