The monopolist solves. maxp(q,a)q C(q) yielding FONCs and SOSCs. p(q (a),a)+p q (q (a),a)q (a) C (q (a)) = 0

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1 Problem Set : The implicit function and envelope theorems. (i) A monopolist faces inverse demand curve p(, a), where a is advertising expenditure, and has costs C(). Solve for the monopolist s optimal uantity, (a), and explain how the optimal uantity varies with advertising if p a (,a) 0. (ii) Suppose the cost of advertising is k(a). Use the envelope theorem and your work from part (i) to characterize the optimal level of advertising (what are the FONCs and SOSCs?). The monopolist solves yielding FONCs and SOSCs maxp(,a) C() p( (a),a)+p ( (a),a) (a) C ( (a)) = 0 p ( (a),a) (a)+p ( (a),a) C ( (a)) < 0 To see how (a) varies with a, suppose we are at a local maximum; that means we know p( (a),a)+p ( (a),a) (a) C ( (a)) = 0 p ( (a),a) (a)+p ( (a),a) C ( (a)) 0 Totally differentiating the FONCs (here we re going to work out the implicit function theorem for this problem) on the first euation yields {p ( (a),a) (a)+p ( (a),a) C ( (a))} a (a)+p a( (a),a)+p a ( (a),a) (a) = 0 and rearranging yields a (a) = p a ( (a),a) (a)+p a ( (a),a) (p ( (a),a) (a)+p ( (a),a) C ( (a))) The denominator is non-negative (which is the second-order sufficient condition), and the second-order conditions must be non-zero (so that we re not dividing by zero). So an increase in a leads to an increase in uantity if p a 0 (that s the easiest thing to assume). Slightly more general would be to assume that or p a ( (a),a) (a)+p a ( (a),a) 0 p a ( (a),a) (a) p a ( (a),a) 1 So that the elasticity of p a ( (a),a) with respect to is larger than 1. (ii) Let V(a) = p( (a),a) (a) C( (a)) be the reduced-form profit function from part (i). Then the firm is trying to solve maxv(a) k(a) a with FONCs V (a) k (a) = 0 1

2 and SOSC s By the envelope theorem, V (a) k (a) < 0 V (a) = [p a (,a)(a)] = (a) so the FONCs are really and the SOSCs are really p a ( (a),a) (a) k (a) = 0 p aa ( (a),a) (a)+p a ( (a),a) a(a) (a)+p a ( (a),a) a(a) k (a) < 0 The uestion doesn t ask for this, but when is the above SOSC satisfied? The simplest set of assumptions would be p aa < 0, k (a) > 0, p a < 0. This ensures the ineuality is satisfied. But then we can t have / a > 0, since that reuires p a > 0. To get less restrictive assumptions that guarantee a critical point a is a local maximum, let s factor a bit to get p aa ( (a),a) (a)+p a ( (a),a)a (p (a) a ( ) (a) (a),a) p a ( (a),a) +1 k (a) < 0 The term in parentheses looks like the elasticity of p a with respect to. So as long as the elasticity of p a ( (a),a) with respect to is greater than 1, p aa < 0, and k (a) > 0, the second-order condition will be satisfied. 3. Suppose there is a perfectly competitive market, where consumers maximize utility v(, α)+m subject to budget constraints w = (p+t)+m, where t is a tax on consumption of, and firms have costs C() = c. Assume that v α (,α) 0. (i) Characterize a perfectly competitive euilibrium in the market, and show how tax revenue t (t,α) varies with t and α. (ii) Suppose α changes and the government adjusts t to hold tax revenue t (t,α) constant. Use the implicit function theorem to show how t(α) varies with α. (It might help to think about the elasticity of with respect to t, ε = t t ) (i) The consumer solves which has local maxima characterized by The firm solves maxv(,α)+w (p+t) v (,α) p t = 0 v (,α) < 0 max(p c) which has solution = 0 if p < c, = (0, ) if p = c, and = if p > c. The market can clear only if v (,α) t = p = c (this is where the firm s supply function and the consumer s demand function intersect). This gives us the function v ( (α,t),α) t c = 0

