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Varian University of California at BerkeleyW. Copyright c , , by W. Norton Ltd. There are many counterexamples. Hence the TRS of a homothetic function has the 4. But we already know that the TRS of ahomogeneous function has the required property. The isoquants look just like the Leontief technology except weare measuring output in units of log y rather than y. Hence, the shape ofthe isoquants will be the same. It follows that the technology is monotonicand convex. It is monotonic and convex.

The derivatives of f x1 , x2 are both positive so thetechnology is monotonic. It ismonotonic and weakly convex. Therefore f is not concave. Except for the border points this is just the complement ofthe input requirement sets we are interested in the inequality sign goes inthe wrong direction. As complements of convex sets such that the borderline is not a straight line our input requirement sets can therefore not bethemselves convex.

Chapter 2. Profit Maximization 6. Hence for any p, themaximum of py over Y O must be larger than the maximum over Y , andthis in turn must be larger than the maximum over Y I. Chapter 3.

Profit Function3. Thereforethe marginal product of factor i can only depend on the amount of factori. In order 9. Chapter 4.

Cost Minimization4. If we have interior solutions forboth xi and xj , equality must hold. However, if you think about it a minute you will see that this It turns out that this corresponds to a constrained maximum and not tothe desired minimum. Check the second-order conditions to verify this. Since the cost function is concave, rather than convex, the optimal solu-tion will always occur at a boundary. It costs 40 to produce units of output,but at the same prices it would only cost 38 to produce units of output.

Chapter 5. Cost Function5. By the properties of the Leontieffunction, we know that if we use x2 and x3 to produce y, we must use 3units of both x2 and x3 to produce one unit of y.

Thus, if the cost of usingone unit of x1 is less than the cost of using one unit of both x2 and x3 ,then we will use only x1 , and conversely. So, we know that if x1 is relatively cheaper, we will use all x1 and no x2 ,and conversely. The cost function must be concave in both prices, so a andb are both less than 1. Utility Maximization7. The con-sumer will choose this consumption point when faced with positive prices.

Chapter 8. Note that if you are going to interpret the Lagrange multiplier as themarginal utility of income, you must be explicit as to which utility functionyou are referring to.

Suppose not. Check that the reverse proposition also holds—i. Just look at the sets ofallocations that are strictly better or worse than the original choice—i. When prices are 2, 4 he spends When prices are 6, 3 he spends But with the constrained grant, he 1must consume at least g1 units of good 1. Incidentally, he will accept thegrant, since with the grant he can always consume at least as much of bothgoods as without the grant.

Chapter 9. Also, let m be income measured in units of z. How do you know thatthere must be another good around? The expenditure function is necessarily a concave function of prices, whichimplies that v p is a convex function. Chapter This is anincreasing function of wealth. According to the Arrow-Pratt measure, u exhibits a higher degree of riskaversion than v.

In this case, higher risk premium would no longer be synonymouswith higher absolute risk aversion. Competitive Markets Also, we see that x1 and x2 are not substitutes at any degree. Hence it prefers to stay out of business.

This is the same as the competitive solution. Marginal costs are constant at c so the monopolistwill want to produce the smallest possible output. Area A is what the monopolist would lose bydoing this. All of the results derived there translate on a one-to-one basis;e.

But for the monopolizedindustry, price exceeds marginal cost, so we want the last term to be posi-tive.


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solutions Intermediate Microeconomics (Hal R. Varian)





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