3 that characterizes the euilibrium uantity. The change in (α,t) with respect to t is (α,t) t = 1 v (,α) < 0 so that the market-clearing uantity is decreasing in the tax. Then tax revenue is t (α,t) which has derivative +t t = t + v (,α) which is ambiguous in sign. If we rearrange a bit, we get t [t ] = 1+ t t }{{} ε,t so that if the elasticity of with respect to t, ε,t, is greater than 1, tax revenue goes up, but it decreases otherwise. (ii) Now we have the euation t(α) (t(α),α) R = 0 Totally differentiating with respect to α yields ( ) t α (α) +t(α) t t α(α)+ = 0 α Re-arranging for t α yields t α = t α +t t The denominator happens to be t [t ], which we already know euals ( 1+ t t numerator is using the implicit function theorem to see how varies with α ). The (α,t) α = v α(,α) v (,α) 0 So the sign of t α is determined by the denominator, and t α is positive if ε,t, is greater than 1, but negative otherwise. 4. An agent is trying to decide how much to invest in a safe asset that yields return 1 and a risky asset that returns H with probability p and L with probability 1 p, where H > 1 > L. His budget constraint is w = pa+s, where p is the price of the risky asset, a is the number of shares of the risky asset he purchases, and S is the number of shares of the safe asset he purchases. His expected utility (his objective function) is maxpu(s +ah)+(1 p)u(s +al) a,s (i) Provide FONCs and SOSCs for the agent s investment problem; when are they satisfied? (ii) 3

4 How does the optimal a vary with p and H? (iii) Can you figure out how the optimal a varies with wealth? (iv) How does the agent s payoff vary with H, L and p? (i) The objective is FONCs are maxpu(w +a(h r))+(1 p)u(w +a(l r)) a pu (w +a (H r))(h r)+(1 p)u (w +a (L r))(l r) = 0 SOSCs are pu (w +a (H r))(h r) +(1 p)u (w +a (L r))(l r) < 0 Marginal utility of consumption is positive, so if H > r > L, there will be a solution to the FONCs. For the SOSCs to be satisfied, u () < 0. If a = 0, we have ph +(1 p)l = r so that the expected return euals the price of the asset. So a is positive if ph+(1 p)l > r, and a is negative if ph +(1 p)l < r. (ii) Let φ(p,h,w) = pu (w +a (H r))(h r)+(1 p)u (w +a (L r))(l r) Optimal a is increasing in p, since φ p = u (w +a (H r))(h r) u (w +a (L r))(l r) > 0 Optimal a is ambiguous in H, since φ H = pu (w +a (H r))a (H r)+pu (w +a (H r)) which is positive for sure if a < 0, but indeterminate if a > 0, since u () < 0 but u > 0. (iii) Well, φ w = pu (w +a (H r))(h r)+(1 p)u (w +a (L r))(l r) is ambiguous, since H r > 0 but L r < 0. (If u () < 0, however, it turns out that if w then a ) (iv) We use the envelope theorem. In particular, let V(H,L,p) = [pu(w+a(h r))+(1 p)u(w +a(l r))] a=a Then = pu (w +a (H r))a H which has the same sign as a. which has the same sign as a, p L = pu (w +a (L r))a = u(w +a (H r)) u(w +a (L r)) Since u() is increasing, this is positive if w +a (H r) > w +a (L r), or a H > a L. So if a > 0, V/ p > 0, but if a < 0, V/ p < 0. 4

5 6. Suppose an agent maximizes maxf(x,c) x yielding a solution x (c). Suppose the parameter c is perturbed to c. (i) Use a second-order Taylor series expansion to characterize the loss that arises from using x (c) instead of the new maximizer, x (c ). Show that for small changes in c, the loss to the objective function is proportional to (x (c) x (c )). (ii) Explain the strengths and weaknesses of using this as a model of subrational decision-making, where agents fail to re-optimize after an unexpected shock. How would you construct such a model? Linearize f(x,c ) in x (c) around x (c ) to get f(x,c ) = f(x (c ),c )+f x (x (c ),c )(x x (c ))+f xx (ξ,c ) (x x (c )) where ξ is between x and x (c ). Since f x (x (c ),c ) = 0 from the FONCs, we can rearrange to get f(x,c ) f(x (c ),c ) = f xx (ξ,c ) (x x (c )) Evaluate at x = x (c), and f(x (c),c ) f(x (c ),c ) = f xx (ξ,c ) (x (c) x (c )) So the loss that the agent incurs by not re-optimizing is proportional to (x (c) x (c )), so the loss is not too large as long as x (c) and x (c ) are close. One strength is that it is simple. We can easily see how the second-derivative is related to the approximate loss in welfare, and the model can be applied to many situations. One weakness is that even though c and c might be close, the optimal policies may be very far apart, or f xx (ξ,c ) might be very large. This makes it hard to judge whether or not the loss is large without looking at a particular problem. I would introduce a fixed cost to changing policies, so that whenever the agent wants to move from x (c) to the new optimum x (c ), they must pay K. Then the agent will only change if f xx(ξ,c ) (x (c) x (c )) K So that the agent is acting optimally to adjust their behavior. 5